/sci/ - Science & Math

[Anonymous No.11165643]

mathlet here, there's a matter I'm curious about. consider the interval [0, 1]. there are infinite fractional numbers within this interval. does that mean there's one fractional number in it for every number within the [-∞, +∞] interval? in other words, can one infinity "beat" another infinity?

[Anonymous No.11165671]

anyone?

[Anonymous No.11165673]

https://youtu.be/SrU9YDoXE88

[Anonymous No.11165683]

>>11165643
No such thing as [-∞, +∞], it's always (-∞, +∞).
But yes, check out cardinality and power sets. In a sense, there are infinities larger than other infinities. It's a Gödel-type situation though, this notion only works under the assumption that the axiom of choice is sensible, if I remember correctly.

[Anonymous No.11165698]

>>11165643
You just discovered zero's paradox... well, kind of. Infinite divisivility is weird.
I would say both those sets would be equivalent, yeah. BUT if you took the infinite divisions between the numbers in then infinite set then that infinity would be larger than the former just counting the whole numbers. Although conceptually I feel all infinite sets are equivalent but you're not allowed to say that outloud or people call you a brainlet.

[Anonymous No.11165709]

>>11165643
You're exactly right OP, well done. I'm on my phone, but search for countable and uncountable infinity, and for the diagonal argument.

[Anonymous No.11165765]

>>11165643
Hilbert's hotel and cardinality of sets. Key part of set theory.

[Anonymous No.11166121]

>>11165683
>No such thing as [-∞, +∞]
why the hell not

[Anonymous No.11166126]

>>11166121

infinity is never a closed set

[Anonymous No.11166131]

>>11166126
>infinity is never a closed set
this doesn't make any sense

[Anonymous No.11166132]

>>11166131

closed set is a set that contains all its limit points, what's the limit of infinity?

[Anonymous No.11166137]

>>11166132
the biggest number, obviously

[Anonymous No.11166140]

>>11166132
you have no idea what you're talking about, do you. [-∞, +∞] is (-∞, +∞) plus minimal and maximal element, endowed with the order topology. it's homeomorphic to [0,1]. you can call it a "two-point" compactification.

[Anonymous No.11166148]

>>11166126
explain infinite sets?

[Anonymous No.11166156]

>>11166140

except -infinity and infinity are not points

[Anonymous No.11166161]

>>11166156
sure they are. why do you think they're not

[Anonymous No.11166171]

>>11166161

what's infinity - infinity?

[Anonymous No.11166174]

>>11166171
undefined. what makes you think that points need to have well defined difference ?

[Anonymous No.11166179]

>>11166174

shouldn't a point be equal to itself?

[Anonymous No.11166189]

>>11166179
not all objects in mathematics can be added or subtracted, that doesn't mean they're not "equal to themselves"

[Anonymous No.11166289]

>>11166189
If we're talking real numbers here, infinity is not a real number. Hyperreals, okay, but that's definitely not what OP was asking.

[Anonymous No.11166296]

>>11166289
>If we're talking real numbers here, infinity is not a real number.
never said it was

[Anonymous No.11166305]

>>11166296
See
>>11166137
That doesn't work. Either you incorrectly assumed infinity being a real number, where there's a concept of "biggest number", or you didn't, but then also gave up the notion of "biggest number" and still made no sense with that post.
I'm not the other guy you argued with btw.

[Anonymous No.11166326]

>>11166305
The extended real line is a perfectly satisfactory topological space. It's just the two point compactification of the reals.

[Anonymous No.11166332]

>>11166326
A circle has no biggest number.

[Anonymous No.11166462]

>>11166332
That would be the one point compactification of the reals, which also exists and is indeed homeomorphic to S_1 though does not have a meaningful way of extending the order relation. We are talking about the two point compactification, which can easily have the order relation extended by defining -∞ <= x <= ∞ for all x.

[Anonymous No.11166529]

>>11166462
https://en.m.wikipedia.org/wiki/Extended_real_number_line
>[...] adding two elements: +∞ and −∞ (read as positive infinity and negative infinity respectively). These new elements are not real numbers.
>not real numbers

[Anonymous No.11166535]

>>11166529
Of course they're not real numbers, you fucking retard. Nobody here claimed they are. They're still numbers in a perfectly valid topology. Please don't tell me you think "non-real" means "fake".

[Anonymous No.11166555]

>>11166535
Ah yeah, forgot we were over that you completely disregard the context OP gave.
Here:
>With these definitions R ¯ {\displaystyle {\overline {\mathbb {R} }}}  is not even a semigroup, let alone a group, a ring or a field, like R {\displaystyle \mathbb {R} }  is one

[Anonymous No.11166563]

>>11166555
We don't need a field structure to define a homeomorphism from the extended reals to [0,1], you utter brainlet. Again, nobody claimed the extended reals are a field. Just admit you don't know undergrad-level topology.

[Anonymous No.11166769]

>>11166563
>can one infinity beat another infinity?
This is clearly a cardinality question, which topology cannot answer by itself.
No need to be insecure about not being able to extract context. Many undergrads struggle with that.

[Anonymous No.11166773]

irrational numbers don't exist
you'll never figure out the primes with your infinity

[Anonymous No.11166774]

>>11166769
Homeomorphism is a stronger condition than bijection, shitposter kun

[Anonymous No.11166780]

Vi Hart answers OPs question here: https://www.youtube.com/watch?v=23I5GS4JiDg and gives Cantor's Diagonal Proof here: https://www.youtube.com/watch?v=lA6hE7NFIK0

Does /sci/ like Vi Hart?

[Anonymous No.11166795]

>>11166774
No. It only shows there are "at least as many infinite numbers in one set as there are in the other".

[Anonymous No.11166848]

>>11165709
This guy is on the right track. There is at least countable and uncountable infinity, and Cantor's diagonal argument is among the simplest ways of proving the difference between the two.

>>11165643
To partially answer your question, OP, consider the set of all possible fractions of integer numbers, that is, the set of rationals [math] /mathbb{Q} [/math] . The set of rationals is a countably infinite set, as evidenced by the following algorithm:

Imagine the subset of all ordered pairs of natural numbers in the plane [math] \mathbb{R}^2 [/math] , i.e. the set [math] \mathbb{N}^2 [/math] . Imagine that you start at the (1,1) point and work your way "towards infinity," counting each unique point in [math] \mathbb{N}^2 [/math] with the natural numbers [math] \mathbb{N} [/math] themselves. Clearly, by moving in diagonal lines and then shifting to the next diagonal once the boundaries of this quadrant of the real plane are reached, it must be possible to sequentially number every point in [math] \mathbb{N}^2 [/math] . Now, consider the x coordinate of this plane to be the numerator of a fraction, and y to be the denominator of that fraction. Clearly, the set [math] \mathbb{N}^2 [/math] contains corresponds 1-to-1 with the set of fractions with integer numerators and demoninators, which is precisely the set of positive rational numbers, and clearly, every such fraction can be numbered distinctly using the set of natural numbers. Therefore, the set of rational numbers is countable.

[Anonymous No.11167047]

>>11166305
>>11166332
>>11166529
>>11166555
I'm the one who started this argument, you've been talking to someone else. all I've said is that the statement
>No such thing as [-∞, +∞], it's always (-∞, +∞).
is bullshit because [-∞, +∞] is a completely valid mathematical object. it's literally just R with two artificially added element: the smallest one and the biggest one, an operation which you can apply to any ordered set. if you think that this somehow implies "there is biggest number" or "infinity is real number", you're a fucking undergrad brainlet who has never seen real math. also fuck you.

[Anonymous No.11167233]

>>11166121
It does exist, unironically look at like the first page of Tookers RH proof, he mentions it and it's not crankery (everything else is though). It's the part about the "extended real numbers"

However, typically when we talk about an interval, we're talking about the normal real numbers. [math] [a,b] \subset R [/math] and since [math] \pm \infty \notin R; \hspace{0.05 in} \pm \infty \notin [a,b] [/math]

[Anonymous No.11167635]

>>11167047
>including an element as the upper and lower limit in an ordered set isn't saying they assume the order
You're not only an undergrad if you think adding elements to an ordered set doesn't submit them to the order relation, you're also a fool. R with two arbitrary elemtns added isn't R anymore.

[Anonymous No.11167819]

>>11167635
>You're not only an undergrad if you think adding elements to an ordered set doesn't submit them to the order relation
where did I say that adding elements to an ordered set doesn't submit them to the order relation ?
>R with two arbitrary elemtns added isn't R anymore.
where did I say that R with two arbitrary elements added is R ?

[Anonymous No.11168227]

>>11167819
I'm saying you deliberately ignore OPs context so you can seem to win this argument, which you are not.

[Anonymous No.11168621]

>>11168227
I never cared about OP's context. you're making me repeat myself, all I've said is that the statement
>No such thing as [-∞, +∞], it's always (-∞, +∞).
is bullshit because [-∞, +∞] is a completely valid mathematical object.

[Anonymous No.11168624]

>open thread
>infinite set autism

[Anonymous No.11168632]

>>11166795
t. guy who doesn't know what a homeomorphism is

[Anonymous No.11169470]

>>11168621
lol, is it so difficult to quote the passage you have problems with instead of autistically screeching?
Yeah, obviously it's not correct in general, but it is true in OPs case.

>>11168632
No u. Find a homeomorphism from a power set to the initial set.

[Anonymous No.11169500]

>>11169470
>is it so difficult to quote the passage you have problems with instead of autistically screeching?
I have literally done this two times already

[Anonymous No.11170235]

>>11169500
No, you think you did, but didn't. While I did like three times.
>in OPs context

[Anonymous No.11170299]

>>11166148
They don't end.