/sci/ - Science & Math

[Anonymous No.11525618]

>about 50% of /sci/ is mathematically illiterate
How can we fix this?

[Anonymous No.11525623]

>>11525618
see >>11525609

[Anonymous No.11525627]

>>11525623
See my response
>>11525609

[Anonymous No.11525629]

>>11525623
See my response
>>11525616

[Anonymous No.11526037]

bump

[Anonymous No.11526046]

>about 50% of /sci/ is mathematically illiterate
citation needed

[Anonymous No.11526056]

>>11526046
There was a poll made and about 50% said 0.999... is not 1.

[Anonymous No.11526155]

How is it equal to 1?
Wouldn't there exist a real number that would always be greater?

[Anonymous No.11526161]

>>11526155
Any real number less than 1 would be surpassed by an element of the set {0.9, 0.99, 0.999, ...}. For example, the number 0.9999999991 is surpased by 0.999999999999999 and hence is not an upper bound of the set.

[Anonymous No.11526201]

>>11526161
Why does it need to have an upper bound?

[Anonymous No.11526215]

>>11526201
It has many upper bounds. For example, 2 is an upper bound because every element of the set is less than 2. 100 is also an upper bound, and so is 1.
Not all subsets of the reals have upper bounds though. For example, the natural numbers, considered as a subset of the reals, has no upper bound.

[Anonymous No.11526225]

>>11526056
Was it Pepeer reviewed?

[Anonymous No.11526230]

>>11526225
Yes, Pepe approved it and Apu put a stamp on it.

[Anonymous No.11526240]

>>11525623
This is a case of the conclusion being clearly right, but the argument being faulty.

[Anonymous No.11526241]

1/3 =/= .33333333

[Anonymous No.11526242]

>>11526201
the set {0.9, 0.99, ...} is clearly nonempty and bounded from above by 2, and the real numbers have the least upper bound property, which states every nonempty bounded set has a unique upper bound u for which any real less than than u is not an upper bound for the set.

[Anonymous No.11526650]

Aren't Large Cardinal axioms a subject of debate?

[Anonymous No.11526656]

>>11526241
lol was it supposed to equal then, 4?

[Anonymous No.11526663]

>>11526650
How is this related to Large Cardinal axioms?

[Anonymous No.11526668]

Wouldn't this be easy? Just multiple them both by 2 and see if the answer is the same.

[Anonymous No.11526691]

>>11526668
Yes, it is easy, and yes, the answer is the same when you multiply both by 2.

[Anonymous No.11526797]

>>11526656
Maybe the decimal system simply can't accurately represent 1/3rd.

[Anonymous No.11526904]

>>11525618
There is no cure for these brainlets.

[Anonymous No.11526915]

>>11525618
Just let it move on in the direction it is already going, and it will reach 99% before long.

[Anonymous No.11526920]

>>11526797
maybe, I guess we'll never know! ¯\_(ツ)_/¯

[Anonymous No.11526928]

>>11526201
clearly 0.999..., whatever it is, should satisfy

0.999... > 0.9
0.999... > 0.99
0.999... > 0.999

and so on. it's a reasonable definition to say that 0.999... is the LEAST number which satisfies all of these inequalities. turns out that the number with this property is 1.

[Anonymous No.11526940]

>>11525618
Mathematical Captchas in order to post. Quality will skyrocket, posting rates will gradually improve.

[Anonymous No.11526966]

>>11526940
Good idea. Mods get on this!

[Anonymous No.11526985]

>>11526904
I wish I was good enough at math so that I could read this with easily.

[Anonymous No.11527008]

>>11526985
It's not something to be read with just a glance. Pay attention to every symbol, try to have a clear idea of everything that's written.

[Anonymous No.11527039]

>>11526904
I think most people will get hung up on your definition (1). I for one understand it perfectly well (and it really doesn't make sense in any other way) but the idea of an infinite sequence of numbers being defined in the first place just seems odd.

[Anonymous No.11527055]

>>11527039
agreed that whoever thinks 0.999... < 1 will not make it past the initial definition

[Anonymous No.11527056]

>>11526904
Could you explain how you substituted the summation in (2) by the term in parenthesis in (3) ?

[Anonymous No.11527063]

>>11527056
0.9 = 1 - 0.1
0.99 = 1 - 0.01
0.999 = 1 - 0.001
etc

[Anonymous No.11527064]

>>11525618
>Someone had to make this shitty meme to justify something that isn't correct
Giving an infinite amount of 9's will never equate to 1. The difference between .9999999999999 and 1 might seem minuscule to us but in the world of science quantity on a minuscule scale matters very much.

[Anonymous No.11527074]

>>11527064
too bad that in mathematics "miniscule" means lesser than anything positive which means zero

[Anonymous No.11527086]

>>11526966
We've kicked the idea around for years, it never happens.

[Anonymous No.11527092]

>>11527086
Watch this anon. Oh /x/! We have a job for you!~

[Anonymous No.11527109]

>>11527039
>>11527055
I mean, just look at the proof - it basically says "0.999... for a sufficient number of nines will approach 1 to any arbitrarily small distance larger than zero". The implication that 0.999... for infinite nines is therefore 1 is just up there in the definition. Of course, it doesn't make sense for it to be any other number, but I think the very idea of considering it to be a defined number is not trivial in the first place.

[Anonymous No.11527111]

>>11526241
Correct. It equals .333...
The ellipses are important, retard. Also tell us what 1/3 DOES equal in decimal notation or fuck off.

[Anonymous No.11527130]

>>11525618
Alright all you chumps who think 0.9999... = 1, answer me this. Given this graph which represents the function f(x), what are the values of:

a) f(3)
b) f(x) when x -> 3

If you think they are the same, you also think x -> 3 is equal to 3. In that case, you have some Calculus I to review sweetie :)

[Anonymous No.11527150]

>>11527130
Lim to a defined number and lim to infinity are entirely different matters.

[Anonymous No.11527180]

>>11525618
>How can we fix this?
Today I learneding how to into add fractions, including mixed numbers.
Does /sci/ think I'm fixed.
Also, tomorrow, I learneding (I hope) how to into multiply fractions.
No, seriously /sci/bros.

[Anonymous No.11527221]

>>11527150
Are they tough? You could rephrase item b) by saying when 1/|x - 3| -> infinity. Not only that, but 0.999... doesn't tend to infinity, the number of decimal places does.

What I'm trying to say is that a limit is a representation of an abstract process that we rationalize as a variable "approaching" a value. The limit of a function never IS something, it indicates that the function approaches a certain value when the independent variable approaches another value (this is also true for the precise definition of a limit, I'm not just saying how I interpret limits). So I think saying that 0.999... = 1 is imprecise and deceitful.

But math isn't my major, so whatever

[Anonymous No.11527236]

>>11526920
No, I'm pretty sure that's it. This whole stupid argument is just an inadequacy in how decimals work.
>>11527111
Decimals cannot show 1/3rd, merely approximate it well enough to work. None of this infinitely repeating shit is real and there's no point to pretending it is.

[Anonymous No.11527279]

>>11526056

PRAISE THE LORD!

That means half the population have not succumbed to the insanity of the GODLESS SODOMY-LOVING FREAKS who believe in "Infinities".

Brothers and Sisters of the ONE TRUE DISCRETE UNIVERSE! I implore you to take up ARMS against those filthy heretics who preach the BLASPHEMY of 0.999...= 1. WE MUST CLEANSE THE EARTH! We shall swing our righteous swords at discrete intervals through discrete space and VERILY CLEAVE the SKULLS of the INFIDELS! BY THE POWER OF THE ALMIGHTY!

DEUS VULT! DESU VAULT! DISS VAULT!

[Anonymous No.11527329]

>>11526904
You GOD CURSED STUPID HERETIC! You are too fucking stupid to understand that your entire proof is based on a false premise. Christ save us all from this filthy stupidity!

You DUMB FUCK! What you are saying is the same as

Pink flying elephant = 1
Then some bullshit
to conclude that
Pink flying elephant = 1

Jesus Wept. YOU BABY KILLER! At the time of reckoning there will be a special place for all you LYING SCUM right at the top of the RIGHTEOUS BONFIRE OF RETRIBUTION!

PRAISE THE LORD! DEUS VULT!

[Anonymous No.11527350]

>>11527279
>>11527329
schizo and cringe

[Anonymous No.11527368]

[Anonymous No.11527395]

>>11527350
Trying saying that to my face on the field of battle, SCUM! No, you wouldn't, because you would be too busy PISSING in your frilly lace panties, the ones you stole from your father.

Godless heathens, cowards and transvestites, all of them.

[Anonymous No.11527553]

>>11527395
>>11527329
>>11527279
cringe larp, try harder faggot

[Anonymous No.11527571]

>>11527368
poggers.

[Anonymous No.11527601]

>>11526650
>>11526650
Yes, but 0.999999 = 1 is provable in RCA - finite analysis. In other words, no axioms pertaining infinite cardinals are required. In fact, 0.999 = 1 is strictly speaking provable in first order Peano arithmetic, although it would be an extremely laborious process, since it requires coding Cauchy Sequences of rationals using integers.

[Anonymous No.11527608]

american education is garbage, you can cram all of undergrad mathematics into k-8

[Anonymous No.11527615]

>>11525618
I don't know, but you are based
af OP. I am sick and tired of scientifically illiterate Redditors asking basic ass questions and arguing with me over shit that they don't even understand.

[Anonymous No.11527642]

>>11527553

One day you shall taste righteous vengeance, crawling worn. As ordained by the Almighty in His divine wisdom. It shall be surely slow and cruel.

[Anonymous No.11527734]

>>11525618
Question: Where is it going? Answer: to its limit. There you go. Calculus in a nutshell.

[Anonymous No.11527772]

No one is saying they are the same but we just count them that way for convenience. :)

[Anonymous No.11527823]

>>11527772
>Math is an absolute discipline with absolute rigour.
>Math is contextual, we just make up shit for whatever purpose suits us at the time, LOL.

Pick one.

[Anonymous No.11528000]

>>11527823
>Pick one
No I'll pick 0.9999...

[Anonymous No.11528169]

>>11527368
>The axiom of infinity defines infinity
No, the axiom of infinity merely states that the natural numbers, as a set, exist, hence infinite(definition: sets that are not finite) sets exist.
>N, S, S* have infinite elements
What the fuck are you talking about? The sets are not finite but all the elements in the sets are finite.
>infinity is defined as the quality which covers all elements of the infinite set N
Where the fuck have you seen it defined like that? It's not even a definition, that's meaningless babble.
The notation [math]\sum_{n=1}^\infty a_n = S[/math] is defined in terms of partial sums, [math]S_k = \sum_{n=0}^k a_n[/math], namely S is defined to be the unique real number (if it exists) such that for all e>0, there exists a natural number N such that for all n>N, |S_n - S|<e. This is the standard definition of an infinite sum [math]\sum_{n=1}^\infty a_n = S[/math] that you'll find in literally EVERY elementary calculus textbook.
>the index location of 0.999... is undefined
That's correct, because 0.999... is not an element of of the set S.
>but 0.999... does exist in S
No it does. You've defined S to only consist of numbers less that 0.999.... By your own definition 0.9999... is not in S.
>S[infinity]
You said you index S by natural numbers. infinity is not a natural number.
Takeaway:
you're a mathematically illiterate retard

[Anonymous No.11528178]

>>11527608
These sort of questions are trying to find the 0.0001% of 6 year olds. I'm sure the whole thing wasn't like that.

[Anonymous No.11528220]

>>11525618
>WORDSWORDSWORDS

/sci/ can't meme.

[Anonymous No.11528235]

>>WORDSWORDSWORDS

>/sci/ can't meme.

[Anonymous No.11528314]

decimals were a mistake

[Anonymous No.11528412]

>>11526985
[math]\forall[/math] - for all
[math]\exists[/math] - there exists
[math]\in[/math] - in
[math]\mathbb{N}[/math] - the set of natural numbers
[math]X\rightarrow Y[/math] - the statement [math]X[/math] implies the statement [math]Y[/math] (if [math]X[/math] is true, then so is [math]Y[/math])

[math]\forall \epsilon>0:\ldots[/math] - "For all (real numbers) [math]\epsilon[/math] greater than zero such that..."
[math]\ldots \exists n\in\mathbb{N}:\ldots[/math] - "... there exists a natural number [math]N[/math] such that..."
[math]\ldots \forall n \in \mathbb{N}\ldots[/math] - "... for all natural numbers [math]n[/math]..."
[math]\ldots n \geq N \rightarrow |a_n-L| < \epsilon[/math] - "... if [math]n[/math] is greater than or equal to [math]N[/math], then [math]|a_n-L|[/math] is less than [math]\epsilon[/math].
Put it all together:
>For all (real numbers) epsilon greater than zero, there exists a natural number [math]N[/math] such that for all natural numbers [math]n[/math], if [math]n[/math] is greater than or equal to [math]N[/math], then [math]|a_n-L|[/math] is less than [math]\epsilon[/math].
If (and only if) this statement is true, then the limit as [math]a_n[/math] is defined to be [math]L[/math].

Have fun, Anon, I hope you enjoy your journey into mathematics with /sci/ as your jumping-off point as much as I have. If you have any confusions, ask me.

[Anonymous No.11528417]

>>11527130
>the function f(x)
You've already lost.
>>11527221
>math isn't my major
That much is immediately apparent.

[Anonymous No.11528452]

>>11528169

The Axiom of Infinity guarantees a single definition of infinity, and that is the size of the set of all Natural numbers N.

Sets S and S* were created bijecting with N, filled with infinite terms (for all n in N) so they're the same size a N. bijections.

[eqn]\sum_{m}^{p} S^*[m] = S[p] [/eqn]
this formula provides that any summation of elements in S*, beginning from the first element of S*, is equal to a same element inside S.
When the limit of p is set to infinity, we are still relying on the definition of infinity established at the top, which is the quality of the quantity of all elements of N.
Similarly because infinity is defined this way, [eqn]S[\infty][/eqn] does not explicitly mean the "∞'th" index of S, but rather the notion that the index which it occupies exists somewhere within ALL of S, and that this index is undefined in the same way that the largest element of N is undefined.
Because S and S* are bijective with N, this also includes all elements of S and all elements of S*.

If all elements of S* are summed together, thanks to the formula we also know that this 0.999... number produced by summing all S* must also exist inside S.

[Anonymous No.11528472]

>>11527221
>limit is a representation of an abstract process that we rationalize as a variable "approaching" a value. The limit of a function never IS something, it indicates that the function approaches a certain value when the independent variable approaches another value (this is also true for the precise definition of a limit, I'm not just saying how I interpret limits).
No a sequence is a representation of a process. Limit is the number the sequence approaches. Does the sequence ever reach this number? Irrelevant.
0.999... is not a process itself. It's a number which a certain process approaches.

[Anonymous No.11528483]

>>11528472
There is no "process"; time is not involved.

[Anonymous No.11528484]

>>11528452
>>11528169
I'll add on that it's imperative that infinity BE defined this way for the function of the sum of S*[m] to actually cover ALL elements of S*, of which there are infinite because it's the same size as N. Nothing but ∞ as the variable for p has the capacity to snag ALL elements of S*.
And since S[p] = sum m->p S*[m], and since the size of S* is infinity elements, just like the size of S and the size of N, there still remains a 1:1 translation from any sum of sequential ordered S* elements to an element inside S.

[Anonymous No.11528603]

>>11526230

[Anonymous No.11528791]

>>11525618
>only 50% retarded on /sci/
You seem to be a member of that set since that number is clearly a lot higher.

[Anonymous No.11528805]

>>11525629
Nice. You're doing god's work anon.

[Anonymous No.11528852]

>>11527130
[math] f(3)=2, [/math]
[math] \lim_{x\to3}f(x)=6. [/math]
Not sure what your point is here. What part of well-ordering and completeness do you not understand?

[Anonymous No.11528858]

>>11527130
>you have some Calculus I to review sweetie :)
coming from a person who doesn't know what a limit is

[Anonymous No.11528859]

>>11527130
>you also think x -> 3 is equal to 3
Yes. That is true by definition. Since the set of all caught sequences with limit 3 is just the definition of 3.

I think you should review some analysis...

[Anonymous No.11528860]

>>11527221
>So I think saying that 0.999... = 1 is imprecise and deceitful.
Do you know the definition of a real number?

[Anonymous No.11528863]

>>11526904
The real issue is that these people do not understand the definition of real numbers...

[Anonymous No.11528870]

>>11525618
1 > 0.9
1 > 0.99
1 > 0.999
1 > 0.9999
1 > 0.99999
1 > 0.999999
...
1 > 0.999999...

[Anonymous No.11528872]

>>11527368
>>11528452
>>11528484
You are completely wrong. Read this >>11518798

[Anonymous No.11528878]

>>11528870
And?
Please define what it means for two real numbers to equal one another.

[Anonymous No.11528884]

>>11528870
Good job completely missing the point of the OP. TELL US WHAT YOU MEAN BY 0.999... or shut up.

[Anonymous No.11528889]

>>11525618
1/3 > 0.3
1/3 > 0.33
1/3 > 0.333
...
1/3 > 0.333...

1/3 != 0.333...

[Anonymous No.11528890]

>>11528889
>1/3 != 0.333...
False by definition.

[Anonymous No.11528892]

>>11525618
The “0.333” argument in base 3:

1/3 = 0.1
2/3 = 0.2
3/3 = 0.1+0.2
1 = 1

Or base 9

1/3 = 0.3
2/3 = 0.6
3/3 = 0.3+0.6
1 = 1

[Anonymous No.11528895]

>>11528890
What the fuck? Are you retarded?

[Anonymous No.11528897]

>>11528895
Please define "=" for two real numbers.

[Anonymous No.11528907]

>>11528897
>Please define "=" for two real numbers.

[Anonymous No.11528908]

>>11528907
shit bait.

[Anonymous No.11528909]

>>11528907
You can't do it?

[Anonymous No.11528926]

>>11525618
you fucking dumbass. No matter how many 9’s you add ITS NEVER GOING TO REACH 1

[Anonymous No.11528931]

>>11528926
>No matter how many 9’s you add ITS NEVER GOING TO REACH 1
And?

Just define what it means for two real numbers to equal each other please...

[Anonymous No.11528942]

>>11528926
OP here, never claimed it ever does. Learn to read.

[Anonymous No.11528953]

>>11528931
1/10=0.1
1/9=1/9
1/8=0.125
1/7=1/7
1/6=1/6
1/5=0.2
1/4=0.25
1/3=1/3
1/2=0.5
1/1=1
5-4=1
0.999...=1-0.00...01

[Anonymous No.11528956]

>>11528953
I asked for a definition, that isn't a definition.

[Anonymous No.11528958]

>>11528956
Two numbers that represent the exact same value

[Anonymous No.11528962]

>>11528958
We can't have that imprecise nonsense language in math.
What does it mean for two real numbers to represent the exact same value?

[Anonymous No.11528963]

>>11528958
Wrong.
Two real numbers are equal if they are the same equivalence class if cauchy sequences, as you will find in any textbooks which defines real numbers via cauchy completion.

If you don't understand what that means you have no place in this discussion.

[Anonymous No.11528965]

>>11525618
>0.999... is the closest possible number to 1
What about 0.FFFFFF... in base 16?

[Anonymous No.11528970]

>>11527236
>None of this infinitely repeating shit is real
an infinite amount of decimals can approximate any real number infinitely close. It’s really not that hard to understand.

[Anonymous No.11528976]

>>11525618
Ban phone posters.

[Anonymous No.11528978]

i think it's time to agree that numbers where a mistake

[Anonymous No.11528979]

>>11526046
Take a random poster. Either they are mathematically illerate or they are not, so it's 50-50

[Anonymous No.11528982]

>>11528970
>approximate
>close
Yes, 0.999... is very close to 1, it’s an approximation

[Anonymous No.11528988]

>>11528982
0.999....9 is an approximation

[Anonymous No.11528990]

>>11528978
No, it's just that 50% of people are born with defective brains which prevents them for ever understanding basic mathematical concepts such as the real numbers.

[Anonymous No.11528993]

>>11525618
people like you are the cancer of /sci/.
if you knew any real /sci/ence you wouldn’t post this dumb 0.999.. shit every day.

reminder: sage all 0.999... threads

have a nice day.
sage.

[Anonymous No.11528995]

>>11528990
>The real numbers
>Basic
Come on now, they are quite difficult to wrap your head around, and I'd say at least half of all math majors would be lying if they say they understand them.

[Anonymous No.11529009]

>>11525618
If I have an infinite number of dice and roll all of them an infinite number of times, will I have any dice that rolled a 6 every single time?

[Anonymous No.11529014]

>>11528995
>I'd say at least half of all math majors would be lying if they say they understand them.
this is actually true

[Anonymous No.11529015]

>>11528995
I find them extremely easy and intuitive ever since I've learned about them in high school. Never met an undergraduate who found them confusing either, although I didn't discuss it with others much either.
That said, you don't even need to know about the real numbers to understand that expressions don't automatically have meaning just because they look like they could.
0.9999.... on it's own is meaningless unless we give it a meaning. This is what all the 0.999...=/= 1 retards fail to understand, as I indicated in my OP.
Whenever pressed to tell what they mean by the expression 0.999... they always start spewing some vague, meaningless garbage about processes and infinity. And it's always abuse of the notion of infinity. They never seem to understand that infinity is not some magical tool that you can throw at everything which solves all the problems. IMO mathematicians should stop using the word infinity at all when talking to laymen because of how confusing it is to laymen to wrap their heads around. Sure, we mathematicians understand that when we talk about infinite sums, infinities in the projective spaces these are all actually finite and well-defined things which only intuitively can be visualized as something infinite, but laymen fail to grasp this subtlety. I think we should only talk about "infinite sets" which mean exactly "sets that are not finite" and not anything else.

[Anonymous No.11529016]

>>11529009
50/50 either it happens or it doesn't

[Anonymous No.11529037]

If I roll an infinite number of 10 faced dices and get all nines, is it equivalent to rolling a one with one dice?

[Anonymous No.11529042]

>>11529009
>>11529037
Completely missing the point. Read the OP pic again.

[Anonymous No.11529049]

>>11529042
I don't care about the OP
I just want to be mischevious

[Anonymous No.11529064]

>>11528990
>basic mathematical concepts such as the real numbers
This absolutely isn't basic. Real numbers are an extremely complex and delicate subject. You have to, at least start, a degree in mathematics to ever be exposed to a rigorous (or even a non rigorous) treatment.

[Anonymous No.11529072]

>>11529015
>I find them extremely easy and intuitive
How are equivalence classes of cauchy sequences "extremely easy and intuitive"?

Constructing the reals is really complicated...

[Anonymous No.11529089]

>>11529064
>Real numbers are an extremely complex and delicate subject.
someone finally said it
those poor kids arguing for 0.999... < 1 have no fucking idea what they have gotten themselves into

[Anonymous No.11529104]

>>11529072
The rational numbers Q have a problem: we don't have convergence of Cauchy sequences. What do?
Answer, create a new space where every Cauchy sequence converges.
How to do it?
The most natural way is simply take Q and add to it some object for every Cauchy sequence that doesn't converge. For clarity we might identify each new object with the Cauchy sequence itself.
Now we think for a while and notice we have a problem. Different sequences of rational numbers which should converge to the same number converge to different cauchy sequences. So we make equal any two Cauchy sequences whose componentwise difference tends to 0.
Now we notice there are 2 different looking objects, equivalence classes of Cauchy sequences that don't converge in Q and elements of Q itself. We want every number of our new number system to look the same, so we identify the elements of Q with constant cauchy sequences. Now every element of our new number system is an equivalence class of Cauchy sequences of rational numbers and it contains Q.
Now just verify that you can do arithmetic in this new number system the same way that you can in the rationals. This is easy.

The whole construction is essentially just following your nose having the problem of convergence in mind. I still maintain it's very easy and intuitive.

[Anonymous No.11529117]

>>11529104
>I still maintain it's very easy and intuitive.
Dude. 99% of the population does not understand what a cauchy sequence is.

Doing this rigorous from ground up are dozens of pages of things only math majors will ever see. How can you call that intuitive?

[Anonymous No.11529121]

>>11529089
Note that the fact that 0.999...= 1 doesn't depend on the real numbers.
A consistent definition of a decimal expansion 0.d_1d_2.... is the supremum **in the rational numbers Q** of the set {0.d_1, 0.d_1d_2, 0.d_1d_2d_3, ...}. For many infinite decimal expansions, this supremum exists. Only when the supremum doesn't exists do we have to resort to the reals. In the case where the supremum exists in Q, taking the supremum in R and in Q gives the same answer.
0.999...=1 just by the definition of infinite decimal expansions in the rationals.

Also
>implying the construction of the reals is difficult
What about this >>11529104 is difficult to you?

[Anonymous No.11529126]

>>11529121
>What about this >>11529104 # is difficult to you?
It certainly is difficult to 99% of the population.

If you think these people debating whether 0.999...=1 even have a highschool degree you are retarded.

[Anonymous No.11529133]

>>11529117
>Doing this rigorous from ground up are dozens of pages of things only math majors will ever see
The whole construction is literally contained in my one paragraph post.
If you want to verify the properties of the reals from this, it can get lengthy, yes, but all the verifications are easy and simply following the nose: doing the most obvious thing you can.
>what a cauchy sequence is
it's just a word, and the meaning is simple: it's just a sequence that seems like it's converging to some point. If you are asked to come up with a rigorous definition for this without reference for the point it should converge to, I believe there is a unique definition that every human being on earth will settle on: that of a Cauchy sequence.

[Anonymous No.11529140]

>>11529133
>The whole construction is literally contained in my one paragraph post.
It's neither rigorous nor from the ground up.

Is a numberphile video on the abc conjecture enough proof for you?
What the fuck.

>it's just a word, and the meaning is simple: it's just a sequence that seems like it's converging to some point. If you are asked to come up with a rigorous definition for this without reference for the point it should converge to, I believe there is a unique definition that every human being on earth will settle on: that of a Cauchy sequence.
I know what it is...99% of people do not.

What the fuck is wrong with you? You realize that almost nobody is a math major?

[Anonymous No.11529146]

>>11529140
>You realize that almost nobody is a math major?
I was just a high school student when I learned all this.
If people are curious enough to wonder about whether 0.999...=1, they're more then ready to learn what the reals are.
>It's neither rigorous nor from the ground up.
Which part of my construction is not rigorous? Literally everything in my post can be translated into formal mathematics by anyone who is familiar with mathematics, even if they never heard of the reals. I assumed knowledge of the rational numbers, of course.

[Anonymous No.11529154]

>>11529121
dude, average 0.999... != 1 shitposter doesn't know what is cauchy sequence, supremum or equivalence class. he might have heard about convergence and limits, but he doesn't actually know the stuff and if you asked him he would tell you just some handwavy bullshit about "approaching but never truly getting there".

he doesn't even understand why are you suddenly trying to take Q and construct something out of it in the first place. reflect upon this for a second.

[Anonymous No.11529178]

>>11529146
From what you wrote, it sounds like all of your intuition comes from already having dealt with the reals. Perhaps you're just really smart, or even a genius, but I don't think the construction of the reals is obvious to many people, even if they are familiar with math. If they've never heard of the reals, then I think it's almost a certainty that they'd struggle with the concept.

[Anonymous No.11529183]

>>11529154
Let's fix this.
>cauchy sequence
A sequence a_1, a_2, .., a_n, ... of rational numbers is called Cauchy if for every rational number e>0, there exists a natural number N such that for all natural numbers n,m that are greater than N, |a_n - a_m| < e.
>supremum
Let A be a subset of the rational numbers. A supremum of A in the rationals is the smallest rational number x (if such exist) such that x is not smaller than all of the elements of A.
>equivalence class
Sometimes we have a set of some objects, and we want to make some objects equal. Then we create a set of subsets of the old set, each subset consisting of elements we declare to be equal.
>he doesn't even understand why are you suddenly trying to take Q and construct something out of it in the first place.
Because mathematicians want to make sense when they talk. That means if they talk about some numbers, they want to explicitly specify what those numbers are so everyone agrees what they're talking about. A good way to construct new number systems is to take old systems everyone agrees exist and make sense and do something with them. In the case of the reals, the rational numbers are taken and modified.

[Anonymous No.11529186]

>>11529183
>A supremum of A in the rationals is the smallest rational number x (if such exist) such that x is not smaller than all of the elements of A.
Meant to say
A supremum of A in the rationals is the smallest rational number x (if such exist) such that x is not smaller than any of the elements of A.

[Anonymous No.11529189]

>>11529146
>I was just a high school student when I learned all this.
Does the average high school student learn this?
No of course not. Are you literally autistic?

>Which part of my construction is not rigorous?
You defined nothing and proved nothing.
If you have ask this question you have no clue about mathematics.

>Literally everything in my post can be translated into formal mathematics by anyone who is familiar with mathematics
Which makes it NOT RIGOROUS.

[Anonymous No.11529192]

>>11529183
>>11529186
Nobody gives a fuck retard.
The shitposters don't have a highschool degree.

You are completely out of touch with reality.
It's also totally non rigorous, so just link a book which actually does it rigorous and from the ground up...

[Anonymous No.11529195]

>>11529183
my point was that if you tell this
>A sequence a_1, a_2, .., a_n, ... of rational numbers is called Cauchy if for every rational number e>0, there exists a natural number N such that for all natural numbers n,m that are greater than N, |a_n - a_m| < e.
to a 0.999... != 1 shitposter, his literal reaction will be "what the fuck is this nigga talking about man"
that's all

[Anonymous No.11529198]

>>11528889
>1/3 > 0.333...
prove it.

[Anonymous No.11529205]

>>11529189
>Does the average high school student learn this?
If they are interested in why 0.99...=1, of course they can learn about it just like I did. You can even use Wikipedia for God's sake. It will explain to you rigorously what mathematicians mean by 0.999... =1
>Are you literally autistic?
I don't think I am.
>You defined nothing and proved nothing.
>If you have ask this question you have no clue about mathematics.
I did not use the language of first-order logic but that doesn't mean I didn't define anything nor that my proof is not rigorous. Show my post to literally any mathematician and they'll understand immediately what I mean, even if you don't mention that this is the construction of the real numbers.
It seems like I have much more clue about mathematics than you do.
>Which makes it NOT RIGOROUS.
No textbook about actual mathematics is written in first order logic. Mathematics is written in english. As you get familiar with the basics, the discussion gets more and more English. Doesn't mean the rigor goes down. Try crashing a conference on algebraic geometry and shouting that they're not being rigorous just because they gloss over some obvious details for the purpose of clearer communication.

[Anonymous No.11529207]

>>11526985
I wish I was good with maths so I could prove this brainlet wrong.

[Anonymous No.11529208]

>>11529192
>It's also totally non rigorous
Yes it is. I assume some part of my definition confused you. Non-rigorous means you failed to see the mathematical meaning in what I said. So which part confused you?
>>11529195
That might very well be correct, but I've seen posters convinced by my posts. Not everyone who claims 0.9999...!=1 is an unfixeable retard.

[Anonymous No.11529211]

>>11529205
>I did not use the language of first-order logic
Irrelevant.
It's still not rigorous.

>they'll understand immediately what I mean
Doesn't make it rigorous.

Have you had ANY formal education about mathematics? Or have you read ANY textbook?

[Anonymous No.11529213]

>>11529207
He's not wrong. It literally just follows trivially form how a decimal expansion is defined. It's extremely elementary and basic. The only reason you don't think so because you don't agree with the commonly accepted definition of what a
decimal expansion that doesn't end represents.

[Anonymous No.11529214]

>>11529208
>Non-rigorous means you failed to see the mathematical meaning in what I said
LOL

are you retarded? That isn't what it means at all.
Use Google.

[Anonymous No.11529220]

>>11529211
Holy fucking shit. Read my post here
>>11529208
>Yes it is. I assume some part of my definition confused you. Non-rigorous means you failed to see the mathematical meaning in what I said. So which part confused you?
So which part of my post confused you?
>Have you had ANY formal education about mathematics? Or have you read ANY textbook?
I guarantee I know way more maths than you and have read way more textbooks than you did. You make the freshman mistake of thinking whenever all the details are not specified explicitly, something is not rigorous. Don't worry, you will get more mathematically mature in the future, provided you continue studying mathematics like I did.

[Anonymous No.11529222]

>>11528982
an infinitely close one, meaning the difference between the platonic value and the decimal representation is infinitesimally small.

[Anonymous No.11529223]

>>11529220
Do you have any formal mathematical training?

[Anonymous No.11529226]

>>11529214
Rigour in mathematics means something different than in colloquial use. Clearly you're a beginner since you haven't noticed this. As I said, try going to an algebraic geometry or algebraic topology conference and protesting that the talks there are not rigorous. See what mathematicians think about that. Unless you are so delusional as to think all their talks never gloss over any details down to the most elementary operations with integers and sets. If that is the case, I suggest you go listen to an actual mathematical talk or read a paper written by a mathematician instead of elementary level calculus textbooks.

[Anonymous No.11529227]

>>11529220
>You make the freshman mistake of thinking whenever all the details are not specified explicitly, something is not rigorous.
That's LITERALLY the definition.

[Anonymous No.11529230]

>>11529223
Yes, I'm a second year mathematics student at Cambridge.

[Anonymous No.11529231]

>>11529226
>As I said, try going to an algebraic geometry or algebraic topology conference and protesting that the talks there are not rigorous.
And every mathematician will agree and tell you the point of a talk is to be non rigorous.

>gloss over any details
That makes it non rigorous.

[Anonymous No.11529234]

>>11529230
I have nearly a master's degree.
So please shut the fuck up and talk to your prof. about this and ask him if your "proof" is rigorous.

[Anonymous No.11529235]

>>11529227
Not in mathematics.
>Often, a written proof is accepted as rigorous although it might not be formalised as yet. The reason often cited by mathematicians for writing informally is that completely formal proofs tend to be longer and more unwieldy, thereby obscuring the line of argument. An argument that appears obvious to human intuition may in fact require fairly long formal derivations from the axioms. In short, comprehensibility is favoured over formality in written discourse.
https://en.wikipedia.org/wiki/Rigour#Mathematical_proof

[Anonymous No.11529238]

>>11529235
Holy shit learn to read...

[Anonymous No.11529239]

>>11525618
Fully factorise x^6-64

[Anonymous No.11529241]

>>11529235
I mean. Do you actually believe that something which takes dozens of pages in a textbook could be rigorously written in a 4chan post?

You are braindead drop out.

[Anonymous No.11529243]

>>11525618
>>11529239
I mean solve* x^6+64

[Anonymous No.11529244]

>>11529234
>I have nearly a master's degree.
Ok, which uni?
>So please shut the fuck up and talk to your prof. about this and ask him if your "proof" is rigorous.
Ok let me ask you a question. Can you give me an example of a mathematical research paper published in the last 10 years that is rigorous according to your standard?
I guarantee you that none of the papers will be written in strictly formal language and all of them will gloss over many details that the reader is supposed to already be familiar with or be able to easily work out themselves.
Yes, my construction of the reals is rigorous provided you know basic set theory, and what rational numbers, equivalence classes and convergence are.

[Anonymous No.11529245]

>>11529243
>Solve this expression.
Come back after you've finished middle school, bud.

[Anonymous No.11529246]

>>11529243
OP here. Over what field do you want me to solve it?

[Anonymous No.11529249]

>>11529244
You have to be trolling here, right?
This is your post >>11529104.
Regardless of whether or not it is a good explanation, makes sense, or if everything follows fairly obviously, no one in their right mind would call that rigorous.

[Anonymous No.11529253]

>>11529241
The real numbers:
Any positive or negative number, including non-whole numbers and irrational numbers, and including zero; any number that can be placed on a 1-dimensional number line.
E.g. 0, 0.9, 0.9999..., 1, 2, e, 3, pi, 4, 100/7, etc

[Anonymous No.11529255]

>>11529246
So OP, apparently you are also mathematically illiterate...

[Anonymous No.11529256]

>>11529249
Which part of the construction confused you then?

[Anonymous No.11529257]

>>11529244
>Ok, which uni?
Some shitty German one.

>Can you give me an example of a mathematical research paper published in the last 10 years that is rigorous according to your standard?
Basically any that you would find on arxiv and meets some basic quality standards.

>and all of them will gloss over many details that the reader is supposed to already be familiar with or be able to easily work out themselves
You tried to summarize something that takes DOZENS of pages to treat rigorously in a textbook into a single 4chan post...
You left out 99.9%.

[Anonymous No.11529258]

>>11529249
OP is mentally retarded

[Anonymous No.11529259]

>>11529255
"Solve" a polynomial obviously means to find the roots of it. Youre the brainlet if you couldnt infer that.

[Anonymous No.11529260]

>>11529253
Surely this is bait right? If not, then:
What is a positive number?
What is a negative number?
What is a whole number?
What is a non-whole number?
What is an irrational number?
Define zero.
What is a 1-dimensional number line, and what is a number that can be placed on it?

[Anonymous No.11529261]

>>11529253
That isn't a definition, but a description.

[Anonymous No.11529263]

>>11529256
Not rigorous doesn't mean confusing...

[Anonymous No.11529265]

>>11529259
It doesn't mean that, no.
And if you're saying we should just infer information, then you should obviously be able to infer the field, dumbass.

[Anonymous No.11529270]

>>11529256
>Regardless of whether or not it... makes sense...

[Anonymous No.11529271]

>>11529265
0=x^6+64
what is x

[Anonymous No.11529272]

>>11529258
I'm just ahead of the curve.
>>11529260
You are correct. The poster you're responding to is a moron.
>>11529263
If what I said is mathematically meaningful and can easily be translated into formal language (which is what you supposedly mean by saying you weren't confused by it), in what sense is it not rigorous?
>>11529257
>Basically any that you would find on arxiv and meets some basic quality standards.
Give me an explicit example.
>You tried to summarize something that takes DOZENS of pages to treat rigorously in a textbook into a single 4chan post...
You left out 99.9%.
Doesn't matter. I did it more quickly because I'm smarter and because I'm not writing a textbook.

[Anonymous No.11529276]

>>11529265
By saying "infer" I mean nothing else could have been meant by the question. Unless you have another possible interpretation of the request "Solve x^6 + 64". Oh what? You don't? Then shut the fuck up.
>be able to infer the field
No because anon could have meant different fields for which the answers would not be the same.
>>11529271
x is a variable :^)

[Anonymous No.11529278]

>>11529271
[math] \pm2, \pm2e^{\pi/3}, \pm2e^{2\pi/3}. [/math]

[Anonymous No.11529279]

>>11529270
If it makes sense and has mathematical meaning, in what sense is it not rigorous?
At this point I'm getting suspicious you're just trolling.

[Anonymous No.11529280]

>>11525618
amazed that this troll thread is still working after so many years. There is only one answer to 0.999... = 1 it's : kys.
/thread

[Anonymous No.11529281]

>>11529278
Exponents should be multiplied by i, and pm sign should be in exponent for the last four terms.

[Anonymous No.11529283]

>>11529272
>and can easily be translated into formal language
Who cares?

>in what sense is it not rigorous?
It leaves out 99.9% of the details. To be rigorous you have to, at the very least, prove and define anything you didn't assume to be known and every definition should be done formally.
Anything else is called "hand waving" in the trade, but that is something you will learn soon, at least when you are reading some research papers or reading some actual textbooks.

>Give me an explicit example.
The most recent paper published related to PDEs. I haven't seen it, but it will almost certainly meet my standards.

>Doesn't matter.
It's literally the only thing that matters. You can't be simultaneously rigorous and leave out almost everything.

[Anonymous No.11529285]

>>11529276
It literally doesn't make sense to say "solve" a polynomial. It is poor use of language, and anon should learn how to construct sentences that make sense.

[Anonymous No.11529287]

>>11529260
>What is a positive number?
a number x>0

>What is a negative number?
A number x<0

>What is a whole number?
a number that can be divided by 1 and have no remainder

>What is a non-whole number?
a number n>x>n+1 where n is any integer/whole number

>What is an irrational number?
a non-whole number with infinitely many digits that don’t follow a pattern

>Define zero.
the smallest positive number and largest negative number. Anything multiplied by it is 0. Adding it to anything has no effect

>What is a 1-dimensional number line, and what is a number that can be placed on it?
A line on which any real number can be placed

[Anonymous No.11529294]

>>11529278
OP is a dumbass

[Anonymous No.11529296]

>>11529278

[Anonymous No.11529311]

>>11529279
First of all, it would be good to make it clear what it means that a Cauchy sequence doesn't converge in the rationals, since this will be very instructive for the new reader.
>The most natural way is simply take Q and add to it some object for every Cauchy sequence that doesn't converge.
This is a fairly misleading line, particularly given the following line. It is helpful for understanding, but not helpful for rigor, since it contradicts what you are actually doing.
>Different sequences of rational numbers which should converge to the same number converge to different cauchy sequences.
What does this mean? What does it mean for two different Cauchy sequence to converge to different Cauchy sequence? It is helpful for you to actually show this using mathematical symbols.
You should define the equivalence class of Cauchy sequences, rather than simply saying "we make equal..."
>Now just verify that you can do arithmetic in this new number system the same way that you can in the rationals
Rigor would be showing this explicitly, given it is new information, and the most important part of the construction. Also, you have to show that it is well-ordered and complete.
Even the sentence,
>We want every number of our new number system to look the same, so we identify the elements of Q with constant cauchy sequences
lacks rigor, because you don't actually state which Cauchy sequence. You should write e.g.
>We identify the rational number q with the sequence q,q,q,q,q,q,...

[Anonymous No.11529316]

>>11529283
>It leaves out 99.9% of the details.
The details are trivial and left for the reader to work out.
>Who cares?
That's what it means to be rigorous.
>To be rigorous you have to, at the very least, prove and define anything you didn't assume to be known and every definition should be done formally.
As I said, I assumed the reader is familiar with basic set theory, as well as with the rational numbers, sequences and convergence.
>You can't be simultaneously rigorous and leave out almost everything.
I only left out the irrelevant trivial details. The meat is there and is rigorous. Most papers and textbooks do this too.
>The most recent paper published related to PDEs
Here is the first in Analysis of PDE's that is not a summary of other results
https://arxiv.org/pdf/2004.01074.pdf
Many trivial steps in the calculations have been skipped, just like I suggested. Low-level details are glossed over.
>>11529294
That was not me. -64 =64 e^(pi*i)
x^6+ 64=0 over the complex numbers is solved by
x in the set{2*e^(pi/6 + (2*pi*k/6)) | k=0,1,2,3,4,5}
Now what is your point?

[Anonymous No.11529320]

>>11529311
>Also, you have to show that it is well-ordered
The reals are not well-ordered bro.

[Anonymous No.11529323]

>>11529287
>A positive number is a number x>0
>A negative number is a number x<0
>Zero is positive and negative.
So 0<0?
Then 0 is not equal to 0.
Nice one, retard.

[Anonymous No.11529324]

>>11529296
Eh, multiply each of them by i.
>>11529294
See >>11529281
Also, I'm not OP.

[Anonymous No.11529327]

>>11529316
>The details are trivial and left for the reader to work out.
Yes. That makes it non rigorous.
You leave out 99.9% of what you want to cover.

>Many trivial steps in the calculations have been skipped, just like I suggested. Low-level details are glossed over.
Obviously. That doesn't make it non rigorous.
Still, things are proven, definitions are formal and excluded stuff is referenced, that is the very least you need to be considered rigorous.
Notice that they don't skip all profs? Or hand wave definitions?

>That's what it means to be rigorous.
Ask your prof...

[Anonymous No.11529328]

>>11529311
Alright, you win. I was just trying to see how far I can push the case that my construction is rigorous. I will concede that it is indeed not rigorous because of how much effort you put into the post.
Other details you're right about.
>What does this mean? What does it mean for two different Cauchy sequence to converge to different Cauchy sequence?
This is probably the most slippy line of my construction. What I meant by it is that I've added several elements to Q which are represented by Cauchy sequences whose pointwise difference tends to 0, indicating that they should converge to the same number, in turn indicating that I've added too many elements and that there's some quotienting out to do :)

[Anonymous No.11529330]

>>11529320
Sorry, didn't know what that word meant. I guess just ordered then, right?

[Anonymous No.11529331]

>>11529323
0 is neither*

[Anonymous No.11529332]

>>11529330
You probably meant totally ordered. Well-ordered means totally ordered with the additional assumption that every subset has a least element.

[Anonymous No.11529336]

>>11529332
Thanks.

[Anonymous No.11529388]

Just for completeness sake and to test my own understanding, I'll try proving the properties of real numbers in the next few posts without looking anything up.
So we define real numbers as equivalence classes of Cauchy sequences of rational numbers.
Let [(a_n)] be the equivalence class represented by the sequence (a_n).
Order:
Define [(a_n)]<[(b_n)] if the sequence (b_n - a_n) is eventually bounded below by some rational number e>0. That means there exists e>0 and a natural number N such that for all n>N, b_n - a_n > e.
Proof that this is independent of equivalence classes.
Suppose [(a_n)]=[(a'_n)]. Then a_n - a'_n ->0. If [(a_n)]<[(b_n)], then pick e>0 and N such that for all n>N, b_n - a_n > e. Pick N' such that for all n>N', | a_n - a'_n | < e/2
Then for all n>max(N, N'), b_n - a'_n = b_n - a_n + a'_n - a'_n > b_n - a_n - | a'_n - a'_n|> e/2, so the < is defined in the left argument. An analogous proof shows < is independent of the representative of the right side.
We've proven < is well-defined. Let's prove it gives a total order.
Let [(a_n)] be different from [(b_n)]. That means a_n - b_n doesn't converge to 0. That means there exist a rational number e>0 such that |a_n - b_n| > e for infinitely many values of n. As a_n and b_n are Cauchy sequences, pick a n N such that for all n,m>N, |b_n - b_m | < e/4 and |a_n - b_m|< e/4. Now take some N' larger than N such that |a_N' - b_N'|>e, or w.l.o.g. b_N' - a_N' > e >0 which is possible since the inequality discussed before holds for infinitely many values of n. Then for all n>N', a_n - b_n = a_n - a_N' - (b_n - b_N') + b_N' - a_N' > e - e/4 - e/4 > e/2>0, so [(a_n)]<[(b_n)] and hence < is a total order.
Will continue this in later posts.

[Anonymous No.11529398]

>>11525618
hiro implements a math captcha that gives basic integrals.

[Anonymous No.11529413]

>>11529287
>>11529331
>Define positive and negative numbers in terms of zero.
>Define zero in terms of positive and negative numbers.
Nope, nothing wrong here, no sir.

[Anonymous No.11529426]

Addition: define [(a_n)]+[(b_n)] to be [(a_n + b_n)].
Proof that this is well-defined:
Suppose [(a_n)] = [(a'_n)]. We want to prove that [(a_n + b_n)] = [(a'_n + b_n)], or equivalently, that
a_n + b_n - a'_n - b_n -> 0
But this precisely what it means for (a_n) to be equivalent to (a'_n). The same argument shows + is well-defined in the right-argument.
We define 0 to be [(0, 0, 0,...)]. Indeed, 0 acts as the additive identity: [(a_n)]+ 0 = [(a_n + 0)] = [(a_n)]. Every element has an additive inverse: [(a_n)] + [(-a_n)] = [(0,0,0,...)] = 0.
Multiplication: define [(a_n)]*[(b_n)] = [(a_n * b_n)].
Proof of well-definedness. Let a_n ~ a'_n. Then a_n - a'_n ->0. We want to show b_n(a_n - a'_n) ->0.
Let e>0. Since b_n is Cauchy, we can find such an N such that for all n>N, |b_n - b_N|<1, hence |b_n| < |b_N|+1. Take N' such that for all n> N', |a_n - a'_n| < e/(|b_N| + 1). Then for all n>max(N, N'), |b_n(a_n - a'n)| < (|b_N| + 1)*( e/(|b_N + 1|) = e so [(a_n*b_n)]=[(a'_n b_n)] and so multiplication is well-defined.
1=[(1,1,1,...)] is the multiplicative identity of R: this is verified the same way as 0 was for addition.
Proof that (R, *, +) is a field:
Most axioms follow applying the same axioms of Q componentwise to the sequences. The only thing we need to prove here (and correct me if i'm wrong) is the existence of multiplicative inverses.
Assume [(a_n)] is not 0.
As R is totally ordered, [a_n]>0 or [a_n]<0. In any case, by the definition of the order, there exists an N such that for all n>N, |a_n| != 0. Define the sequence (b_n) by setting b_i=0 if i<N and b_i = 1/a_i if i>=N. Then the sequence (a_i * b_i) is eventually all 1's thus [(a_i)]*[(b_i)] = [(1,1,1,...)] = 1.
What else do I need to prove here?

[Anonymous No.11529878]

>>11525618
Whenever someone tries to explain in understandable terms why 0.999... would = 1, there's always a step in the argument where, for seemingly no reason whatsoever, they lose their supposed rigorousness and accuracy and decide that in this case in particular it's ok to fudge it.

Usually they will base their beliefs off of the rumors on the history of beliefs they don't understand of people they've never met, but when they TRY to make it understandable in the way Euclid was understandable, they fail horribly. Either it's a coincidence that everyone who's tried to explain this to me can't think straight, or I'm not the one who can't think about it.

[Anonymous No.11529893]

>>11529878
It's because you're a moron.

[Anonymous No.11529897]

>>11529878
if you want it in Euclid's terms:

draw the number line. mark 0 and 1.
I suppose you know what 0.9 represents: divide the interval between 0 and 1 into 10 pieces of the same size, then 0.9 is the last mark.
0.99 you divide into 100 pieces. and so on.

0.999... is the smallest point on the number line which is larger than all 0.9, 0.99, 0.999 etc.

(here larger/smaller mean "lies on the left/right", I think you get that)

[Anonymous No.11529901]

>>11529878
Read a book on the foundation of analysis.

[Anonymous No.11530446]

Fuck you OP. You mindless, pond scum tier, Godless heathen. Your arrogance ensures only one thing: That you will burn in HELL for A DISCRETE AMOUNT OF TIME!

What you conceited fools fail to recognize is this: The foundations of human logic are built upon nothing more than human intuition. You are incapable of understanding that simple fact for two reasons:
1) You lack the cognitive capacity to think deeply.
2) You are a sodomite.

0.999... DOES NOT EQUAL 1

Indeed, 0.999.. does not even exist. It's a fantasy. One might as well write:

Pink Flying Elephant = 1.

That's the crux of your so called "logic". YOU GODDAMNED FOOL! You and your ilk have set back mathematics for centuries. May you be cursed with daily violent diarrhea for the rest of your miserable life.

DEUS VULT!

[Anonymous No.11530596]

>>11528872
i'm sorry your dada dropped you on your head as a baby but you're wrong.

[Anonymous No.11530631]

>>11529222
there is no measure of closeness in infinity.
it either is or isn't infinite.
it's explicitly a binary method.
no calculation has ever required, or will ever require infinity. Given the binary notion of it, the fact it never has been means it always hasn't been.

if you have infinite decimal places, you're required to fill all infinite decimal places to get "close" to 1.

much like $0.99 isn't literally a dollar, but close to it; so is 0.999... not literally 1, but close to it.

[Anonymous No.11530648]

>>11529413
What else then?
Sage

[Anonymous No.11530655]

>>11530648
>the largest negative number

[Anonymous No.11530681]

>>11530631
>it either is or isn't infinite
uh what? he didn't say it was infinite or that it was close to infinity but not infinity, or that it was semi-infinite. It's a finite value, it's just unbearably close to another value.

[Anonymous No.11530686]

>>11530648
the number that is neither negative nor positive?

That way, you're putting it in terms of how you defined numbers before, because a number that isn't less than or greater than 0 must be equal, it's a consistent definition, lol.

[Anonymous No.11530691]

>>11525618
>The best scientific minds and mathematicians express themselves clearly and orderly
Have you ever read any cutting edge papers at all?

[Anonymous No.11530742]

>>11530681
you might find this hard to believe but infinity as defined by the axiom of infinity can be shown to be a literal "finite" end.
the amount of 9's in 0.999... is infinite for it has as many 9's as natural numbers, but EVERY decimal place is also a real number decimal [math]\frac{9}{10^n}[/math] for any n. The scope of this merely means that you can't define "which" number is the greatest number.

again, there is no closeness aspect within infinity, so there can always be a "greater" amount of 9's imaginable for any instantiation of "0.999...", but regardless of what greater imaginable amount there are, it's still within the scope of infinity.

0.999... < 1, simply because every decimal 9 can be bijected with every real number in the set of all natural numbers.
Another way of saying that is ALL the natural numbers, assigned to their n'th decimal place (aka 1,2,3 -> 0.9, 0.99, 0.999) will only equate to an infinite amount of 9's. You'd need a larger """number""" than ALL the numbers to accommodate anything greater than 0.999..., which relative to relative to this scale between 0.000... -> 1.000..., would be the number 1.

For example, if infinity were falsely described as an actual number and that this number existed at a point on the numberline far ahead of all the other natural numbers, when scaling 0->∞ to bijected mappable decimal values between 0->1, then 1 and ∞ represent the same point.
This inherently leads to the false assumption that 0.999...=1 because it has ∞ decimals.

But again, that would be a false attribution of infinity, and not what it actually is. Infinity as presented in the Axiom of Infinity is an attribute that accounts for ALL the natural numbers. from 0->the greatest real number.
What is the greatest real number?
simple: [math]undefined[/math].
it's not defined.

[Anonymous No.11530768]

>>11530742
>0.999... < 1, simply because every decimal 9 can be bijected with every real number in the set of all natural numbers.
Non sequitur.

>You'd need a larger """number""" than ALL the numbers to accommodate anything greater than 0.999..., which relative to relative to this scale between 0.000... -> 1.000..., would be the number 1
Circular argument.

>This inherently leads to the false assumption that 0.999...=1 because it has ∞ decimals.
It has infinite decimals regardless of whether inf is on the number line. You are confusing the members of the sequence with the sequence itself.

>Infinity as presented in the Axiom of Infinity is an attribute that accounts for ALL the natural numbers.
The axiom of infinity does not present infinity as anything.

[Anonymous No.11530781]

>>11530742
>>11530681


Yet another way of looking at it is two sets.
[math]A: [ \frac{9}{10}, \frac{99}{100}, \frac{999}{1000}, ... ] [/math]
[math]B: [ \frac{1}{10}, \frac{1}{100}, \frac{1}{1000}, ... ] [/math]
Every n'th element of A requires the addition of the same n'th element from B to equal 1.
Since infinity is the trait which encompasses all natural numbers, if you made the statement "A and B have infinite elements", it also carries the meaning that ALL of A and B's elements can all be attributed to any real natural number n, but also the number "0.999..." exists inside A, requiring the addition of it's B counter-part.

[Anonymous No.11530789]

>>11530768
the axiom of infinity is the basic definition of infinity and the only guaranteed definition of infinity.

[Anonymous No.11530799]

>0.999... is actually less than 1 according to the definition of infinity
>mathlets perpetuated a lie that 0.999...=1 cause they never actually even learned what infinity was

[Anonymous No.11530845]

I DONT CARE ABOUT WHATS TRUE OR USEFUL I JUST WANT TO BE A GOOD BOY AND LISTEN TO THE PROFESSOR AND GET GOOD GRADES

[Anonymous No.11530853]

math is a liberal art.

lmao

the absolute state of mathematics

[Anonymous No.11530945]

>>11526904
its faster to just use the geometric series formula

[Anonymous No.11530952]

>>11526904
>therefore lim_n -> inf 1/10^n = 0
nobody disputed this
>Therefore 0.999... = 1 - 0 = 1
IF THE LIMIT IS THE SAME THING AS EQUALITY REGARDING DECIMAL EXPANSION. IT IS NOT.
Christ you people are thick.

[Anonymous No.11530959]

>>11529287
>a number that can be divided by 1 and have no remainder
fuck my sides

[Anonymous No.11530973]

God cursed heathens, sodomizing each other while babbling about infinities.

CRUSADE FOR A DISCRETE UNIVERSE WHEN???? LET'S HAVE AT THE HEATHENS!

DEUS VULT!

[Anonymous No.11530984]

>>11530973
Your mom was also babbling while I was sodomizing her.

[Anonymous No.11531108]

>>11530973
I love you don't stop

[Anonymous No.11531142]

>>11530789
>the axiom of infinity is the basic definition of infinity
It's not a definition of infinity. You obviously haven't even read it. Dumb LARPER.

[Anonymous No.11531144]

>>11530781
>Since infinity is the trait which encompasses all natural numbers, if you made the statement "A and B have infinite elements", it also carries the meaning that ALL of A and B's elements can all be attributed to any real natural number n
How?

>but also the number "0.999..." exists inside A
How?

[Anonymous No.11531150]

>>11530952
>IF THE LIMIT IS THE SAME THING AS EQUALITY REGARDING DECIMAL EXPANSION. IT IS NOT.
It literally is, that is how you construct the real numbers. You retards don't even know what a real number is.

[Anonymous No.11531156]

>>11530973
the idea of discrete space-time is an abomination, the fascination that christfags and /pol/ have with it is inexplicable given that the physical reality of infinities suggests some form of theism is tenable (they don’t exist but finitists are still brainlets). Tooker understood this

[Anonymous No.11531288]

I think with this thread it's safe to say we've reached an understanding that the anons claiming 0.999...=/= 1 are actually mentally handicapped who through the whole thread didn't even read the OP still fail to understand the basic definition that mathematicians use for an infinite decimal expansion.
Learn something every day!

[Anonymous No.11531323]

>>11531150
Ok, what do you do to the number 0.888... to make it real?

[Anonymous No.11531329]

>>11529222
>an infinitely close one, meaning the difference between the platonic value and the decimal representation is infinitesimally small.
yeah, exactly. an infinitesimally small difference is not the same as a difference of 0. infinitesimally small != 0.

[Anonymous No.11531334]

>>11531323
Could you rephrase your question to be coherent?

[Anonymous No.11531336]

>>11531329
It literally is the exact same thing. 1/inf=0.

[Anonymous No.11531345]

>>11528970
That was my point, yes. I think saying that 0.333... literally is 1/3rd is false, but it's only different by an undefinable degree. For most purposes, that's good enough.
>>11531336
Sounds like bullshit people made up to make things simpler, so they could stop worrying about shit so infinitely and indefinably insignificant that it doesn't matter.

[Anonymous No.11531356]

>>11531345
>Sounds like bullshit people made up to make things simpler
You just described math. Yes, we have defined things in such a way that 0.999...=1, since that actually makes sense in most contexts.

[Anonymous No.11531362]

>>11531356
Sure. It's good enough and works, but that's no reason to forget that it's bullshit. That's really the end of any argument on the subject, isn't it?

[Anonymous No.11531369]

>>11531362
How is it bullshit?

[Anonymous No.11531374]

>>11528962
See >>11528958

[Anonymous No.11531390]

>>11531369
Take the 1/3rd example. 0.333... is simply the result of the decimal system endlessly failing to represent 1/3rd. But hey, we still need something to represent 1/3rd so we use it anyway. Infinity's not even really a number, so who cares if it differs slightly at some non existent infinite decimal point?

At some point, it's best to give up and work with what you have. You smooth it over with bullshit, because that's how you keep it working.

[Anonymous No.11531488]

>>11530984
I imagine you found that quite gratifying, taking into account she has been dead for ten years. You sick fuck.
>>11531108
I love you too. In the platonic sense, of course.
>>11531156
YOU WILL BURN! So will Tooker, whoever the fuck he is.
>>11531288
YOU WILL BURN TOO!
>>11531323
You cant make something unreal real. I can write logic equations which prove that Blue Whales have formed major civilizations in the Sahara desert. Doesn't matter how logical the proofs are, its still bullshit.
>>11531356
>>11531390
Praise the Lord! A believer in the one true FAITH! May the peace of the Almighty be with you, Brother.

And there we have it. A mighty victory for God and Reason! We have battled the pagans and the infidels and emerged VICTORIOUS!

Brothers and Sisters! Our next step is to burn any book containing any recurring decimal bullshit. Maybe even sack Constantinople at the same time. DEUS VULT!

[Anonymous No.11531507]

>>11531144
Here is the set of natural numbers.
N: [1,2,3,4,...]
It is said that the size of this set is ∞, according to the axiom of infinity.

Let A be a set populated by the partial sums of [math]\sum_{n=N[1]}^{\infty}\frac{9}{10^n} [/math]
the initial starting condition of the sum is the first element of N, indexing from 1.
The first element of N just also happens to be 1.
A: [0.9, 0.99, 0.999, 0.9999, ...]

As the sum goes from 1 to ∞, and ∞ is what describes the collection of ALL natural numbers, the sum is said to account for ALL elements of N, and as the result of the sum populates A, then A has an easily understood bijection with N. Each element of N can be paired with each element from A. N has ∞ size, so A also has ∞ size, the same size as N.


>but also the number "0.999..." exists inside A
>>how?
>>11527368

[Anonymous No.11531522]

>[1, 2, 3]
>Size: 3
>Contains 3 as an element.

>[1, 2, 3, 4, 5, 6, 7].
>Size: 7
>Contains 7 as an element

>[math]\N [1,2,3,4,5,6,7,...][/math]
>Size: __
>Contains __ as an element

>infinity is an indeterminable finite natural number
Bullshit or truefacts?

[Anonymous No.11531702]

>>11531507
If S is in bijection with N, then either ∞ is in N or the infinite sum is not in S. So which one is it?

[Anonymous No.11531756]

>>11531345
"the n-th digit is d" is a simple logical formula which actually involves only fractions
0.333... means that the n-th digit is 3 for all n
the only number which satisfies this is 1/3

[Anonymous No.11531766]

Can someone explain to me concisely why the number 0.0000 .... 0001 can't exist?

As in.
>0.1
Add a 0 before the 1
>0.01
Add a 0 before the 1
>0.001
Add a 0 before the 1
>0.0001
Repeat for infinite steps
>0.00000....001

Why is this a mathematical impossibility? I've seen other "algorithmically" built numbers like this before.

[Anonymous No.11531771]

>>11531766
>Repeat for infinite steps
The sequence converges to zero, so by definition of equality it is equal to zero.

0.0000 .... 0001 is just a stupid way to say 0.

[Anonymous No.11531772]

>>11525618
0.999... = x
9.999... = 10x
9.999...- 0.999... = 10x - 0.999...
9 = 10x - 0.999...

But from the first line 0.999... = x, so.

9 = 10x - x
9 = 9x
9/9 = x
1 = x

Do they seriously not teach you guys this in middle school?

[Anonymous No.11531775]

>>11531766
it can be easily proven that any number x > 0 must satisfy x > 1/10^N for some integer
this clearly contradicts the meaning of 0.000....001

[Anonymous No.11531829]

>>11531775
>x must be greater than 0
>u-uh i mean x must be greater than 1/10^n

[Anonymous No.11531836]

>>11531772
Did you not read the thread, Unbeliever?

That fallacious so called "proof" has been disproven, by the grace of God. For years, decades, perhaps centuries, Humanity has suffered long under the LIES, DECEIT and OUTRIGHT MAKING STUFF UP OUT OF THIN AIR!

We suffer no longer. 0.999... DOES NOT EQUAL 1 We are free now. Free from the TYRANNY of "LOL, I will make shit up as I go" mathematics. As a side benefit we are also free to conquer Jerusalem and retake the HOLY LAND for the GLORY of GOD! DEUS VULT!

[Anonymous No.11531837]

>>11531829
exactly

[Anonymous No.11531853]

>>11531829
[math]N = \lceil \log_{10}(\frac{1}{x}) \rceil[/math]

[Anonymous No.11531877]

>>11531837
Why must x be greater than 1/10^n?
sounds like u mad 1/10^n exists.

[Anonymous No.11531885]

>>11531877
[eqn]N = \lceil \log_{10}(\tfrac{1}{x}) \rceil \\
N \geq \log_{10}(\tfrac{1}{x}) \\
10^N \geq 10^{\log_{10}(\tfrac{1}{x})}\\
10^N \geq \tfrac{1}{x}\\
x \geq \tfrac{1}{10^N}[/eqn]

x > 0 is the only thing that was used

[Anonymous No.11531887]

>>11531885
I have no idea what your goofy formula is supposed to mean. Make sense pls.

[Anonymous No.11531893]

>>11531887
y becauz u mentally challengd lol

[Anonymous No.11531896]

>>11531887
>I have no idea what your goofy formula is supposed to mean
>I don't understand logarithms
underage b&

[Anonymous No.11531960]

>>11531893
>>11531896
It has no relevance to infinity or a smallest decimal.

i can't imagine it has relevance to anything actually.

brainlet math for brainlets.

[Anonymous No.11531978]

>>11531960
it has /significant/ relevance to smallest decimal

it proves that no such thing exists

[Anonymous No.11531979]

>>11531960
y becauz u ar an fagit lmao

[Anonymous No.11532148]

>>11531507
>It is said that the size of this set is ∞, according to the axiom of infinity.
Incorrect, the axiom of infinity simply says that a set exists containing all natural numbers.

>>but also the number "0.999..." exists inside A
>>how?
>>11527368
This states that 0.999... is a member of S without stating why. Then it repeats the incorrect definition of infinity as if that explains anything.

Again, how is 0.999... a member of S? If you can't answer this question your entire argument fails.

[Anonymous No.11532155]

>>11531323
>8/9 is not real

[Anonymous No.11532160]

>>11531390
>Take the 1/3rd example. 0.333... is simply the result of the decimal system endlessly failing to represent 1/3rd.
How is failing to represent 1/3?

[Anonymous No.11532167]

>>11531522
0.9 < 0.999...
0.99 < 0.999...
...
0.999... < 0.999...

Absolutely brilliant. A new technique in math was born today, with unbelievable results.

[Anonymous No.11532174]

>>11532167
>NOOOOOOO YOU CAN'T DO THAT IT ONLY WORKS WHEN 1 IS ON THE RIGHT HAND SIDE

[Anonymous No.11532185]

Reminder that 0.999...=1, as a mathematical statement, when you unravel the definitions, means precisely that
For all rational numbers e>0, there exists a natural number N such that for all natural numbers n>N, |10^(-n)|<e.
That's literally just a restatement of 0.999...=1 and you're a brainlet if you still think the above restatement is not true.

[Anonymous No.11532207]

0.999...=1
---------------------
is equivalent to (by the definition of an infinite decimal expansion)
----------------
The Cauchy sequence (0.9, 0.99, 0.999, ...) is equivalent to the Cauchy sequence (1,1,1,...)
-------------
is equivalent to (by the definition of equivalence relation in the real numbers)
--------------
The sequence (1-0.9, 1-0.99, 1-0.999, ....) tends to zero. This is precisely the sequence which is 10^-i at the i'th place.
-------------
is equivalent to (by the definition of "tends to 0")
-------------
For all rational numbers e>0 there exists a natural number N such that for all natural numbers n greater than N, |10^-n|<e
-------------
Proof of the last proposition:
-----------
Given a rational e=p/q >0, as n->10^n is an increasing function, we can find such an N that 10^N > q. Then 10^-N < 1/q< p/q and for all natural numbers n>N, since 10^n > 10^N, 10^-n < 10^-N < p/q = e so 10^-n < e. As was desired.
As the proposition I've just proved above is equivalent to 0.999..=1, we see that indeed the real number 0.9999.... is equal to 1.

Simple kindergarten level maths. Why are the 0.999...!= 1 morons so confused about this?

[Anonymous No.11532225]

>>11532207
>Why are the 0.999...!= 1 morons so confused about this?
I will give you an example:

>>11532207
>The sequence (1-0.9, 1-0.99, 1-0.999, ....) tends to zero.
yeah it TENDS to zero but it's not like it ACTUALLY GETS THERE so 1 - 0.99.. is never zero, checkmate

[Anonymous No.11532253]

>>11532225
>but it's not like it ACTUALLY GETS THERE
Never said it does.
> so 1 - 0.99.. is never zero
That doesn't follow. Read my post again, retard.

[Anonymous No.11532285]

>>11532225
Imagine not knowing what "equals" means.

[Anonymous No.11532297]

Obligatory post

[Anonymous No.11532767]

Bump.

[Anonymous No.11532777]

Sorry for this unrelated question, but I didn't want to start a new thread.

I'm trying to make a secure password for a winrar archive, 24 bit, numbers and lowercase letters. How many permutations would there be?

[Anonymous No.11532813]

>>11532777
I gotchu.

Each bit has 34 characters available. 24 bits must be filled with one of 34 characters. Hence, there are 24^34 = 84 quattuordecillion permutations.

If you want a strong password with no repeats, that will be P(34,24) = 81 nonillion permutations.

[Anonymous No.11532832]

>>11532777
Go look up the formula on Wikipedia

[Anonymous No.11532906]

>>11532813
>>11532832

Thanks very much for replying to me. I am indeed retarded. Would you consider it a strong password?

[Anonymous No.11532909]

>>11532813
and just to clarify, some letters and numbers do repeat, which means of course that not every letter or number is necessarily used

[Anonymous No.11532924]

>>11531323
Point eight repeating is 8/9. Stop being a retard, retard.

[Anonymous No.11533499]

The only thing that equals 0.999... is 0.999...

0.999... APPROXIMATES 1, but it does NOT equal it.

Case closed. Next.

[Anonymous No.11533533]

>>11532148
I'm sorry you're too retarded to understand, but i'm pretty sure its already reduced to simplest terms. Any further understanding is a requirement of you to learn what's being discussed and how to read the formulas. There are only three aspects you really need to know about.
1. The axiom of infinity, google it
2. How to perform a sum function in calculus, google it. The [math]\sum[/math] is called "sigma", but "calculus sum" should be a sufficient search term.
3. Array indexing in computer terminology. I use it in place of traditional set notation. [math]\mathbb{N}_1 = N[1] [/math]. Indexing usually begins at the 0'th element, for example
>Array S: [1,2,3,4]; S[0]=1
which is why I might reference "indexing from 1" instead, so S[1]=1 rather than S[1]=2 if indexing from 0.

the key part of the proof formula in >>11527368
essentially says that an element from S*, when added together with all prior S* elements before it, is equal to that same element in S.
If S* can be shown to construct "0.999..." by infinitely adding all it's infinite terms together, then the formula holds that this construction also exists inside S, which is in bijection S*, where both S and S* are also in bijection with N, where N is the set of all natural numbers with the set size of infinite.

There are more implications to the proof than merely understanding "0.999..." exists inside S, so it'd be nice if you got around to understanding the whole scope.

[Anonymous No.11533566]

>>11532167
>>11532174
Since 0.999... exists in a set bijected with the set of Natural real finite numbers,
[1->∞] = [0->0.999...]
You can arbitrarily map [1->∞] to fucking [0->1] if you want, but why stop there?
What happens when you map [1->∞] to [0->2]
It can merely assumed that 1 must exist in that mapping because it's between 0 and 2, but at the same time there is no defined arithmetic with ∞. There is no "∞/2 = 1" in the mapped range [0->2], or rather to say it better since ∞ is mapped to 2, there is no "∞/∞ = 1".
If you were smart about it, you could probably make the case that [math]\frac{\infty}{\infty} =1 [/math] should be a true statement.

Point is, you can arbitrarily map [1->∞] to fucking anything, including [0->1], if you at least show a proof.
However i've already provided a proof that the maximum value that can appear bijected with the only defined guarantee'd infinity within the range [0,1] is merely 0.999..., so the ACTUAL scope of that range is also merely [0->0.999...]
If you want a range that has 1 in it, then fucking add 1 to that range
[1->1.999...]

[Anonymous No.11533568]

>>11532297
>lim ahead of sum
Sum already contains a lim function in it, brainlet.

[Anonymous No.11533572]

>>11533568
Stop speaking chinese

[Anonymous No.11533575]

>>11533566
(0 -> 0.999...) < (1 -> 1.999...) <
(2 -> 2.999...) < (3 -> 3.999...) < ...

Is this really so hard to understand?

[Anonymous No.11533749]

>>11525618
I try to avoid these .999... /= 1 threads because they are are a blight on this already shitty board and at least half of the posts saying they are not equal are bait, but fuck it.

If 1 /=.999... then what is the value of x = 1- .999...?

Hint: either x = 0 or your number .999... has finitely many nines!

>inb4 0.000...001
That's not a valid construction.

The fact of the matter is that decimal representations of real numbers need not be unique.

[Anonymous No.11533755]

>>11533499
>0.999... APPROXIMATES 1, but it does NOT equal it.
Then what is 1-0.999...?

[Anonymous No.11533771]

1/3=.333...
3×3=9
1/3×3=.999...
1=.999...

[Anonymous No.11533778]

>>11533749
but do you expect non trolling brainlets who cant understand 0.999... = 1 to understand the difference between a number and a representation of it?

[Anonymous No.11533782]

>>11533533
>I'm sorry you're too retarded to understand, but i'm pretty sure its already reduced to simplest terms.
There's nothing that has to be reduced. You didn't make an argument for how 0.999... is an S, you just stated it is. This means your entire argument fails.

>1. The axiom of infinity, google it
You're the one who hasn't even read it. Take your own advice.

>essentially says that an element from S*, when added together with all prior S* elements before it, is equal to that same element in S.
Yes, but that doesn't imply all elements of S* added together is equal to an element of S. There is no last element of S*, so there is no element of S corresponding to the entire sum. I've explained this mistake several times to you and you have failed every time to respond to it.

>If S* can be shown to construct "0.999..." by infinitely adding all it's infinite terms together, then the formula holds that this construction also exists inside S
No, it doesn't, as I've already explained. Handwaving is not an argument.

>There are more implications to the proof than merely understanding "0.999..." exists inside S, so it'd be nice if you got around to understanding the whole scope.
Implications of a flawed proof are irrelevant.

[Anonymous No.11533790]

>>11533778
Call me naive but I think that this could actually be beneficial for people who legitimately have some inkling of an interest in math. I some of the people who have put in some (fundamentally misguided) effort into arguing that .999... /= 1 do, in fact, have some interest in mathmatics. I could very easily be wrong though.

[Anonymous No.11533797]

>>11533566
>[1->∞] = [0->0.999...]
What exactly is the bijection here?

>You can arbitrarily map [1->∞] to fucking [0->1]
That is what you already did since 0.999... is 1.

>It can merely assumed that 1 must exist in that mapping because it's between 0 and 2
You could just define a mapping instead of making up assumptions.

>but at the same time there is no defined arithmetic with ∞. There is no "∞/2 = 1" in the mapped range [0->2], or rather to say it better since ∞ is mapped to 2, there is no "∞/∞ = 1".
If you were smart about it, you could probably make the case that ∞∞=1 should be a true statement.
What does this have to do with the mapping?

>Point is, you can arbitrarily map [1->∞] to fucking anything, including [0->1]
It depends what you mean. You certainly can't map from natural numbers to every real between 0 and 1. The amount of reals between any two different numbers is uncountable. You're just making up vague statements without saying anything relevant to my post.

>However i've already provided a proof that the maximum value that can appear bijected with the only defined guarantee'd infinity within the range [0,1] is merely 0.999...
Where?

Nothing you said even begins to explain why >>11531522 is valid but >>11532167 is invalid.

[Anonymous No.11533804]

>>11533749
The construction of infinity via the axiom of infinity, the only guaranteed infinity, pretty much allows that infinity is actually an "indeterminable arbitrarily large finite number".
so saying 0.999... has finitely many 9's isn't exactly incorrect.

[Anonymous No.11533805]

>>11533790
Just realized that I had a stroke halfway through typing that. Should say:
"I think that some of the people who have put in a (fundamentally misguided) effort into arguing that .999... /= 1 do, in fact, have some interest in mathematics."
"

[Anonymous No.11533816]

>>11533782
[eqn]\sum_{m=1}^{p} S^{*}[m] = S[p] [/math]
What about this formula do you disagree with?
If we take it to any p, it holds true.
Test for p=4
S*[1] + S*[2] + S*[3] + S*[4] = 0.9 + 0.09 + 0.009 + 0.0009 = 0.9999
S[4] = 0.9999 as well

What do you disagree with here? Which assumptions or solving of the formula do you find disagreeable?

[Anonymous No.11533824]

>>11533782 #
[eqn]\sum_{m=1}^{p} S^{*}[m] = S[p] [/eqn]
What about this formula do you disagree with?
If we take it to any p, it holds true.
Test for p=4
S*[1] + S*[2] + S*[3] + S*[4] = 0.9 + 0.09 + 0.009 + 0.0009 = 0.9999
S[4] = 0.9999 as well

What do you disagree with here? Which assumptions or solving of the formula do you find disagreeable?

[Anonymous No.11533831]

>>11533816
>What about this formula do you disagree with?
Nothing, if p is restricted to natural numbers. Otherwise it's false since S[p] doesn't exist.

>What do you disagree with here?
Nothing, why do you think this shows 0.999... is an element of S?

For any natural number p, the sum from m=1 to p is not the sum of all elements of S*. So this can never equal 0.999...

[Anonymous No.11533836]

0.999... does NOT equal one. If this is not patently obvious then you are brain dead.

There is no point trying to talk about this to people who have less intellectual processing power than a Scyphozoa. Their stinking, filthy, tiny little minds are closed. The time for rational discussion is over. The time for direct military action has begun.

Brothers and Sisters. Sharpen your swords. Join with me in a HOLY CRUSADE to liberate mathematics from the Infidels. Yes. It is time to take back the text books from the clutches of the GOD CURSED SODOMITES! It is time to proclaim the universal truth of a DISCRETE Universe!

DEUS VULT!

[Anonymous No.11533847]

>>11533790
Naive probably not, but very optimistic for sure. Most of the (non trolling) people here arguing for 0.999... != 1 do not seem humble enough.

[Anonymous No.11533852]

>>11533804
What I mean to say is that that if our x = 1-.999... is non-zero, then our sequence of nines after the decimal place must terminate, specifically [math] x \neq 0 \implies \exists N \in \mathbb{N}[/math] st. [math].999... = 0.d_1d_2d_3...d_N[/math].

If you give me a value of x, I can always find an N satisfying the above property.

This is equivalent to what has already been said; however, I think it might be a helpful way of approaching it.

[Anonymous No.11533854]

>>11533836
xd bro. funnier every tim

[Anonymous No.11533858]

>>11533847
>Most of the (non trolling) people here arguing for 0.999... != 1 do not seem humble enough

Exactly. They lack grace before the sight of God. Their arrogance has blinded them. That is why they must burn. DEUS VULT!

[Anonymous No.11533861]

>>11533824
>>11533831

[Anonymous No.11533862]

>>11533852
The reason for putting it this way being that finding any such N would be a contradiction to anyone's notion of what 0.999... "means" in the sense that you have found a terminating digit.

Note that (as I said earlier), the "number" 0.000...0001 as a solution to 1-.999... is inherently contradictory for this reason.

[Anonymous No.11533866]

>>11533831
But what about
[eqn]\sum_{m=1}^{\infty} S^*[m] [/math]?
Do you disagree that this equals "0.999..."?
It's imperative to understand the meaning behind the invocation of ∞ here, and that this is the same ∞ provided within the scope of the axiom of infinity:
>"guarantees the existence of at least one infinite set, namely a set containing the natural numbers."
This is strictly understandable as
>[math] \mathbb{n} : [1,2,3,4,5,...] [/math]
and that the size of this set is ∞.

A: [1,2] is size=2
B: [1,3,7] is size=3
[math]\mathbb{N} : [1,2,3, ...] [/math] is size=∞

2 describes ALL elements in A.
3 describes ALL elements in B.
∞ describes ALL elements in [math]\mathbb{N}[/math]

So when the use of ∞ is applied to p for that formula, what it's saying is that it's accounting for ALL the infinite elements.
The sum strictly produces "0.999..." that has ∞ decimal places because ALL elements of S* had been accounted for.
The formula shows that this is a member of S at S[∞], which is to say it is an index inside ALL of S, which boils down to "it is inside S".

[Anonymous No.11533877]

>>11533866
What he is saying is that S[∞] does not exist because ∞ is not a natural number.

[Anonymous No.11533880]

>>11533852
The axiom of infinity allows for indeterminable finite numbers to exist inside N. These are numbers that are explicitly finite for the assumption that they worked under finite arithmatic (aka x+n, n-y), but at the same time are indeterminable. There are as many indeterminable finite numbers in the set of all Naturals as there are determinable finite numbers, for example: [eqn]\lim_{n\to\infty} \infty-n[/eqn]
And what exactly means indeterminable? Well its right there
>(aka x+n, n-y)
If we assume x, n, and y are all finite numbers, then what the fuck is x+n?
x+n = ???
an indeterminable finite number.

[Anonymous No.11533881]

>>11533858
n u r still an fag ror

[Anonymous No.11533891]

>>11533877
[math]\mathbb{N} : [ \underbrace{1,2,3,4,5,...}_{\infty} ] [/math]
Infinity describes ALL the natural numbers.

I would believe you have more of an issue with the notion of "truly" accounting for ALL members of an infinite set, but the sum's historical usage of being taken to infinity within it's limit ([math]\sum_{n=1}^{\infty}[/math] would seem to indicate that there really is no problem using infinity this way.

[Anonymous No.11533916]

>>11533866
>Do you disagree that this equals "0.999..."?
No.

>It's imperative to understand the meaning behind the invocation of ∞ here, and that this is the same ∞ provided within the scope of the axiom of infinity:
You don't seem to understand that saying an infinite set exists is not a definition of infinity. Not that it matters since it's yet another irrelevancy.

>So when the use of ∞ is applied to p for that formula, what it's saying is that it's accounting for ALL the infinite elements.
Just define p already. Is p the index of the set? Or is it a size of a set? Inf is not an index of the set. There is no infinity index. That's the same as saying there is a final index of the set, when the whole point is that there is no final member. If p is the size of a set then there is no guarantee S[p] exists.

>The formula shows that this is a member of S at S[∞]
How? You keep saying this but it's not a mathematical argument.

Let me show you an analogy to your argument:

Let A = [1, 1.5, 1.75]
Let B = [1, 0.5, 0.25, 0.125]

Sum from k=1 to n of B[k] = A[n] for n=1,2,3

Sum from k=1 to 4 of B[k] = 1.875

Therefore A[4] = 1.875

Where did the proof go wrong?

[Anonymous No.11533926]

>>11533880
>The axiom of infinity allows for indeterminable finite numbers to exist inside N.
No it doesn't, and there is no such thing as an indeterminable number.

>for example:
>limn∞∞−n
That's a limit of an indeterminate form.

[Anonymous No.11533929]

>>11533891
>Infinity describes ALL the natural numbers.
What do you mean by "describes?" And how does this show S[∞] exists?

[Anonymous No.11533933]

>>11533880
Another example of an indeterminable finite number is TREE(3).
It has to be a finite, natural number because it is an iterative count. 1,2,3,4-> etc.
but each digit of the number is not known, and it's probably physically impossible to find a medium which can store every single digit. It has more digits than there are assumable atoms in the universe. And yet even despite this gargantuan number, it's still finite and there still less than infinity and simply an arbitrary member of [math]\mathbb{N}[/math]

[Anonymous No.11533949]

>>11533916
Your sets A and B are not in bijection with each other. A has 3 elements, B has 4.
They aren't the same size.
Bijection requires the size.
A: [6,9,56]
B: [1,2,3]
You can biject these two sets because they're the same size.
[math]A: [ \stackrel{6}{\downarrow} , \stackrel{9}{\downarrow} , \stackrel{56}{\downarrow} \\ B: [ 1, 2, 3 ] [/math]
What are you trying to pull...?
N, S and S* are all the same size. They are bijected. One element from N can pair with exactly one element in S* and S.

[Anonymous No.11533957]

>>11533949
>Your sets A and B are not in bijection with each other. A has 3 elements, B has 4.
OK, make A = [1, 1.5, 1.75, 2]

The proof is still wrong. Good job, you made a completely irrelevant distinction. Try again.

[Anonymous No.11533959]

>>11533957
you are way too retarded to understand the proof and therefore way too retarded to argue against it.
Your idiocy is not an argument.

Shut the fuck up stupid bitch.

[Anonymous No.11533961]

>>11533959
Not an argument, try again.

Also, you failed to define p. Why can't you explain basic parts of your "proof?"

[Anonymous No.11533983]

If your penis is 1.5 times the length of the coconut it is in, put the camera on the other side an post it.

[Anonymous No.11533999]

>>11525618
There is no way to mathematically express the citation of a non sequitur, i.e. the fact that every mathematical proof I've seen commits non sequitur of the type circular reasoning, i.e. they all presuppose ahead of time that .999...=1 in one way or another.
Nor is there a way to mathematically express a negation that is not an affirmative expression, i.e. your arguments for why .999... would be equal to 1 simply are not compelling to anyone not retarded. Since it is entirely conceivable that .999...!=1, and since there are no real-world implications for .999... equaling 1, aside from making the succeeding calculation or two as if it does (when the metaphysical truth of it not equaling 1 is not relevant), what reason could you possibly have for insisting it equals 1 that isn't dogmatic or even evil?
>.666... is a whole 7
You might as well be insisting that your waifu pillow is a real gurl.

We should be wary of people learning to express themselves through language like math lest the layman confuse your stupidity for sense.

[Anonymous No.11534006]

>>115339
We should be wary of people on adderall

[Anonymous No.11534027]

>>11533961
lmao didn't you see how the S* set was constructed?
[eqn]\lim_{k\to\infty} \frac{9}{10^k} [/eqn]
S* = [0.9, 0.09, 0.009, ...]
N = [1, 2, 3, ...]
Do you not see that both are countable based on bijection with N?
The 3rd element of N is 3. The 3rd element of S* is 0.009, which is 3 decimals. A 9 is at the 3rd decimal digit.
An element n in {N} is strictly paired with the same element n is {S*} which was constructed from the n'th element of {N}, and this applies to {S} too.

Just cause your A and B sets aren't strictly related doesn't mean N, S and S* are also unrelated. Cut it out.

[Anonymous No.11534032]

>>11533771
>1/3=.333...
False. Decimals just can't do 1/3rd. Let's not make excuses for an imperfect number system.
>>11533755
That's a stupid question. 0.999... is shit that just exists on a piece of paper. Infinity is an abstract concept.

[Anonymous No.11534074]

>>11534027
Then the equation doesn't hold for p=inf and your argument fails.

[Anonymous No.11534077]

>>11534032
>False. Decimals just can't do 1/3rd.
This is not an argument. Do long division of 1/3 and you get 1 as a repeating remainder, which means 1/3 = 0.333...

>That's a stupid question. 0.999... is shit that just exists on a piece of paper.
Then why are you saying 0.999... approximates 1? An approximation has an error. You can't give the error is because it's not an approximation.

>Infinity is an abstract concept.
So is every other number.

[Anonymous No.11534091]

>>11534074
So you're saying that it's impossible to account for all elements of an infinite set?
Nothing short of setting p to ∞ has the capacity to account for all the elements.

But that also must mean it's also impossible to do infinite arithmetic. Like ramanujan's [math](1+2+3+4+...) = -\frac{1}{12}[/math]
Or even "infinite sums" become questionable.
What do you suppose [math]\sum_{n=1}^{\infty} [/math] means, then?

[Anonymous No.11534099]

>>11534091
>So you're saying that it's impossible to account for all elements of an infinite set?
No, where did you get that?

>Nothing short of setting p to ∞ has the capacity to account for all the elements.
What do you mean by account for? All elements are indexed by a natural number. No element is indexed by inf. Instead of hiding behind vague terminology, just make a logical proof.

>But that also must mean it's also impossible to do infinite arithmetic.
Why?

>Or even "infinite sums" become questionable.
Why?

>What do you suppose ∑∞n=1 means, then?
It means the limit of the partial sums as n approaches infinity.

[Anonymous No.11534129]

>>11534099
everything you sais was retarded lol. Anyway.

Okay, here's a trivial proof for well ordered sequential sets as [math]\mathbb{N}[/math] is.

G: [1,2,3]
H: [1,2,3,4,5]
J: [1,2,3,4,5,6,7]
G, H and J contain an element that is also used to describe their size.
G's size is 3. G contains an element equal to 3.
H's size is 5. H contains an element equal to 5.
J's size is 7. J also contains an element equal to 7.
Since [math]\mathbb{N} : [1,2,3,...] [/math] with a size equal to ∞, it holds that N contains an element equal to ∞.

If we constructed an unlimited amount of "V" sets via [math]\lim_{k\to\infty}\sum_{n=1}^{k} 1[/math]
Producing:
[math]V_1 : [1] \\ V_2 : [1,2] \\ V_3 : [1,2,3] \\ ... \\ V_{\infty} = [1,2,3,...] [/math]
Do you agree that [math]V_{\infty}[/math] and [math]\mathbb{N}[/math] are in-fact the same set?
Do you agree or disagree with [math]V_{\infty}[/math] existing?

[Anonymous No.11534139]

>>11534129
>Since N:[1,2,3,...]N:[1,2,3,...] with a size equal to ∞, it holds that N contains an element equal to ∞.
Wrong and doesn't follow from what precedes it. What is the successor of inf? If it has no successor, it's not a natural number by definition and is not in N.

>If we constructed an unlimited amount of "V" sets via limk∞∑kn=11
What does that mean? Does that mean every set V is a partial sum and corresponds to a natural number k? Then no set V can be equal to N. Or you could simply define a set V = N without all this useless obfuscation. Either way, you completely fail to prove anything.

[Anonymous No.11534153]

>>11534139
I'm just trying to figure out what you don't understand about the proof because you seem unable to raise any real objections beyond handwaving and saying "no" without explanation or counter-proof.

Gonna have to put you in the troll category. It's literally a 1 page proof too, it's not hard to follow.

[Anonymous No.11534155]

>>11534153
>I'm just trying to figure out what you don't understand about the proof because you seem unable to raise any real objections beyond handwaving and saying "no" without explanation or counter-proof.
Don't project your failings onto others. I have explained the error and you have failed to provide a proof. The burden of proof is on you.

[Anonymous No.11534167]

>>11534155
Your disagreement with P being allowed to be ∞ is not an explanation of my error, and all the post together must account for your judgement fallacy which you still haven't explained.
>>11534091
>>11534027
>>11533933
>>11533891
>>11533880
>>11533866
>>11533824
>>11533575
>>11533566
>>11533533
>>11534129
So if your issue with all this is really outside the scope of having retrieved an answer from these posts, your issue is not with what has been provided, but what you presume with your own pre-assumptions, which you haven't been able to explain.

Regardless if you're a troll or simply genuinely not knowledgeable enough of the topics, this ends here.

[Anonymous No.11534188]

>>11534167
>Your disagreement with P being allowed to be ∞ is not an explanation of my error
I already explained your error here >>11533782 and you have failed with every post to respond to it:

>Yes, but that doesn't imply all elements of S* added together is equal to an element of S. There is no last element of S*, so there is no element of S corresponding to the entire sum.

[Anonymous No.11534190]

>>11533999
Ive provided a clear proof here:
>>11532207
Where do you see circular reasoning?

[Anonymous No.11534232]

>>11534188
You presume my usage of the axiom of infinity is incorrect, and you fail to sway from that assumption. That's not my fault. You're just stubborn.

I think the only honest requirement to understand the proof, much less the rest of my posts, is to first go into it with the assumption that infinity does not exist and is yet to be defined. This is covered at the very beginning of the proof by then first defining what infinity is, so that it may be used as per that definition.

If you go into the experience without assuming or even knowing what infinity is, then you are instantly shown at the very beginning of the proof what infinity is and how it is defined.

You made several attempts to say infinity is not defined by the Axiom of Infinity, but I clearly showed it IN THE VERY LEAST through this post: >>11533866
>It's imperative to understand the meaning behind the invocation of ∞ here, and that this is the same ∞ provided within the scope of the axiom of infinity:
>>"guarantees the existence of at least one infinite set, namely a set containing the natural numbers."
>This is strictly understandable as
>>N:[1,2,3,4,5,...]
>and that the size of this set is ∞.
Which simply means you're wrong if you continue to resist. If you disagree about the size of N, the set of Natural numbers, being ∞, then you are simply wrong. Nothing but ∞ can describe the size of N.
Anything less than ∞ would obviously need to be a determinable finite number, which is no more or less retarded than claiming the size of N is 10,000 and that there are only 10,000 natural numbers.

I'm not sure why you believe the "Axiom of Infinity" contains no relevant information to define what infinity is, but it's morbidly negligent and retardedly dismissisive of you to insist it doesn't.

[Anonymous No.11534313]

>>11525618
Assuming that number is correct I would suggest: Going away from artificially complicating math by using obfuscating language, i.e. if you want to be understood, scientists should make an effort to break down their stuff as much as they can without losing information in the process. There will be a lower bound to this, but this will first and foremost destroy such pseudoscientific fields as economics.
Then have everyone do probabilistic math for a year and really ingrain this into people.

[Anonymous No.11534360]

>>11534027
If S is in a bijection with N, then either ∞ is in N or the infinite sum is not in S.
So which one is it?

[Anonymous No.11534372]

>>11534360
was already covered in the posts.
>>11534129
Just cause it may be novel does not mean it's invalid.

if you disagree then provide a proof.

[Anonymous No.11534390]

>>11534372
>was already covered in the posts
So which one is it? You can't have both.

[Anonymous No.11534397]

>>11534390
>You can't have both.
Ignore this

[Anonymous No.11534410]

>>11534390
read the linked post broski. It's like halfway through the post. Look for the [math]\mathbb{N}[/math] symbol.

May be interesting to add the note that calculable computer numbers treat infinity as a finite number around [math]1.7×10^{308}[/math]. Even wolframalpha can appear to run into a similar issue, but it tries it's best to print out an error instead.

Point is, this assumption of infinity being an arbitrarily large real number has been a deducible notion since cantor created the Axiom of Infinity.

[Anonymous No.11534470]

>>11534410
0.999... is in S
S is in bijection with N

therefore there exists some natural number "n" such that 0.999.. = S[n].

then S[n+1] > 0.999..., actually you have an unlimited amount of elements in S which are bigger than 0.999...

I don't think this is what you want bro....

[Anonymous No.11534504]

>>11534470
When n=∞, theres nothing left to do.
∞+1 is still just ∞.

Fact of the matter is, although Axiom of Infinity may not come out and explicitly define exactly what infinity is, it is instead structured in such a way that a variety of reasonable definitions can be extrapolated from it.
One of them is: "∞ is an arbitrarily large, indeterminable, but functionally finite number." This definition holds for the formula and the existence of 0.999... inside S. Although not entirely similar to the way a computer number might think [math]1.7×10^{308} = \infty[/math], which is merely a restriction of the way floating point numbers are stored in computer memory, and also that this 1.7blahblah number is in fact determinable. The datatype this number computes in is merely restricted to 1.7blahblah, but still someone had to make the decision for this number to return "infinity" rather than literally doing anything else, and that notion exists as evidence of the possibility to invoke ∞ within the set of Naturals, aka "it's he biggest finite number".

another definition can be, "∞ is the property that encompasses ALL elements of an Infinite Set". This definition can also hold true for the formula, by way of using ∞ as the limit in the sum, it essentially assumes ALL elements are accounted for. In terms of S*, this creates the 0.999... with ∞ decimals, and in terms of this number being a member of S, the index location of S[∞] remains valid in a feintly arbitrary way by extending the "index location" of this specific element to instead meaning that, since ∞ is a notion of "completeness" to account for ALL elements of an infinite set, S[∞] must mean less of a specific real number indexed location but instead that the location arbitrarily exists inside S as means of also saying that S is complete and all elements have been accounted for.

Ironically neither of these disagree with the notion that ∞ is endless. It's messy.

[Anonymous No.11534517]

>>11534504
"there exists a natural number n such that n + 1 = n"

just listen to yourself

[Anonymous No.11534533]

>>11534517
[math]\sum_{n=1}^{\infty}[/math] presumably doesn't stop summing, but if it did stop, it would have to stop at ∞.
Much like [math]\sum_{n=1}^{10}[/math] has to stop at 10.
Asking what the 11th iteration of the sum to 10 is, is meaningless and outside the scope of the equation.
Asking what the ∞+1 iteration of the sum to ∞ is, is also just as meaningless and beyond the scope.

There are two options to believe in here. Either infinite sums must continue endlessly and never produce a reasonable result,
or
infinite sums can, in the scope of imagination (which is where infinity also exists btw) "properly" reach ∞ and end summation.

One of these virtually invalidates calculus's usage of infinite sums.

[Anonymous No.11534551]

>>11534533
>infinite sums can, in the scope of imagination (which is where infinity also exists btw) "properly" reach ∞ and end summation.
that doesn't imply ∞ is a natural number

[Anonymous No.11534563]

>>11534129
{1} is a finite set
{1,2} is a finite set
{1,2,3} is a finite set
...
{1,2,3,...} = N is therefore a finite set

nice logic

[Anonymous No.11534565]

>>11534551
The reasonable implication of ∞ being a natural number was provided in >>11534129

Whether or not it is also has no bearing on the hostorical usage of infinite sums which use infinity as a limit, where otherwise only a natural number limit should exist.

[Anonymous No.11534591]

>>11534504
>∞+1 is still just ∞.
what is ∞ - 1

[Anonymous No.11534618]

>>11534591
An indeterminable finite number. A number assumed to be finite, but not having a determinable value means whothefuckknows. Ironically it shares that trait with ∞, so ∞-1 is also ∞.
Again, computer numbers treat this in the same case when you add up beyond [math]1.7×10^{308}[/math]. If the computer returns "infinity", no reasonable subtraction from that value will pull it back into normal real numbers. It just gets stuck at infinity.

tree(3) is also an indeterminable finite number. It's assumed to exist, but it's so big that it's actually unproveable to show every digit of it.
Theres probably a philisophical question to be answered in whether or not TREE(3) is as good as infinit, and probably a deeper question in whether or not there could be any functional use for such a gargantuan number as Tree(3), much less Infinity.

[Anonymous No.11534632]

>>11534618
is is true that whenever n < ∞, then n+1 < ∞ ?

[Anonymous No.11534672]

anyway, infinity is stupid.

[Anonymous No.11535170]

>>11534232
>You presume my usage of the axiom of infinity is incorrect, and you fail to sway from that assumption.
Go look up the axiom of infinity and compare it to what you said about it. You're being swayed is of no consequence since the words speak for themselves. Also, this isn't even relevant since your incorrect description is irrelevant to the argument. It's just filler. The important flaw is this:

>Yes, but that doesn't imply all elements of S* added together is equal to an element of S. There is no last element of S*, so there is no element of S corresponding to the entire sum.

Your continuous failure to respond to this indicates you know your argument is incorrect. Your post completely ignores this and tries to distract with an irrelevant point about the axiom of infinity. You fail again.

>I think the only honest requirement to understand the proof, much less the rest of my posts, is to first go into it with the assumption that infinity does not exist and is yet to be defined.
It is defined though. You can prove anything you want if you presume the ability to redefine anything you want, but that's simply semantics. If you want to prove something meaningful then use the definitions that already exist. But remember this doesn't matter since your "definition" is not relevant to the argument.

>You made several attempts to say infinity is not defined by the Axiom of Infinity, but I clearly showed it IN THE VERY LEAST through this post: >>11533866 #
That's not a definition. Saying a chihuahua exists and is a dog is not a definition of a dog. But remember this doesn't matter since your "definition" is not relevant to the argument.

>Which simply means you're wrong if you continue to resist. If you disagree about the size of N, the set of Natural numbers, being ∞, then you are simply wrong. Nothing but ∞ can describe the size of N.
I didn't disagree with that, I disagree with your assertion that this defines infinity.

[Anonymous No.11535187]

>>11534372
>Just cause it may be novel does not mean it's invalid.
It's invalid because it's not a logical argument.

4 > 1
4 > 2
...
4 > 4

By your logic I just proved 4 > 4

[Anonymous No.11535206]

>>11534410
>May be interesting to add the note that calculable computer numbers treat infinity as a finite number around 1.7×103081.7×10308.
So what?

>Point is, this assumption of infinity being an arbitrarily large real number has been a deducible notion since cantor created the Axiom of Infinity.
An approximation of necessity is not a deduction.

>>11534504
>When n=∞, theres nothing left to do.
∞+1 is still just ∞.
By definition that means inf is not natural.

>Fact of the matter is, although Axiom of Infinity may not come out and explicitly define exactly what infinity is, it is instead structured in such a way that a variety of reasonable definitions can be extrapolated from it.
>One of them is: "∞ is an arbitrarily large, indeterminable, but functionally finite number."
Please explain how you extrapolate this from the axiom of infinity.

>another definition can be, "∞ is the property that encompasses ALL elements of an Infinite Set".
What does encompasses mean? This is not a definition, it doesn't seem to have any specific meaning at all. Your just saying an infinite set is infinite, trivial.

>In terms of S*, this creates the 0.999... with ∞ decimals, and in terms of this number being a member of S, the index location of S[∞] remains valid in a feintly arbitrary way by extending the "index location" of this specific element to instead meaning that, since ∞ is a notion of "completeness" to account for ALL elements of an infinite set, S[∞] must mean less of a specific real number indexed location but instead that the location arbitrarily exists inside S as means of also saying that S is complete and all elements have been accounted for.
If you extend the set you're creating a new set and you're not talking about S.

[Anonymous No.11535217]

>>11534565
As has already been pointed out to you, that's not a valid implication. So stop citing it.

[Anonymous No.11535226]

>>11531766
>repeat for infinite steps
>terminates with 1
pick one

[Anonymous No.11535243]

>>11534618
Want to get your hot take on what 1-.999... is equal to then

[Anonymous No.11535507]

>>11525618
[math] \sum_{k = 1}^{\infty} \frac{1}{2^k} < 1 [/math] prove me wrong.

Observe:
1/2 < 1
1/2 + 1/4 = 3/4 <1
1/2 + 1/4 + 1/8 = 7/8 < 1
...
[math] \sum_{k = 1}^{\infty} \frac{1}{2^k} = \frac{2^\infty - 1}{2^\infty} < 1 [/math]

[Anonymous No.11535553]

>>11535507
[math]\frac{2^{\infty}-1}{2^{\infty}} = \frac{2^{\infty}}{2^{\infty}} - \frac{1}{2^{\infty}} = 1 - 0 = 1[/math]