I recommend you reading this: https://thephilosophyforum.com/discussion/1554/definition-of-arithmetic-truth#Item_24

I had a similar problem. In fact there are formal systems in which every true arithmetical statement can be proved. But there are unprovable true sentences in that stronger formal system of course.

I dont think these problems can't be solved within mathematics since math is based on axiomatic systems.

I did not get your answer. Some of the sentences were not clear, sorry.
Will read the link you sent.

In any system complex enough to do arithmetic - that is, any system complex enough to allow for counting - there will be some truths that cannot be derived from the axioms.

Can all the truths which can be established under an axiomatic system be discovered by mathematics? — guptanishank

Quite the reverse: there cannot be an axiomatic system from which we could derive all of mathematics.

I understand the reverse statement which you said.

But, my point is that how do we even know if a mathematical statement is true or not without an axiomatic system?

I think I just understood that the theorems have to be formed from the axioms themselves. As such, all of them can be found through mathematics?

I think my understanding of the theorems is quite naive to be honest.

Simply, the sentences say that they are unprovable; since they are indeed unprovable, they are true.

What?
How can something unprovable be true?

Why not?

Because true statements have to be proven as true?

Because true statements have to be proven as true? — guptanishank

Earlier you said "I always thought that we had some preexisting fundamental truths, and mathematics attempts to map these truths to formal statements."

Could you clarify what you mean by this? Are you accepting that there are unproven truths, but arguing that true statements are statements that are proven to "map" these "preexisting fundamental truths"?

Yes, in a sense that is what I meant.
That there might be unproven truths, but we try and map all the truths to statements in mathematics which can be proved or unproved.
Basically that there are some pre existing fundamental truths. Mathematics represents them in formal language.
I wanted to know if it could completely represent them, besides other things.
In math however if a statement is unproven, we cannot "know", if it was one of these fundamental preexisting truths or not.

In math however if a statement is unproven, we cannot "know", if it was one of these fundamental preexisting truths or not. — guptanishank

Sure, but it might still be true. For example, if we take the current number system, and assume that we hadn't yet measured the value of Pi, would it not be the case that "the first three digits of Pi are 3.14" would be true, even if we didn't know it? A lucky guess can be true, can't it?

Yeah, I am not denying that. Not at all!
In fact one could prove a statement to be true, even outside the axioms I guess.

The question however is if all the truths about mathematics, say for example arithmetic can be discovered? Would discovery imply provability as well? Because certainly you would need to prove them one way or the other to establish them to be true.

In this way, we can go on. And now, the question arises if mathematics itself is an imperfect, incomplete language. Because all the statements have to be based on axioms. No perfect set of axioms can ever be obtained. Therefore all the true statements which math attempts to map to, cannot be discovered.

I thought that only applied to the Riemann hypothesis?

Since it is nonsensical to imagine proving the consistency of an axiomatic system that captures arithmetic, it is equally nonsensical to imagine the existence of true yet un-provable statements.

Godel sentences are not "true but unprovable" for this reason. For to assume that they are true is to beg the question of consistency, an assumption without which it is impossible to assign any meaning to godel sentences, for they are no longer necessarily non-derivable.

When reading popular accounts of Godel's theorem, there is always this whiff of a shady magical trick being pulled before the reader's eyes. And this magical trick is when authors like Douglas Hofstadter attempt to sell mystery to the reader by saying to the effect "forget about this boring and logically impossible-to-verify disclaimer about logical consistency that we cannot meaningfully assert, or the related fact that completed infinity doesn't really exist - *cough* look at this weird "self-referencing" Escher picture!"

I think @andrewk did a thread on the old forum going through Godel's original proof, so he might have some good input here.

I have a hazy understanding of Godel's theorems, and there are two incompleteness theorems. The first one states, roughly:

(1) Consider a consistent formal system F that allows the expression of arithmetic truths, then there is a statement in F which cannot be proved or disproved (ie F is incomplete)

And the second one:

(2) Consider a consistent formal system F that allows the expression of arithmetic truths, then F cannot (syntactically, using elements and rules in F) imply F's consistency.

Theorem 1 is done through construction. Godel figured out a way to uniquely encode every element of the formal system F (mathematical entities) with a number. Many steps in the proof later, he assigned such a number to the statement $G_F$ "This statement is unproveable in F". Then if F allows derivation of $G_F$, F derived something unproveable, so F is inconsistent. If F does not derive $G_F$, then $G_F$ is true. To establish $G_F$'s truth I think he had to go to a bigger system than F (think 'more arithmetical truths' than 'simple arithmetic'). This kind of makes sense, since he's trying to prove something about the system as a whole - specifically whether a statement of F's consistency implies $G_F$.

Theorem 2 precisely concerns the aforementioned idea of 'having to go beyond the system to establish the system's overall properties'. Specifically, 'having to go beyond the system to prove the system's consistency', since the second theorem is 'A consistent formal system F (that contains simple arithmetic) does not allow the derivation of F's consistency within F'. But I don't have any intuitions about its proof since my model theory is pretty weak.

What are the implications of Godel's theorems for mathematics? Well, when they came out they were a massive 'fuck you' to the Hilbert Program, which was a desire to axiomatize all mathematics. As collateral damage, it screwed over the idea of formalism in philosophy of math - since there are now mathematical truths which cannot be ascertained through string manipulation rules of very general axiomatic systems (like ZFC).

What does it mean for the actual practice of mathematics? Well that depends on the discipline. It has little to no consequences for applied mathematics, it has big consequences in proof theory and mathematical logic. The interesting thing about the theorem for me is that the practice of mathematics, what it means to reason mathematically about mathematical entities, was largely unperturbed - though it did rain on the parade of having a 'complete axiomatic system of all mathematics', and was in essence a no go theorem for that aspiration.

I think this is because the desire for axiomatisation isn't removed by Godel's theorems, you still want to be as precise as you can about mathematical entities. But when you're familiar with the entities in a problem class in mathematics, you don't think in terms of syntactic operations in that class. This is evinced further by the majority of papers with proofs in them not providing a formally valid proof - just 'enough' of the proof that a skilled reader can construct it in their head.

Also, the role of conjecture and heuristics in mathematics wasn't changed by Godel's theorem. People still publish conjectures and heuristics - statements of interesting problems and informal ways of thinking about them.

The idea that Godel's theorem destroys mathematics in some sense is largely due to poor outreach about it. It's in the same ball park as 'quantum weirdness' for generating misapprehensions about a science. I think if it was presented in its philosophical and historical context, and these presentations contained assessment of the impact of the theorems on subfields outside of proof theory and model theory, it wouldn't be seen as a cataclysmic event for mathematics.

I actually like it. For me it gives some kind of internal evidence within mathematics that mathematical progress takes on a quasi-empirical character, like Lakatos and others have argued.

Thank you fdrake and others!

Just one last thing:
Where does the law of excluded middle fit into all this?
A statement must be either true or false.

So if it is unprovable, within a formal axiomatic system, and you cannot decide it's truth value even by going outside the system, what value do you assign to that statement?
How does this fit within the context of Godel's theorems?
@sime: I don't understand why that would be nonsensical.
Are you implying that every statement in mathematics can be shown to be either true or false?

Thank you fdrake and others!

Just one last thing:
Where does the law of excluded middle fit into all this?
A statement must be either true or false.

So if it is unprovable, within a formal axiomatic system, and you cannot decide it's truth value even by going outside the system, what value do you assign to that statement?
How does this fit within the context of Godel's theorems? — guptanishank

LEM is irrelevant, since Godel's Incompleteness theorems don't use it, that is to say, his proof is entirely constructive and syntactic without invoking ~~P -> P.

Recall that Godels results weren't at all surprising to Intuitionists who rejected LEM twenty years in advance of the publication of his incompleteness theorems precisely because they rejected the the assumption that logic has transcendental significance beyond the step-wise empirical construction of its formulas in accordance with intuition. Why on the basis of this intuition ought it be expected that for any well-formed formula P in the language of an axiomatic system that we must derive P or ~P?

Well, when they came out they were a massive 'fuck you' to the Hilbert Program — fdrake

Indeed, historians have recently discovered a cache of previously unpublished letters between Gödel and Hilbert. I take the liberty of summarizing them here.

Godel: Fick dich!

Hilbert: Oh no mein freund. Fick DICH!!

Gödel: Fick dich to the n-th power!

Hilbert: Und deine Mutter auch!

etc.

So if it is unprovable, within a formal axiomatic system, and you cannot decide it's truth value even by going outside the system, what value do you assign to that statement?
How does this fit within the context of Godel's theorems? — guptanishank

Godel's theorem does not say that we cannot prove the statement by going outside the system. Indeed, for the unprovable statements often considered in this context, they can be proved by moving to a meta-system that is larger than the original one and consistent with it. The problem is that there will then be new statements in that meta-system that cannot be proved or disproved (are 'undecidable') in that system. So we need to move to a meta-meta-system to decide those. No matter how many times we do this, the system we end up with will contain new undecidable statements.

Ah! So that is the secret of Kurt Godel's theorems.

Indeed, historians have recently discovered a cache of previously unpublished letters between Gödel and Hilbert. I take the liberty of summarizing them here.

Godel: Fick dich!

Hilbert: Oh no mein freund. Fick DICH!!

Gödel: Fick dich to the n-th power!

Hilbert: Und deine Mutter auch!

etc. — fishfry

Hahahaha

I don't know how the law of excluded middle relates to Godel's theorem, or how rejecting it relates to Godel's theorem. One way to find out how it relates would be to study Godel's incompleteness theorems in the context of intuitionist interpretations of mathematics. Here is a starting place, but I can't guide you any more.

Thank you for that link! This one after my exams.
But I got the essentials here.
I don't really care about the law of excluded middle.
But I just came to know about the assumption that Godel made and it's made me a little happier.

Godel's theorem does not say that we cannot prove the statement by going outside the system. Indeed, for the unprovable statements often considered in this context, they can be proved by moving to a meta-system that is larger than the original one and consistent with it. The problem is that there will then be new statements in that meta-system that cannot be proved or disproved (are 'undecidable') in that system. So we need to move to a meta-meta-system to decide those. No matter how many times we do this, the system we end up with will contain new undecidable statements. — andrewk

I thought Gödel's theorem only applied to non-negative integers. Why would it have anything to say about mathematics in general or anything else?

It applies to any system that contains arithmetic. This is why it's so important. If you want a system to do maths in, it should definitely contain some kind of arithmetic, so Godel's theorems - or analogues of them - will apply.

Godel's theorem does not say that we cannot prove the statement by going outside the system. Indeed, for the unprovable statements often considered in this context, they can be proved by moving to a meta-system that is larger than the original one and consistent with it. The problem is that there will then be new statements in that meta-system that cannot be proved or disproved (are 'undecidable') in that system. So we need to move to a meta-meta-system to decide those. No matter how many times we do this, the system we end up with will contain new undecidable statements. — andrewk

Is there a transcendental-logical or arithmetic system that could account for everything or is this just stating the set of all sets that is also a self-containing set paradox?

No Posty, if the system contained enough arithmetic it would still be subject to Godel's theorems.

Yes, but, my understanding is that a theorem that can't be proved within a system can be proved by a meta-system as mentioned. So, can this process go on forever or is Godel's Incompleteness just a proof of a hard limit to this process, thus giving rise to some asymptotic behavior of the ability to prove arithmetic truths?

A la, Penrose, if one believes in mathematical Platonism and such, then there seems to be a final system that could account for all proofs in it, no?

If you're prepared to contemplate the uncontemplatable there is arguably a loophole in that Gödel's incompleteness theorem only applies to logical languages with countable alphabets. So it does not rule out the possibility that one might be able to prove 'everything' in a language with an uncountably infinite alphabet.

Such a language would be incomprehensible to we mere humans, or any other finite beings, but one can fantasise about possible infinite mega-beings or deities that might be able to reason in such an alphabet, and hence potentially even 'know everything'.

So, Hilbert was right, just not as we have come to understand it or are even able to within our current framework...

It seems intuitively obvious if you consider QM in infinite Hilbert Space or anything in infinite Hilbert Space. Degrees of freedom fly out the window, etc. etc. etc.