Apparently I've been accused of playing 'language games'

[Anonymous No.22664468]

I have a serious problem. The Council of Mathematicians has kicked me out of /sci/ because I've been caught playing 'language games' with Tarski and Witt. As a result, my work is no longer mathematical. It's been sold off as scrap to the philosophers. That's why I'm here, to auction off the pieces of my theory to the highest philosophical bidder.
Item 1: Gödel's First Incompleteness Theorem.
This fine piece was hand crafted by Gödel himself, and represents a fine tradition rivaled only by Swiss watchmakers. Please feast your eye on this beauty. Isn't she lovely? Can I get a revolution in epistemology for her? Yes. Thank you. One revolution. Two revolutions in epistemology? One revolution in epistemology and one revolution in ontology? yes. One revolution in epistemology and one in ontology. Two metaphysical revolutions total.

[Anonymous No.22664483]

Has logic really ever done anything besides give us more philosophical problems

[Anonymous No.22664486]

>>22664483
the answer to your question is yes or no

[Anonymous No.22664502]

>>22664486
Sounds like another problem to me

[Anonymous No.22664510]

>>22664483
Gave us computers and the internet godels theorem is literally the first computer program. You can use the power of logic to watch unlimited free porn!

[Anonymous No.22664514]

>>22664483
Did Alice raise any philosophical problems?

[Anonymous No.22665610]

>>22664483
it proved the existence of god, for one

[Anonymous No.22665653]

[Anonymous No.22665675]

>>22664468
It's sad to see rationalists still trying to pass their rationalism as non delusion.

[Anonymous No.22665683]

>>22665675
https://girard.perso.math.cnrs.fr/LLcup.pdf

[Anonymous No.22665694]

>>22665683
Since Plato rationalists have been clinging to their fantasy that somehow rationality is a the best tool to talk about truth and reality, whereas the only result they get is ... well absolutely nothing. Those morons still haven't produce any truth, any claim about reality which is not contingent on whatever era and conventions the rationalists have been living in, and even worse they still haven't even found a method to discriminate between delusion and truthful claims.

[Anonymous No.22665819]

>>22665610
Why is law of non-contradiction true?

[Anonymous No.22665824]

Gödel's theorems are such a reliable midwit filter. Whenever a thread about this topic pops up, 99% of the posters replying didn't understand the statements.

[Anonymous No.22665961]

>>22665824
the funniest thing is none of these midwits know that absolutely no mathematicians (except possibly for logicians, but i'm pretty sure neither do they) care about these theorems and just go on happily doing mathematics

[Anonymous No.22665971]

>>22665961
Gödel becomes important in the discussion of consciousness. His theorems are evidence in favor of substance dualism. An AI will always be limited by Gödel while a conscious human being has the freedom of choice and creativity to escape these limitations.

[Anonymous No.22665985]

>>22665971
Lol. Two posts down from
>Whenever a thread about this topic pops up, 99% of the posters replying didn't understand the statements.
you get
>An AI will always be limited by Gödel while a conscious human being has the freedom of choice and creativity to escape these limitations.
So prove to me whether the Continuum Hypothesis is true or false. Or are you an AI?

[Anonymous No.22665997]

>>22665961

Godels theorem is important for both math and philosophy and if you aren’t aware of the academic environment of his time it might be lost on you how massive they were.

A vet could keep on fixing broken cheetah bones or whatever and not give a fuck about Darwin but it doesn’t make either’s work less important

[Anonymous No.22666008]

>>22665985
Your deterministic reply was predictable, NPC.

[Anonymous No.22666009]

>>22665985
Not that anon.

> So prove to me whether the Continuum Hypothesis is true or false. Or are you an AI?
The continuum hypothesis rests on the false assumption that there are different sizes of infinity, which follows from the delusion of undefinable numbers. An undefinable number is one which can not be represented by a finite definition. Every number that we interact with is definable, whether it be rational or irrational, and we can use approximations in the real world to solve problems. But undefinable numbers are completely useless. That which we cannot interact with does not exist by definition. But Cantor’s diagonal argument produces a diagonal argument which is undefinable, whereas if we work with actual real numbers, then the list is composed purely of definable numbers. So the diagonal number isn’t actually a real number, and the notion of larger infinities is complete nonsense.

[Anonymous No.22666012]

>>22666008
Use your "conscious human being has freedom of choice and creativity to escape these limitations" to prove or disprove the Continuum Hypothesis. Do it. Unless you're an AI

[Anonymous No.22666014]

>>22665985
its not that the tool is necessarily inconsistent, its that its use is nonsensical

[Anonymous No.22666019]

>>22666009
>doubles down on Godel retardation with Cantor crankery
Lol. /lit/ never changes.

[Anonymous No.22666024]

why are we all using language? its clear that it binds us conceptually, so then why use it?

[Anonymous No.22666033]

>>22666019
>crankery
Not an argument. The belief in undefinable numbers will never lead to any useful result in the real world. Believing in something which is undefinable is actual crankery, yet you defend it like a religious man. How ironic.

[Anonymous No.22666038]

>>22666024
society is made by normies for normies and women and those people hate life wherein there is not much talking (mostly because, precisely , languages are the best tools to circle jerk during the day and doing armchair philosophy, before joining the weekly orgy where suddenly none of the intellectuals care about truth, reality and ethics, weird huh?)

[Anonymous No.22666043]

>>22666012
It's funny how you don't even recognize the flaw in your question. Your reasoning truly is limited by Gödel, just as predicted.

[Anonymous No.22666045]

>>22666024
language is useful. But people use it in such ways that it becomes useless

[Anonymous No.22666047]

>>22666033
Your argument doesn't even work if you accept your bullshit about undefinable numbers
>whereas if we work with actual real numbers, then the list is composed purely of definable numbers
So just apply the diagonal argument to your new list of real numbers you find acceptable. Producing the diagonal is clearly has a definition and it would still not be on the list by Cantor's argument

[Anonymous No.22666051]

>>22666038
well but my point is sort of self defeating. Its a cynical remark on how limited our understanding could be just because language is obviously conceptually limited, note that this whole thread is in language.
>>22666045
well it has uses, but its usefulness is only shown in boundaries which it itself formalized

[Anonymous No.22666063]

>>22666047
Yeah except the diagonal number is NOT definable, since it cannot be UNIQUELY REPRESENTED by a FINITE definition. It’s somewhat trivial to create a countable list of real numbers by ordering the numbers in increasing definition size. But even then, you would have repeated numbers throughout the whole list. It would take an infinite time to go through and whittle the list down until it has no repeats, but of course there would be no easily noticeable pattern of the reals at that point, and you could not predict, say, the 1000th digit of the diagonal number. So at no point would the diagonal number actually be definable. But the real numbers themselves are still countable because they can be represented by finite definitions. I’m simply saying that the undefinable number does not exist at all and is completely useless to us.

And that is precisely why the Continuum hypothesis will never be solved. Because it is useless.

[Anonymous No.22666070]

>>22666063
>Yeah except the diagonal number is NOT definable
It clearly is definable. You take the nth decimal position of the nth number in the list. If your list is defined then the diagonal is. What makes Cantor's actual diagonal undefined is that his list includes undefined numbers so that the diagonal has to take the nth decimal of an undefined numbers. But your new diagonal would be working on a list without undefined numbers so that the diagonal would be defined. Even when you accept your crank shit about undefinable numbers being bad you still fail to show that Cantor's diagonal doesn't work.

[Anonymous No.22666088]

>>22666070
Ok, so make a list of definable reals right now, then give me the 10th, 100th, and 1000th digits of the diagonal number. I will then ask for the nth digit where n is a random number between 100 and 1,000. If you respond before you die, I will admit defeat. Note that your definition of the list must allow for ANYONE to find the nth digit of the diagonal number in finite time, otherwise it’s not actually definable.

[Anonymous No.22666094]

>>22666088
>Ok, so make a list of definable reals right now, then give me the 10th, 100th, and 1000th digits of the diagonal number.
Why are you asking me lol. This is you in the same post I was responding too
> It’s somewhat trivial to create a countable list of real numbers by ordering the numbers in increasing definition size
Your dumbass can't even remember what you wrote one post ago.

[Anonymous No.22666119]

>>22665961
genuinely not the case. mathematical logicians deal with completeness and incompleteness all the time, from proof theory to model theory.

[Anonymous No.22666120]

>>22666038
Interesting. What would life be like without language though? Wouldn’t we be even more primitive? I’m interested in what sort of world arrangement you propose.

[Anonymous No.22666121]

>>22666009
good post. have you read much feferman and/or weyl?

[Anonymous No.22666125]

>>22666094
Well even though we can say “let the definitions of the numbers increase in size,” it’s hard to actually define how this increase happens. When you reach the point at which your definitions include 1,000 symbols for example, we may start with all numbers with 1,000 digits, but then what? In what order shall we proceed? All sums? Then all products? But how shall we order the numbers in the definitions themselves? Any finite expression within mathematics that represents a number must be accounted for, and though it’s easy to say that we can arrange these definitions in increasing order, it’s virtually impossible to actually define this arrangement such that every person could re-create the exact same list with that definition. And so the list itself would not be definable, and neither would the diagonal number, even though every number in the list is definable.

You see, even if we allowed all definitions in English, then we would run into Richard’s paradox, which is bullshit, because then the diagonal number would both be in the list and not in the list, because IT IS DEFINED TO BE DIFFERENT THAN ITSELF, hence the paradox (contradiction). So either the number is in the list, because it is a contradiction, or it is not in the list, because it’s not a number at all.

[Anonymous No.22666131]

>>22664483
>>22664486
Every day I browse this forum I realize I'm suffering in a prison I can leave at any time but willfully remain. I hate this place.

[Anonymous No.22666132]

>>22666121
no, i only watched a few videos of Wildberger and did the rest of the thinking for myself. I do not necessarily agree with everything Wildberger has said, by the way.

[Anonymous No.22666138]

>>22666125
>Well even though we can say “let the definitions of the numbers increase in size,” it’s hard to actually define how this increase happens.
Hahaha. You realize the whole point of Cantor's argument is to show that the list can't exist right? Which means the real numbers are a larger infinity than the naturals. And when confronted with another stupid consequence of your reasoning you pivot to saying the list can't exist WHICH JUST PROVES CANTOR RIGHT. If you cannot find a one-to-one onto function(aka a list) from the naturals to the reals then the reals are a larger infinity.

[Anonymous No.22666152]

>>22666132
if you are interested in this kind of thing i highly recommend solomon feferman's writing. he makes very formal the notion of the continuum hypothesis not being a "definite" mathematical (or even logical) problem, with a very clear writing style devoid of any mysticism. he also argues that cantorian infinity in mathematics is not even applicable to the natural sciences, i.e. it's enirely dispensable

[Anonymous No.22666163]

>>22666138
There’s absolutely nothing wrong with a list of definitions of real numbers with increasing size of the definitions. We just don’t know how to define how the definitions are ordered. Nevertheless, every definable real number would be included in the list, and it would actually appear infinite times, as every number can be represented in infinitely different ways. So the definable numbers ARE countable, and uncountability is the result of undefinability. This is true regardless of how you find uncountability. It is ALWAYS the result of undefinability, which is useless and impossible to interact with, and so we may as well say it does not exist, and study problems that actually matter.

[Anonymous No.22666165]

>>22666138
>If you cannot find a one-to-one onto function(aka a list) from the naturals to the reals then the reals are a larger infinity
Nta but I think he’s arguing that this problem is not well-posed, rather than the non-existence of such a function constituting proof of the hypothesis.

[Anonymous No.22666177]

>>22666163
>There’s absolutely nothing wrong with a list of definitions of real numbers with increasing size of the definitions. We just don’t know how to define how the definitions are ordered.
You fucking moron you just said they were listed in increasing size of definitions. But ignoring that here you're saying you have a list of the definable real numbers. So just apply the diagonal argument which would result in a definable number since none of the numbers in the list are undefinable. The result would be a number not on your list thus proving the list isn't exhaustive of the real numbers.
>So the definable numbers ARE countable
Which means there is a one-to-one onto function from the naturals to the reals. This is also called a list.

[Anonymous No.22666182]

>>22666165
>Nta but I think he’s arguing that this problem is not well-posed
If he was saying that he wouldn't have said this earlier
>The continuum hypothesis rests on the false assumption that there are different sizes of infinity
He's a run of the mill Cantor crank that is deeply confused about all aspects of the proof and logic in general.

[Anonymous No.22666187]

>>22666177
> The result would be a number not on your list thus proving the list isn't exhaustive of the real numbers.
But that’s impossible, since every finite representation of a real number is in the list. EVERY DEFINABLE NUMBER IS IN THE LIST. If its definition has 334,781,980, it is STILL IN THE LIST. The only numbers that are NOT in the list are undefinable numbers. So if the diagonal number is not in the list, then it is undefinable.
> Which means there is a one-to-one onto function from the naturals to the reals. This is also called a list.
Sure, we just don’t know what that function is. But even if you used it and tried to create the diagonal number, the diagonal number would still be undefined from the logic that I just gave. It would be defined in terms of English but it wouldn’t be able to be defined by pure math symbols, and it would be completely useless in the real world, and will never help us solve anything. And that’s supposing that you had the infinite mind to comprehend the one-to-one function in the first place.

Here’s a challenge for you: show that the prime numbers are countable :)

[Anonymous No.22666197]

>>22666187
>The only numbers that are NOT in the list are undefinable numbers. So if the diagonal number is not in the list, then it is undefinable.
The diagonal number is easily defined like I did here
>You take the nth decimal position of the nth number in the list.
and since it's not on the supposedly exhaustive list you've found the contradiction. Congratulations the assumption that the list of definable real numbers exists must be wrong.
>Sure, we just don’t know what that function is
The diagonal argument works on any function.

[Anonymous No.22666204]

>>22665997
i am well aware how important Godel's work was at the time (Entscheidungsproblem) and i am not attempting to diminish its overall importance, what i mean is 99% of working mathematicians don't care about it and don't interact with it in any way in their daily work
>>22666119
i specifically mentioned logicians as a caveat as i am not familiar with the kind of work they do. and i was not suggesting completeness is a meaningless concept, i was specifically talking about Godel's results (however, for all i know they might be important in modern work)

[Anonymous No.22666207]

>>22666197
If the diagonal number could be represented by finite math symbols, then it is trivially in the list. It follows that it is not definable.

[Anonymous No.22666210]

>>22666207
>If the diagonal number could be represented by finite math symbols, then it is trivially in the list. It follows that it is not definable.
Or it follows that the assumed list doesn't exist. And since I already gave a trivial definition of the diagonal guess what is left. You're wrong

[Anonymous No.22666218]

>>22666210
> And since I already gave a trivial definition of the diagonal guess what is left
But you haven’t. It’s not definable until you can represent it in finite terms and it only represents ONE number. You should be able to write it on a piece of paper, send it to thousands of mathematicians, and they should all be able to reproduce the exact same number. That’s what definability means. This is what you do not understand.

[Anonymous No.22666225]

>>22666210
> Or it follows that the assumed list doesn't exist
why can’t a list of valid math expressions which increase in size “exist” ? For every definition with n symbols, there is a limit to how many definitions you can make with n symbols, so the list necessarily exhausts all finite math expressions which represent real numbers, which would be all definable numbers.

[Anonymous No.22666227]

>>22666218
>But you haven’t. It’s not definable until you can represent it in finite terms
Because you haven't given a specific list to apply the diagonal argument to yet. Hint: you can't
>and it only represents ONE number
It's trivial to show that the diagonal argument results in a unique real number when applied,

[Anonymous No.22666239]

>>22666227
> Because you haven't given a specific list to apply the diagonal argument to yet. Hint: you can't
Just because I can’t give the exact definition of the list doesn’t mean that the definable numbers aren’t countable. The list is undefinable and so is the diagonal number. The reals are countable.

[Anonymous No.22666241]

>>22666225
>why can’t a list of valid math expressions which increase in size “exist” ?
Besides Cantor's diagonal being applied to that same list? The more informative explanation gets into the weeds with "definabilty" being a second-order quantifier over the real numbers but to be honest my grasp on that is limited.

[Anonymous No.22666244]

>>22666239
>Just because I can’t give the exact definition of the list doesn’t mean that the definable numbers aren’t countable.
No it's the fact that Cantor's argument applied to any list results in a diagonal not on that list.

[Anonymous No.22666255]

>>22664514
>Still she haunts me phantom-wise
>Alice moving under skies
>Never seen by waking eyes..
Not sure. She did however raise her dress, at least in LC's dreams..

[Anonymous No.22666262]

>>22666241
>>22666244
We’re working with a list of math symbols, not all English definitions of numbers. The diagonal number, even if you could definite it properly, is defined in English. So it does not contradict the list. If it could be represented by, say, 3,000 math symbols arranged in a particular order, then all you would have to do is go to the part of the list that contains 3,000 symbols in the definitions, and if the diagonal number’s definition were valid, then it would be in the list. There is a finite amount of permutations you can make with n symbols, so it would necessarily be in the list. This proves that the diagonal number is not definable by math symbols.

[Anonymous No.22666267]

>>22666262
>This proves that the diagonal number is not definable by math symbols.
Or it proves that your ASSUMED list is not definable in a finite number of math symbols. Which it isn't and the contradiction is found

[Anonymous No.22666268]

>>22665653
Huh?

[Anonymous No.22666278]

>>22666267
ruh roh raggy!

[Anonymous No.22666283]

>>22666278
Read more closely. It never says the definable numbers are a countable set. There is no one-to-one onto function from the naturals to the definable numbers. Feel free to provide it if you think you have one. Be warned I'm just going to use Cantor's diagonal on it

[Anonymous No.22666297]

>>22666283
THE DIAGONAL ARGUMENT DEFINES THE NUMBER IN ENGLISH, NOT MATH SYMBOLS, YOU STUPID FUCK. IF YOU HAD TO REPRESENT IT IN MATH SYMBOLS, IT WOULD NECESSARILY BE INFINITE SINCE YOU WOULD HAVE TO REFERENCE EVERY SINGLE NUMBER IN THE LIST, SO ITS DEFINITION WOULD BE INFINITE AND NOT FINITE, WHICH MEANS IT IS NOT DEFINABLE

[Anonymous No.22666303]

>>22666297
>IF YOU HAD TO REPRESENT IT IN MATH SYMBOLS, IT WOULD NECESSARILY BE INFINITE SINCE YOU WOULD HAVE TO REFERENCE EVERY SINGLE NUMBER IN THE LIST, SO ITS DEFINITION WOULD BE INFINITE AND NOT FINITE, WHICH MEANS IT IS NOT DEFINABLE
Lol how do you think the real numbers are defined? If you use the Cauchy sequence definition the reals are infinite sequences defined at every point in the sequence that obey a convergence criteria. Referencing an infinite number of things is not what makes something undefinable. It's like saying odd numbers are undefinable since you would have to reference every number to see if it was not divisible by 2.

[Anonymous No.22666312]

>>22666303
Because the only way to represent the diagonal number in math is by showing that it is different than the numbers in the list, one by one. So you would have an infinite sum with highly complicated math to use every number in the list, calculate the nth digit of that number, and then change it so that it is another digit. So the diagonal number, if represented by math symbols, would necessarily be infinite, since its definition would include every single definition of the real numbers. So it is undefinable.

[Anonymous No.22666318]

>>22666312
>So you would have an infinite sum with highly complicated math to use every number in the list, calculate the nth digit of that number, and then change it so that it is another digit
You would have provided the one-to-one function that defines the list. All the diagonal would have to do is take nth decimal position of the nth number. So a very finite definition(unless your provided function doesn't exist.)
>So the diagonal number, if represented by math symbols, would necessarily be infinite
No it wouldn't

[Anonymous No.22666334]

>>22666318
You’re conflating the English definition and the representation of the number in pure math symbols. You could not represent the diagonal number in pure math symbols in finite terms, though it could be infinite. And the reason it could be infinite is because its definition is infinite, so it would only reference finite definitions of numbers, so there would be no contradiction.

[Anonymous No.22666352]

>>22666334
The only undefinable part of this is your claimed bijective function from the naturals to the definable numbers. This isn't the reason it's undefinable but lets apply to your own reasoning to this function(or any function). Since all functions reference an infinite amount numbers are the definitions of all all functions infinite? Do you believe functions even exist?

[Anonymous No.22666379]

>>22666352
It doesn’t matter if the objective function “exists” or not. The definable reals are still countable, and the diagonal number is necessarily undefinable (in math symbols). You can expand definability to the English language, but English isn’t formal so it’s hard to define definability. But supposing that you could have some computer program that checks for definability on English expressions, you would still run into Richard’s paradox, which shows that you can’t just say “let there be a number that’s different than all numbers defined in English ,” because then it would actually be referencing itself, and therefore defining itself to be different than itself. But you can use the diagonal argument on the definable reals because your diagonal number isn’t actually definable (in math), it’s definable (in English). And that’s assuming that we know the function in the first place. I’m not actually sure if it’s “possible” to define that list in English, I’ve never tried. But even if you could, the real numbers would still be definable and the Continnum hypothesis would be a waste of time and intellect.

[Anonymous No.22666391]

>>22666379
>It doesn’t matter if the objective function “exists” or not
Lol yes it does. The whole point of Cantor's argument is that it doesn't. If you given up claiming a bijection from the naturals to the definable reals exists you've lost even after I gave you the benefit of accepting your definable numbers only bullshit.
>The definable reals are still countable
THEN GIVE THE FUNCTION

[Anonymous No.22666398]

>>22664468
how were you kicked off of an anonymous image board just larp as someone else lmao

[Anonymous No.22666423]

>>22666391
You can’t give the function for the prime numbers. But you can assume that it exists. So let’s assume that the function between the naturals and (definable) reals exists. Now you, the retard, will say “then I will just use my heckin Cantor’s argument.” Which produces a number not in the list. Now you, the retard, says “therefore the list is impossible.” But the list is possible and the diagonal number doesn’t have to be in the list. I don’t agree that the diagonal number was in the list in the first place, because I don’t recognize it as a definable real number. Then you, the retard, will say “but I clearly defined it,” then I will say, “but when I say definability, I refer to formally defined finite expressions of math symbols, not English definitions.” The diagonal number is not in the list because it is undefinable. Even the experts admit that definable numbers are “at most countable.” It follows that different sizes of infinity comes from undefinability. But undefinability is useless bullshit and the fact that you defend it so much proves that you are the only crank here.

[Anonymous No.22666430]

>>22666204
>i specifically mentioned logicians as a caveat as i am not familiar with the kind of work they do. and i was not suggesting completeness is a meaningless concept, i was specifically talking about Godel's results (however, for all i know they might be important in modern work)
you are making a valid point, but to stress how important the result really is and the indirect impact it does have on the rest of mathematics, just try to picture how different modern mathematics would be if godel's theorem didn't hold. mathematical logic, at least, would be a completely different ball game.

[Anonymous No.22666440]

>>22666268

[Anonymous No.22666452]

>>22666423
>You can’t give the function for the prime numbers.
Huh? This is trivial just apply whatever prime checking function you want to the naturals until you you accumulate enough primes to get to the number that is entered into the function.
>But the list is possible
No it's not. The diagonal argument proves it isn't
>The diagonal number is not in the list because it is undefinable.
It is easily definable. I've done it several times. You have yet to provide the definition of your bijective function from the naturals to the definable reals.
>Even the experts admit that definable numbers are “at most countable.”
But they don't say they're a countable set. Almost like they know the diagonal argument could be applied to such a set and it's bijection.

[Anonymous No.22666470]

>>22664468
it's a shame kurt seemingly did not pay any attention to the categorical developments that were happening in the latter part of his life. wonder how CT would have impacted his world view vis a vis set theoretic platonism.

[Anonymous No.22666477]

>>22666452
> No it's not. The diagonal argument proves it isn't
Create computer program that checks for valid math expressions. Then make it go through all permutations of math symbols to check for all valid definitions. Take the process to infinity. Why would it not be able to produce such a list? Every definable number would be in the list.
> It is easily definable. I've done it several times.
LMAO, I love how I predicted this exact response, and you ignored my refutation of this exact argument:
> then I will say, “but when I say definability, I refer to formally defined finite expressions of math symbols, not English definitions.”

> But they don't say they're a countable set.
Then what the FUCK does it mean to say they are countable?
> Almost like they know the diagonal argument could be applied to such a set and it's bijection.
The diagonal argument traditionally assumed that the list includes undefinable numbers in the first place. It cannot be applied to the definable reals in the same fashion.

[Anonymous No.22666490]

>>22666477
>Create computer program that checks for valid math expressions
Again why are you asking me? You're the one saying such a thing exists. I DO NOT BELIEVE A FUNCTION OR COMPUTER PROGRAM FOR DEFINABLE NUMBERS EXISTS. If you have one put it forward and achieve mathematical glory.

>Then what the FUCK does it mean to say they are countable?
They don't say they're countable in the image you linked.
>Because formal languages can have only countably many formulas, every notion of definable numbers has at most countably many definable real numbers.

>The diagonal argument traditionally assumed that the list includes undefinable numbers in the first place
It still works on your list of definable reals like I've shown repeatedly.

[Anonymous No.22666496]

>>22666490
yeah, you’re just retarded. I can’t fix that. Let me know when the undefinable numbers will solve any problem in the real world.

[Anonymous No.22666504]

>>22666496
Let me know when Cantor cranks are given their due in the math hall of fame. Keep fighting the good fight lol. Godel and Cantor bring out the schizos everytime, you even managed to tie the two together.

[Anonymous No.22666510]

>>22666504
>being regarded by other mathematicians is more important than actually solving problems
lol thanks for making it obvious for everyone to see

[Anonymous No.22666521]

I am neither one of these people but FYI the idea that non-definable numbers (i.e., numbers which cannot be arbitrarily approximated by some effective/computable procedure) don't exist isn't actually that controversial within philosophy of mathematics academia. plenty of working logicians share this view. for instance, per martin lof (i don't know whether this is genuinely his view but it is heavily implied in the essays of his that i've read)
indeed modern mathematics has whole heartedly embraced cantorian set theory, but some research suggests that plenty of modern mathematics can be done without it - in this sense, it is argued that cantorian higher infinites may be used as a short cut, but are not things that have a fundamental existence like many would contend the natural numbers do

[Anonymous No.22666544]

>>22666430
>just try to picture how different modern mathematics would be if godel's theorem didn't hold
i don't need to picture anything, my point is precisely what you said: that these are historically very important results that shaped 20th century mathematics in certain ways, however almost no working mathematician cares about them today. a comparison would be something like the theory of quadratic forms -- it gave birth to algebraic number theory and class field theory, but no number theorist today particularly concerns himself with it

[Anonymous No.22666549]

>>22666521
It’s not a matter of debate. Until they are shown to be useful, they do not exist. It’s like putting magical space fairies in physics theories to account for things we can’t explain. You can’t use that theory to make any predictions or solve any problems. It’s useless. Cantor and his zealots are way behind the curve. Mathematicians are wasting their time and their intellect. This is a big problem and should be resolved.

[Anonymous No.22666556]

>>22666549
lollol, if they can't exist then what are you talking about?
if they can exist then how do you know they don't?

[Anonymous No.22666559]

>>22666556
> lollol, if they can't exist then what are you talking about?
this guy believes in magical space fairies! lmao!

[Anonymous No.22666563]

>>22666549
also its not like physics for there are standards which disallow "useless" concepts

[Anonymous No.22666567]

>>22666521
>I am neither one of these people but FYI the idea that non-definable numbers (i.e., numbers which cannot be arbitrarily approximated by some effective/computable procedure) don't exist isn't actually that controversial within philosophy of mathematics academia
He was just using undefinable numbers to try to attack Cantor. I went on to show that Cantor's argument still applies to the definable reals. Since you claim to know something about this you know how problematic his claim of a set of definable reals is. Definability is a second-order concept
>but some research suggests that plenty of modern mathematics can be done without it - in this sense, it is argued that cantorian higher infinites may be used as a short cut, but are not things that have a fundamental existence like many would contend the natural numbers do
I don't even believe the natural numbers exist. Whenever some shows me two things they're really just showing me one thing and another one thing. The point is that Cantor's infinities are consistent under the current foundations of math. If you want to go and make up your own based on what you feel is real go ahead.

[Anonymous No.22666568]

>>22666559
oh yeah that is what i said. No wonder you spew barely sensical nonsense

[Anonymous No.22666575]

>>22666563
Right, mathematics is just a game to them. Mental masturbation.

[Anonymous No.22666576]

>>22666567
have you discussed dedekind cuts?

[Anonymous No.22666577]

>>22666549
>You can’t use that theory to make any predictions or solve any problems.
Math is not a science. And even if it was science isn't made true by it's usefulness.

[Anonymous No.22666579]

>>22666575
so? That is what maths should be

[Anonymous No.22666583]

>>22666576
I talked to him about Cauchy sequences above. He didn't even respond. He's a normal Cantor crank without even a real analysis class under his belt.

[Anonymous No.22666585]

>>22666567
fair enough i'm with you, i realise the other guy is a nutjob

[Anonymous No.22666588]

>>22666577
>>22666579
Why do universities teach mathematics as a “science” then? My degree says Bachelor’s of Science in Mathematics. Why didn’t they tell me which parts were a game and which parts were a science?

[Anonymous No.22666590]

>>22666585
>if you don’t believe in these useless concepts that lead to problems that just waste everyone’s time then you are a nutjob
clown world

[Anonymous No.22666591]

>>22666588
oh it says science, well im sorry i have to admit you got me there. Didn't think of that one

[Anonymous No.22666593]

>>22666590
no, you're argument is flawed. you can choose not to believe in higher infinities but that is a choice, there is no flaw in cantor's proof.

[Anonymous No.22666596]

>>22666585
>Why didn’t they tell me which parts were a game and which parts were a science?
You should of learned that if you actually had a BS in math. Math is based on logical deduction from an axiomatic foundation. There is no guarantee any of it exists in the real world and there is no inductive(mathematical induction is totally different) empirical tests to prove anything in math.

[Anonymous No.22666602]

>>22666593
Cantor’s proof assumes undefinable numbers in the list. I do not.
https://math.stackexchange.com/questions/4308285/why-does-cantors-diagonalization-argument-fail-for-definable-real-numbers

[Anonymous No.22666604]

>>22666588
do you like empiricism? If so, what do you see as its usefulness?

[Anonymous No.22666605]

math is invented

[Anonymous No.22666607]

>>22666549
>studying things that we don't have complete knowledge about means it's LE RELIGION
People like that shouldn't have a voice in an educated society. They want to drag science and math down to their level simply because they're too stupid to understand these fields.

[Anonymous No.22666611]

>>22666607
I mean they make the rest feel smart, I like that. Everyone is useful.

[Anonymous No.22666612]

>>22666602
The first and only answer to that post
>The short answer is that “x is definable in ZFC” cannot be defined in ZFC
Your set of definable real numbers doesn't exist so the Cantor diagonal can't be applied to it. So there is no bijective function from the naturals to it and it is not countable

[Anonymous No.22666614]

>>22666604
The concept of usefulness should be clarified. I value the advancement of the human race, so math games are not useful to me. But someone who enjoys math games will find it “useful” to entertain himself. And that’s fine, I cannot force anyone to stop playing math games. But people should be aware of what it is and stop calling people who disagree with them “cranks.”

[Anonymous No.22666620]

>>22666607
What made you think I didn’t understand what was going on here? Cantor made a proof that implicitly assumed the existence of undefinable numbers. And I point out that this assumption has lead to problems that won’t be solved because undefinable numbers are useless concepts to begin with. You can’t test them in any way. You can’t interact with them. They do not exist.

[Anonymous No.22666622]

>>22666614
>And that’s fine, I cannot force anyone to stop playing math games. But people should be aware of what it is and stop calling people who disagree with them “cranks.”
But you didn't just say you don't like Cantor's infinities and don't think they're useful. You claimed they were false. Which makes you wrong and a crank

[Anonymous No.22666624]

>>22666614
The measure of "advancement" is conventional: it would have different meaning in middle ages. So it could be argued that there is no advancement in absolute sense.
If you use empiricism you cannot speak of its usefulness unless you assume it. Only proof of it is going to be again empirical evidence.
I'm sorry but i dont see such methods as respectable, and am not wiling to believe.

[Anonymous No.22666632]

>>22666622
When did I say they were “false” ? Please show me.
>>22666624
Stop being so obtuse. The discovery of calculus was a mathematical advancement which clearly led to many technological advancements.

[Anonymous No.22666645]

>>22666632
and how do you know that? Its almost like the measure changes, when "advancement" is made. Im not trying to be obtuse, its just that in order to see how weak the "advancement" measure is we have to do this. With maths its even funnier for it doesnt even claim to anyhow correspond to the real world.

[Anonymous No.22666646]

>>22666622
>>22666632
Nevermind, I said it in the beginning. Yes I personally view it as false, because I think it’s useless. But of course if you like math games then you will call it “true.” I can’t stop you from doing that. But it won’t help you solve problems in the real world.

[Anonymous No.22666651]

>>22666646
why are you retarded, "usefulness" here i assume means empirical usefulness. That however rests on assuming its usefulness. so its a joke

[Anonymous No.22666657]

>>22666632
>When did I say they were “false” ? Please show me.
Your very first post >>22666009
>The continuum hypothesis rests on the FALSE assumption that there are different sizes of infinity
which I responded to in my next post >>22666019 by calling you a crank for the first time. You can't even keep your own bullshit straight in your own mind. You've done this several times, asking me to provide you arguments.

[Anonymous No.22666665]

>>22666651
Usefulness is determined after the fact. You must make the assumption. You must assume that the universe will continue to operate in a causal manner. You must assume a lot of things to go forward. And you will never be able to truly justify it until after the fact. That’s how evolution works. So is it a mistake to look at the past and see what works, and continue to do that? I don’t know, but that’s what I will do.

[Anonymous No.22666685]

>>22666665
ok you are probably just trolling me now.
Since your opinion rests on circular argument, it really says nothing about reality. I dont know where you see progress in that, its almost like believing in God, and therefore in all the other shit associated with that

[Anonymous No.22666689]

>>22666685

[Anonymous No.22666695]

>>22666689
does this somehow make what i said not true?

[Anonymous No.22666706]

>>22666695
All opinions rest on that which cannot be proved. But to prove something literally means to test it, so you can’t justify a belief until after it has been tested through experience. We have biological tendencies and we learn from experience so we find it easy to have more faith in certain beliefs, but in the end it’s still faith. So I don’t know what point you’re trying to get at here.

[Anonymous No.22666724]

>>22665610
So God exists because..... God must exist??

That's just so retarded

[Anonymous No.22666730]

>>22666706
my point is that unlike in maths which is just formal and makes no claim about describing reality, natural science actually tries to say that even though it rests on circular argument, it somehow describes reality. It rests on circular argument because unless one assumes usefulness of empiricism, he cannot show its usefulness(because he would have to use empirical evidence to show the usefulness). Now however you are wrong opinions can rest at somethinng that is provable, just not in the sense you mean. This is doable in formal stuff like maths. So for example one says assuming the set of conditions A, one can deduce set of conclusions B, which provably follow. This is provable and rests only on stuff it says it assumes.
To prove something doesnt mean to test it(here i think you mean empirical evidence)
You cannot use empirical evidence with me(i explained why) so biological tendencies which are known empirically are useless. So is "learning" from "experience".(are you saying that if experience was different, then truth would be different?)
In maths its not necessarily faith for one can acknowledge that there are assumptions(without saying that these actually are true)

[Anonymous No.22666746]

>>22666730
I recommend learning another language, then you might know these things. To say that a thing can be proven really just means that it can be tested. You can test all math that is useful. You can construct a right triangle and if the hypotenuse is not the square root of the sum of the squares of the legs, then you have just disproven the Pythagorean theorem. The axioms of math and logic are also subject to being proven, but they are proven every day, so we continue to believe in them. We evolved to believe in them. If we evolved under different circumstances, then we would not believe in them.

[Anonymous No.22666769]

>>22666746
Well if you want to speak of language im all for it. As you know meanings in natural languages evolve through use, and words may have multiple meanings, they also have something called contextual meaning, which here is clear when speaking of formal stuff like math.

Here you are wrong when you say that proving means testing you are only considering empiricism, which i said that rests on circular argument, so it says absolutely nothing(do you understand why?)

the example of the triangle is a good one to see where you are wrong, by your wrong logic, if one tested in euclidean space, he would "prove" one thing, and in noneuclidean he would test again and "prove" a different thing.

Last thing again comes from misunderstanding of formal proof and try to put empiricism in its place. The fact that one believes in something says very little about truthfulness of what he believes in.

[Anonymous No.22666783]

>>22666769
we’ll see what scientific discoveries you make with your non-empiricism. Time will prove me right, and your beliefs will be made extinct.

[Anonymous No.22666793]

>>22665610
>A God-like being posses all positive properties

[Anonymous No.22666801]

>>22666783
lol even here you assume the usefulness, why are you so stubborn. Again the concept of usefulness is problematic here, but we should not resort to such terrible means and sleep well at night. Either dont use such weak methods, or know how laughable your activities are.

[Anonymous No.22666807]

>>22666793
The meaning of "positive" is so disconnected from common language in his proof that it would have been more intellectually honest to use a made up word instead. Of course to Godel his proof probably seemed more real than the outside world so maybe I should have said if he wasn't insane it would have be more intellectually honest to use a different word.

[Anonymous No.22666808]

>>22666801
If I assumed nothing then I would die, dumbass.

[Anonymous No.22666817]

>>22666783
lets get nice and technical here
circularity could be expressed as set equality
consider two sets of claims A and B, if A implies B that means that A contains B a way.
Therefore circularity just means A=A

Now empiricists do this: "empiricism is usefull" = "look at the evidence of its usefulness"

By doing this they say nothing about reality and when they try to deduce following statements from this A=A they say again nothing about reality.

>>22666808
Interesting, why do you feel the need to assume stuff?

[Anonymous No.22666822]

>>22666689
What is this? Trilemma for ants?

[Anonymous No.22666833]

>>22666817
> Interesting, why do you feel the need to assume stuff?
If I assumed nothing, then I’m afraid that I would die. I can’t “prove” that the laws of logic and causality and physics will continue as they always have. But because my ancestors made this assumption, and survived, I also carry this tendency, and so far it has worked, so I see no reason to set it aside. Again, that’s simply how I am designed to operate. Only the future will prove me right or wrong. But for my sake I must assume that I am right before I can act in the first place. If I fail, so be it.

[Anonymous No.22666844]

>>22666024
Language is a disease of mythology.

[Anonymous No.22666862]

>>22666833
yet you know none of this and you base it on circular argument. Ahh yes interesting choice indeed. Also progress measure is conventional and not absolute therefore there is no absolute progress.

[Anonymous No.22666878]

for proper autists, you make like this https://www.youtube.com/watch?v=cLXQnnVWJGo

[Anonymous No.22666880]

>>22666862
define “know”

[Anonymous No.22666882]

it touches on the conventionality of progress

[Anonymous No.22666886]

>>22666880
ok I get your point. Let me satisfy you only partially. I will say what it doesnt mean. It doesnt mean that when you notice empirical evidence, you now have knowledge of anything beyond that. (ie no underlying structure). So that is not knowing.

[Anonymous No.22666887]

>>22664483
they hated him because he told the truth

[Anonymous No.22666894]

>>22666886
I would never say that I “know” anything. That word only signifies extreme confidence and familiarity with the idea. Empiricism is justified because it WORKS. When it stops working it will stop being used.

[Anonymous No.22666910]

>>22666894
Ok I probably have not explained it very well.
The point that empiricism works relies on itself.
Consider a situation: I want to know what color are swans(because im autistic), now I ask a statistician and he shows me that with over 99% certainty adult swans are white(this was true historicaly in some contexts). He says that he has seen evidence of this by not seeing any other color swans. Now why is that useful? Well he says that he has seen evidence of such method used and they being considered as useful and so on.

This doesnt tell us anything about reality, for even at that time if sample was chosen differently he would notice black swans.(and there is even bigger problem with empiricism but i will not get into that)

[Anonymous No.22666921]

>>22666910
What’s the problem, exactly? I think there’s a conflation of pragmatism and empiricism going on here, btw

[Anonymous No.22666937]

>>22666921
I would say that the problem is that you believe that there is some "usefulness" or "progress" which justifies use of empirical evidence. This is from my point of view an illusion, I think that if you didnt use the framework of empiricism, you would not see any usefulness or progress.
Just like it was fashionable to believe that society is very smart useful and progressive 100 years ago, and today we would point out many issues(ie our illusion of progress), its just fashionable today to believe that we are the right ones. I dont see no progress here .

[Anonymous No.22666956]

>>22666937
well that’s a different topic entirely. Obviously everyone has their own views of progress. Ted K certainly doesn’t think we are progressing. But in my opinion, the closer we get to genetically engineering humans to be smarter, more adaptive, and more life-affirming, the better.

[Anonymous No.22666971]

>>22666956
but that is my point, that any such measure is essentialy retarded

[Anonymous No.22667131]

>>22664468
Thoughts on this book?
https://en.wikipedia.org/wiki/Where_Mathematics_Comes_From

[Anonymous No.22667582]

>>22665610
Define "positive".

[Anonymous No.22667685]

>>22666131
I come here because I’m bored. No other reason.

[Anonymous No.22667747]

>>22665961
There are similar problems in computation despite us using computers widely, it's a guaranteed nobel prize if anyone solves one

[Anonymous No.22667804]

>>22667747
if you mean P/NP and other stuff in computation theory, they have little to do with completeness/consistency (properties of theories)

[Anonymous No.22667900]

>>22667804
they're more closely related than you might think - look at descriptive complexity

[Anonymous No.22668682]

>>22666956>>22666956
>be smarter, more adaptive, and more life-affirming, the better.

those are the code words by atheists for ''hedonism''

[Anonymous No.22668684]

>>22666783
>>22666910

what is a fact is that an empirical proof is an oxymoron created by atheist rationalists desperate to be passing as empiricists. reminder that empiricists are never rationalist and don't carry any experiment.

[Anonymous No.22669121]

>>22666391
The burden of proof is on you to show that a bijection between the naturals and reals is impossible and Cantor’s method is insufficient by the aforementioned reasons surrounding definability.

[Anonymous No.22669175]

>>22666549
The point of mathematics is neither to make any predictions nor solve anything. It must be eminently useless, lest it lose its beauty. You call it religion, I grant this, by your asinine definition of religion, and in turn call you vulgariser.

[Anonymous No.22669181]

>>22667582
In Goedel's case it is simply aesthetic. Everything that may be considered good, like just, beautiful, wise, understanding, everlasting.

[Anonymous No.22669363]

The facefag is just the next logical step in the protestant mind virus.

[Anonymous No.22669375]

>>22666131
You are 100% a newfag.

[Anonymous No.22669712]

>proves that the Gödel sentence cannot be proven to be true
>... oh and btw the Gödel sentence itself must be true because ...
What did he mean by this?

[Anonymous No.22669934]

>>22668682
hedonism means mindlessly chasing pleasure, dumbass. How does being more intelligent and affirming of life, which even includes suffering, equate to hedonism?
>>22669175
the first 1,000 math discoveries were not just beautiful, they were also useful and falsifiable

[Anonymous No.22670252]

>>22669712
it cannot be proven to be true within peano arithmetic, but it is trivially true when we look at it metalinguistically
the point of godel's theorem is that particular formula systems are incomplete - not (necessarily) that there are universally unprovable things. for example, you can add the Godel sentence to peano arithmetic to get a new formal system within which the original Godel sentence is trivially provable - but this new formal system will have its own Godel sentence thus will be incomplete in its own way.
you wanna know another sentence that isn't provable within a given formal system (containing enough of peano arithmetic)? the sentence stating that the system is consistent. well peano arithmetic is obviously consistent (at least many would contend), so how about we just make the consistency of peano arithmetic an axiom, i.e. by adding the aforementioned consistency sentence to peano arithmetic? sure, let's do that. then we get a new system which cannot prove it's own consistency - but of course, since we believe peano arithmetic is consistent, so must this new one! so let's add the consistency sentence of this system as an axiom. how about we keep repeating this and take the limit of the process, so we obtain a new formal system with a denumerable amount of new axioms stating the consistency of ever more powerful systems. hmm, you're probably wondering what we one can do with this new system. well let me tell what you can do, or more precisely i shall let alan turing tell you! turing proved that in the system constructed in the way just described, one is able to prove all true "universal" statements about the natural numbers - that is, statements of the form "for all natural numbers n, P(n)" for some (i believe unbounded, iirc) proposition P. pic related is mfw first finding this theorem

[Anonymous No.22670272]

>>22670252
So basically it's over for formalists? Platonism won?

[Anonymous No.22670292]

>>22670272
it's over for formalists who want to capture the entirety of mathematics within a single formal system, but it isn't over for formalists in general. the theorem doesn't tell us anything about the ontology of mathematical objects, it only tells us that there is a necessary syntactic limitation to all sufficiently powerful formal systems.
incidentally though, godel himself seemingly did view the theorems as a kind of victory for platonism as you are suggesting

[Anonymous No.22670306]

>>22670292
Gödel only applies in first order logic. What are the problems with second order logic? Why hasn't it been pursued to the same extent as FOL?

[Anonymous No.22670339]

>>22670306
if only you knew how incomplete things really are

[Anonymous No.22670370]

>>22669934
They were only useful in accident, not in essence. I could go more in depth into why this is but I don’t want to get too deep into idealist metaphysics in this thread.
>affirming life
Sorry, but this is not a rigorous concept or you could define it without circularity. Your philosophy is half-baked and self-refuting.

[Anonymous No.22670393]

>>22664468
But did the council grant you the rank of master?

[Anonymous No.22670409]

>>22670370
yawn

[Anonymous No.22670424]

>>22665610
Math cope for god not existing is even stupider than normal christcuckery.

[Anonymous No.22670536]

>>22664468
>with Tarski and Witt
Witters would be more disgusted by your language games than anyone. Go to Gulag. Do not pass Go, do not collet 200 IQ points.

[Anonymous No.22671252]

>>22670424
You can't even read it.