I am not dismissing it at all!
The thing is that truth can ONLY be defined circularly.

Tarski was brilliant and recognized that and still gave a meaningful definition.
As far as I know it.

If he was indeed able to give a non circular definition, then hats off to him for achieving that.
The google result circular definition of truth, brings up tarski on the first 2-3 results.

I meant nothing emotional when I made those statements about Tarski. Just, what seemed like a matter of fact to me.

Oh it is not?
Could you give me a link to more resources then?
It looks so obviously circular. Something depending on itself is circular.

Oh, you mean the one where it is based on formulae that you can write in the language?
Yup, that is circular as well, I think.

I am well aware of Tarski's definition.
It is circular.

This semantic vs syntactical is still a a little confusing.
Is there a definition for truth as well? Non-circular?

The thing is I have a very different platonic notion of "Truth".
I realize that in mathematics, the notion is entirely different and totally based on semantics, which is why I am still having a hard time wrapping my head around this.

Thank you so much though!. Please allow me to analyze and read carefully all that you have written and come back for more questions if any.

No statement has any truth value by itself. It's just a string of symbols. 2 + 2 = 4 doesn't mean anything until we say what are '2', '+', '=', and '4'.

Statements are syntax. A statement is a string of symbols manipulated according to formal rules. No meaning.

A model is an interpretation of some statements. An interpretation is a domain, or universe, in which the statements are to be interpreted. Then you map each symbol to some object in the domain. Like '2' refers to the number 2, where the number 2 is "out there" in Platonic land. But it's hard to argue that the number 2 doesn't have (abstract) existence so we'll just say it exists. And '2' is a symbol that refers to it, as are '1+1' and so forth.

Now some statements are true in some models and not in others. For example the statement "Every number has an additive inverse" is false in the natural numbers but true in the integers.

Now if a statement is true in EVERY model, then it has a proof. That's Gödel's completeness theorem. — fishfry

Thank you for this. I will study this in more detail.

The axioms are more like definitions?

Does the notion of truth then pertain to consistency?(I guess that's what you meant when you said "Every") That the model under which I am trying to say if the statement is true is consistent with all my other models out there, and so on recursively.

What's at the end of the recursion?
There must have been something very basic model, which was chosen first and said that the proof has to be true in at least this model.

I think I understood you, but can't explain it in words yet. Have a diagram in my head.

Gödel's completeness theorem says that in first-order predicate logic, a statement is true in every model of the system if and only if it has a proof in that system. — fishfry

So to recap, if I am understanding this right.
1) Every statement has a definite truth value (under every model of the system). This is a semantic notion.
2) Provability is syntactical. We have a set of assumptions assigned as true, some operators defined on some symbols, and say that every statement within it can be proved or not proved?
3) Under Godel's theorems not all true statements within an axiomatic system can be proven, but we know they are true by going "outside" the system, and imposing a larger meta framework, and proving them in there?

Have I understood it correctly so far?

Truth is a semantic notion. Given an axiomatic system, we choose some interpretation of the symbols, and then we see if the statement is true or false under that interpretation. — fishfry

How do you "see" this without proving? Mathematical statements can be notoriously hard to "see" if true or false.

But if it is unknown it is alright? And what if it's truth value changed according to the axiom framework under consideration?

Provability and truth are two distinct notions?
How would you know if a statement is true without the proof?

What do you do with a statement unprovable within the axioms? Do you say that it is neither true nor false?

One thing:

How do we know that the statement we are proving outside the axioms is the same statement as inside the axioms?
— guptanishank

I don't know what that means. Can you give an example? — fishfry

Well, we discussed that if a statement is unprovable, then we go outside the ambit of the axiomatic system to establish a larger meta system which can prove or disprove that statement, in the above discussion.

A statement is comprised of all the axioms above it.
— guptanishank

That doesn't correspond with my understanding of what a statement is. A statement in a formal system is simply a well-formed formula that may be true or false. For example "2 + 2 = 5" is a statement. "2 + 2 =" isn't. — fishfry

Of course 2+2 = isn't a statement, but 2+2 = 4, comprises of all the axioms needed to define 2, +, = and 4

So in a way we are proving different statements to be true, even though they may look the same.
Any thoughts?
— guptanishank

Don't know what you mean. Example? — fishfry

Let's say under ZFC we could not prove 2+2 = 4, as true. Then to prove Godel's theorems we would still need to establish that statement as true or false eventually. The thing is that truth or falsehood of the statement will vary according to the axiomatic framework we consider.

One thing:
I think Math IS an incomplete language.

How do we know that the statement we are proving outside the axioms is the same statement as inside the axioms?
A statement is comprised of all the axioms above it. So in a way we are proving different statements to be true, even though they may look the same.
Any thoughts?

It seems very counter intuitive to me to use two different axiomatic systems for two different proofs of the same statement.
Could it not be that a statement is True under one axiomatic system and False under another then?

Even then, as a realist I can argue, that if I were to travel along a circle, I could go on forever. So infinity in nature or reality is not so unfathomable.

But a computational process, could go on forever? The symbols might be finite, but they are referring to something quite plausible.

But recursion need not be finite. Surely that is possible.

Informal systems?

Gödel was a Platonist — fishfry

I did not know that about Godel.

Yeah, I meant how do the platonists who do think this way, think about it?

A Platonist is someone who thinks that "out there" in the Platonic world there are sets, and that in that world of sets, Choice is either true or else it's false. There's a definite answer. — fishfry

But the world is every changing. How do they account for that?
What if Choice being true or Choice being false shifted in value. That could very well be true as well.

In that case you would only be able to say that "Choice so far points to True".

So it does not rule out the possibility that one might be able to prove 'everything' in a language with an uncountably infinite alphabet. — andrewk

Maybe not prove everything perfectly?
Say we have a language which says that this statement "currently points to being true".
In the sense a language where you lose accuracy and precision a little, but it points to everything being true or not.

I think such a language can exist.

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.

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

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

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?

I thought that only applied to the Riemann hypothesis?

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.

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.

Because true statements have to be proven as true?

What?
How can something unprovable be true?

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.

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