/sci/ - Science & Math

[Anonymous No.10001115]

let's say i made some concrete mathematical claim, some really fucking wacky one like

>Claim: the distribution of prime numbers maps to the distribution of digits in pi according to this formula: *insert formula*

then i said
>Proof: I demonstrate that this claim is not provable from the axioms of math
>*insert proof of unprovability here*
>...
>Q.E.D.

why? because if it were not true, you could algorithmically find a counter example in some finite process, right?

so if a well-formulated claim is demonstrated to be unprovable, then it's true automatically then?

[The Lord No.10001128]

>>10001115
>because if it were not true, you could algorithmically find a counter example in some finite process, right?
that makes perfect sense to me

[Anonymous No.10001138]

>>10001115
Misunderstanding unprovability theorem.

Godel says SOME claims are not proveable. If you can show your statement is indeed unprovable within some axioms, that could indeed be interesting. However, most (some?) theorems under axioms like set theory can be proven.

Also saying "a counter example hasn't been found YET" is not a proof.

[Anonymous No.10001145]

>>10001138

>Also saying "a counter example hasn't been found YET" is not a proof.
not saying that. i'm saying "this can't be proved" is itself a proof, provided the claim is well-formulated.

> If you can show your statement is indeed unprovable within some axioms, that could indeed be interesting.
exactly

anyhow i'm stealing this from
https://youtu.be/O4ndIDcDSGc
12:00 onwards

[Anonymous No.10001163]

>>10001145
I see what you're saying.

The key is proving something is unprovable under all axioms, that's pretty difficult. And then if that is "well-formulated" it implies truth. That hardly applies to most mathematical statements.

[Anonymous No.10001167]

>>10001163
what about the axiom of choice?

hasn't it been shown that the axiom of choice is not provable within the normal axioms minus the axiom of choice?

(sorry, physicsfag here, probably out of my depth)

[Anonymous No.10001169]

>>10001115
What do you mean? 1 = 2 is unprovable and also wrong

[Anonymous No.10001177]

>>10001169
the problem is that you coudln't provide a _proof_ that 1=2 is an unprovable claim. (at least not a valid one)

most claims have a proof or counterexamples, but few have proofs that the statement is unprovable. godel cooked up some unprovable statements that were demonstrated to be true...

so if it were possible to prove that a certain claim was not provable (which you haven't done for 1=2), then i'm arguing you've implicitly demonstrated that the claim is true already

[Anonymous No.10001182]

>>10001177
on second thought maybe i just don't understand this shit at all... fucking godel

i guess what i mean is that 1=2 has a way of evaluating whether it's true or false, and that's demonstrable. so i guess i mean "unprovable" means "you can't prove it" AND "you can't prove it wrong"

[Anonymous No.10001185]

>>10001177
>the problem is that you coudln't provide a _proof_ that 1=2 is an unprovable claim
Who cares? That's not a problem. Most statement in the universe are unprovable. I don't need a proof that there's no proof that 1=2. Nonsense statements are objectively and obviously nonsense.

[Anonymous No.10001186]

>>10001177
What's wrong with this proof?
If 1 = 2 were provable, then, as 1 != 2 is also provable, we have a contradiction. Hence 1 = 2 is not provable.

[Anonymous No.10001188]

>>10001115
How does unprovability relate to the truthness of a statement?

[Anonymous No.10001190]

>>10001182
So maybe you mean unfalsifiable?

[Anonymous No.10001194]

>>10001190
that's a "science" word, OTOH i mean "unprovable" in the sense of godel

>>10001188
well godel constructed a statement, and proved that it was both unprovable and true. and the way the proof works is basically that godel constructed something that said:

"this statement is unprovable"
which must be true because if it were false, you could prove it, making it true...

so it does have some relation

[Anonymous No.10001203]

>>10001194
Unprovable only refers to the non-existence of a proof, not that it can't be disproved.

[Anonymous No.10001214]

>>10001203
if it could be disproved, though, by some counterexample or some logical proof, that means it is provably false, right?

(all my logic is being stolen from the video here: >>10001145)

[Anonymous No.10001219]

>>10001214
Yes, 1 = 2 is provably false and hence unprovable. Otoh, statements can be independent of a set of axioms, which means they can neither be proven nor disproven from those axioms - like CH and ZFC. I can't watch that video rn as I have a class in 5 minutes or so.

[Anonymous No.10001227]

>>10001219
well he claims that if you could prove the Riemann Hypothesis was unprovable, then that implies it's true via the argument i gave in OP

[Anonymous No.10001237]

>>10001145
>>10001163
something being unprovable does not mean it's automatically true, it just means knowledge is constrained for whatever reason. For example, any negative existence statement is unprovable depending on how solipsistic you want to get, even if they seem to break some rule of logic. That doesn't automatically mean they exist. Another simple exercise is to to simply think of the negation of your claim. If A is unprovable, then ~A is also unprovable(or disproveable) since it being disproved would prove A (Law of Excluded Middle; I realize this has some problems but ONLY in statements that can be neither true nor false not those that are in fact true, as you were claiming was implied) and it being proved would disprove A. Finally, I think you have a confusion that comes because of that popsci channels just doing surface-level explanations of concepts and having to make them as interesting as possible, sometimes damn near lying. Their summation of infinity video is equally bad. Either way, the point of godel's theorem was that any complicated enough logical system has to have some assumption in its foundation axioms or it has to be circular. If any statement in math was unprovable, it'd most likely be about one of the basic axioms, not something like the distribution of prime numbers.

[Anonymous No.10001264]

>>10001237
thanks for the post.

i agree that this youtube video must be cutting some corners logically; and your and others' explanations were helpful

gödel shit is still cool anyhow

thanks

[Anonymous No.10001297]

>>10001264
No problem. By the way, another thing you might find interesting about godel in relation to this is his completeness theorem. If any statement S in a language L is unprovable in that language, then there is always another language L' which satisfies L and in which S is provably false. This is how Euler's fifth postulate was proven unprovable.

[Anonymous No.10001308]

>>10001297
Euclid's*

[Anonymous No.10002451]

>>10001115
Let T be a theory with the property that any unprovable statement is true. If p is any unprovable statement in T, then its true by hypothesis. We now consider its negation ¬p. It is either provable or unprovable. In any case, ¬p is true, which would make p and ¬p are both true. Hence, T is inconsistent. We conclude that any theory with the property listed above is inconsistent.

Notes:
1. We're assuming the theory actually has an unprovable statement, it may very well have none but that would mean any statement and its negation are provable (because if one of them isn't, then it has an unprovable statement) which would make the theory inconsistent anyway.
2. The definition of inconsistency says that a theory is inconsistent if a statement and its negation are provable within the theory and what I used to conclude T was inconsistent was that p and ¬p are both "true". However, if p and ¬p are both "true", then we can use this and the fact that a->(b->a) to prove any other statement. I won't give a proof of that because it gets pretty symbolic pretty fast and this post is long enough already.
3. We're assuming a bunch of things about T. For example, that it admits modus ponens and has the usual material implication. It ultimately doesn't matter because we're not assuming anything that set theory (ZF) doesn't already have.