There are some theorems which are true but can not be proven.

>>9069518
If they are true but cannot be proven true is not the same thing as we cannot know if they are true.

>>9069560
What theorem can we know is true without proof?

there is no truth, only proof

>>9069879
the axiom theorem

>>9069879

There is no example of one, it is saying that it exists

> Euclid says there are infinite primes
> "but how can there be a prime bigger than the highest prime we know?"

>>9069518

Wrong, there are true statements that cannot be proven within any single given system, but there is more than one possible system and what can be proven in one is not the same as what can be proven in another.

>>9069957
>There is no example of one, it is saying that it exists
Then how can you know it's true?

>>9069518
Truth is undefinable in mathematics, this is known as Tarski' Theorem (see google).

The Godels theorems says that in a recursive theory that can formalize arithmetics (i.e. which is able to talk abut numbers, elementary functions on numbers and recursion), both following claims hold:

1° if the theory is consistant (i.e. does not prove 1=0) then there are statements which are neither provable or reftuable (i.e. you cannot prove their negation)

2° It impossible to prove that such a theory is consistant using the tools of the theory, unless the theory is actually inconsistant.

NAmely there is a meta application form the set of formulas to the numbers called godel encoding (converting formulas; which are just character strings, into numbers is rather trivial in terms of computer algorithms), and a formula with one parameter, D(x) such that if #p is the godel number of the (formalized)sentence , then D(#p) means "p is provable". For every sentence, it is possible, form a proof of D(#p)-> p, to produce a proof of p itself (Lob's theorem), as a result, since not(D(# (0=1))) is equivalent to D(#(0=1))-> (0=1), you get the aforementioned result

>>9069879

Suppose we have the Peano axioms but without the axiom of induction.

In this model, we wouldn't be able to prove that a + b = b + a for natural numbers a and b because there is no induction and thus no way to prove that it would hold for all numbers. However, we know that this statement "should" be true, but we have no way of prove this to be the case.

(note: We can show that a + b = b + a for every particular instance of a and b but not for the generalized version).

see also: https://en.wikipedia.org/wiki/Robinson_arithmetic

>>9070047
1st Godel theorem will hold in any formal system in which you can do the following.
You have a map " # "which takes a formula or a property written in a formal,language and returns a unique number (converting strings into integers is not a problem).
If F is a a property, i.e. formula of one parameter and if there is a property F' such that for every proprety g, F'(#g) <-> F'(#(g(#))), then F has a fixed point:
i.e F(#e)<->e for some sentence.

e just happens to be F'(# F' ).

With this construction, we can build sentences which talks about themselves

for instance if F(x) means "x is not the number code of a true formula" then F' (#F') says "my number is not the code of a true formula" (a paradox)

If F(x) means "x is not the number code of a provable formula" then you have a formula "my code is not he code of a provable formula" and so on

>>9070071
You meant
F'(#g)<->F(#(g(#g)))

>>9069984

How can you know their is a prime bigger than the highest known prime?

Because it was p r o v e n dumbfuck

>>9070102
Which theorem was proven to be known without proof?

>>9070109

You wouldn't have to show that a statement is true but can't be proven

you would have to prove that a true statement exists which cannot be proven

>>9070112
This is math, not blind faith. I'll believe it when I see it

The Incompleteness Theorems are almost entirely irrelevant to any mathematics outside of mathematical logic

Stop worrying about them so much my dudes

>>9070112
existence without construction is the cancer killing mathematics

>>9069518
>There are some theorems which are true but can not be proven

What, like the Riemann hypothesis? So you think telling people they can't will prevent them from doing so?

>>9070146
the incompleteness theorems are constructive

>>9070152
Not me, Gobel proved that it's literally impossible to prove some statements in any given math system

>>9070170
see
>>9069879

>>9070181
No if the system is inconsistent , it will be possible to prove any statement, and in addition if you prove the system cannot prove any statement from within, then in fat you can build a proof of a sentence and its negatin with the very proof of consistency you 've come with. And then every statemet becomes provable.

File: 1487814680418.png (324KB, 720x540px) Image search: [Google]

324KB, 720x540px

>>9069518

>>9070188
https://math.stackexchange.com/a/1484929/181407

>>9070059
If [math]\mathcal{Q} \not \vdash (\forall x)(\forall y)(x+y=y+x)[/math], let that formula be [math]\varphi[/math], and [math]\mathcal{Q}[/math] is consistent, then [math]\mathcal{Q}+ \neg \varphi[/math] is also consistent, and thus has a model. This would obviously be a non-standard, and most likely non-recursive model. But remember truth is relative to the model, the feeling we have about truth of that statement [math]\mathcal{Q}[/math] is because we have the standard model in mind.

>>9070230
*statement in \mathcal{Q}