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?

1) Every statement has a definite truth value (under every model of the system). This is a semantic notion. — guptanishank

That's a bit muddled.

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.

But of course we would hope that if a statement is true in some models but not in others, there would NOT be a proof. Because proofs are syntactic. They apply to every model. So the only statement that can have a proof is a statement that's true in every model. [Unless our axiomatic system is badly behaved].

Statements are syntactic. Models are semantic. "All numbers are even" is a statement that's neither true nor false. It's true in the model consistent of the set of even numbers. It's false in the model consisting of all the whole numbers. We would HOPE that there's no proof of "all numbers are even" because it is NOT true in all models.

This is all a huge area of mathematical logic. I can't really hope to do any of this justice. If you are interested, there are a lot of courses on formal logic on EdX and Coursera and other MOOC providers. I definitely recommend a course in elementary logic, it will clarify a lot of things for you.

2) Provability is syntactical. We have a set of assumptions assigned as true, some operators defined on some symbols, — guptanishank

Yes.

and say that every statement within it can be proved or not proved? — guptanishank

Well maybe every statement or its negation can be proved. If so, the system is called complete. If not, it's incomplete. Some systems are complete and others are incomplete. The modern formulation of Euclidean geometry, for example, is complete. Every statement is either provable or not.

3) Under Godel's theorems not all true statements within an axiomatic system can be proven, — guptanishank

A little muddled. There are no true statements by themselves. Incompleteness just says there are statements that can't be proven nor disproven. There is no reference to truth.

but we know they are true by going "outside" the system, and imposing a larger meta framework, and proving them in there? — guptanishank

You can disprove them that way too. Take the Continuum hypothesis, CH. We know that it's independent of ZFC. So we can now work in ZFC plus CH, or ZFC plus not-CH. Math doesn't care. They're both equally valid. You can choose whichever you like.

The quest is to find out WHICH is "true," CH or not-CH. And what we mean by "true" is that there is some preferred or intended model of set theory; and we want to know if CH is true in that model. Or in the Platonic world if we believe in such a thing. This is all very nebulous and I'm very far from being an expert.

The point here is that we can make a larger system in which CH can be proved; and another system in which not-CH can be proved. The question of what the "true" answer is, starts with trying to explain what that would even mean, and then trying to answer it. Very smart people have been working on all this for over a century. We won't be able to nail it down here.

Have I understood it correctly so far? — guptanishank

I've tried to place into context some of your thoughts, to the limit of my own understanding and ability to explain it. You'll have to tell me if I've been successful.

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.

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.

↪fishfry The axioms are more like definitions? — guptanishank

That's a very good question.

In set theory, which is regarded as the foundation of math, we think of the axioms as statements accepted without proof in order to get our formal system off the ground. The axioms are chosen on the basis of naturalness and general agreement that they represent what we think about sets.

On the other hand, in non-foundational areas of math, axioms really are used more as definitions. As an example, in abstract algebra we talk about groups, rings, and fields. In real analysis we talk about metric spaces and topological spaces. In each case we list a set of "axioms," which are taken to be the defining property of the object in question.

So we're not saying the axioms are true. We are saying that anything that satisfies the axioms deserves the name we're giving it. Anything that satisfied the axioms for a toplogical space will from now on be known as a topologica space.

It's not about truth. It's about classification.

Now we can take this point of view and retrofit it to set theory. A "system of set theory" is any collection of mathematical objects that satisfies the axioms of set theory.

In this sense we can in fact view even the axioms of set theory as being definitional rather than foundational. The axioms of set theory are simply the defining properties of the things we call sets.

Does the notion of truth then pertain to consistency? — guptanishank

No. Consistency is syntax. Truth is semantics.

A formal system is consistent if it does not have a proof of both P and not-P for some statement P. Syntax. It's about the existence or nonexistence of proof, which are just sequences of statements derived according to inference rules. A computer could crank out a proof.

Truth is about meaning. You assign meaning to your symbols to see whether it's true. "It is raining outside." A string of meaningless letters and punctuation. Onece you define "raining", "outside," and "It", and "is", and so forth, then you can look out your window and determine if it's raining outside.

If by raining you mean sunny and by outside you mean inside, then the meaning of the statement changes. Meaning is semantics.

Symbols are syntax. Meaning is semantics.

(I guess that's what you meant when you said "Every") — guptanishank

Not clear what you're referring to.

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. — guptanishank

That diesn't make sense to me. There's no recursion. You're overthinking this.

Formal manipulation of strings is syntax. Proofs. Consistency. Completeness. Assignment of meaning is semantics. Truth.

What's at the end of the recursion? — guptanishank

There's no 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. — guptanishank

Oh yes. This is the concept of the "intended model." Given a set of axioms, it often has many models. Mathematicians are thinking of a particular model and not the weird models. Again this is about the intuition of what the axioms are about.

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

There are a lot of good explanations around the Internet, lots of Wiki pages, etc. There's too much content being covered in this conversation to be summarized easily. Don't rely too much on just what I've said, I'm not an expert.

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.

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". — guptanishank

That doesn't preclude you from learning what the logicians think about it. I might be a vegetarian but I could still go to butcher school and learn how to cut meat. I just wouldn't eat it. I'm only explaining a point of view in response to your questions. I'm not demanding that you agree with any of it.

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. — guptanishank

Ah. Well, no. In mathematical logic we spend a lot of time worrying about these things. Working mathematicians generally don't spend any time thinking about them at all. A group theorist proves theorems about groups, investigates the different types of groups. attempts to classify all the possible groups, and so forth. It's not part of their discipline to define truth or think about syntax and semantics. It doesn't come up. [To be accurate, there are some questions in group theory that depend on mathematical logic, and to that extent, group theorists are interested in logic. But only to solve problems in group theory, not because they care about mathematical logic].

We're talking about mathematical logic. We are not talking about math in general. You can get an undergrad math degree without ever spending five minutes thinking about syntax versus semantics. Probably a Ph.D. too in most mathematical disciplines.

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

Ah yes well in the 1930's, Alfred Tarski defined truth.

https://en.wikipedia.org/wiki/Semantic_theory_of_truth

This stuff is way over my head. A standard example is:

'Snow is white' is true just in case snow is white.

I have never taken the trouble to learn the presumably deep theory underlying this idea. They talk about the object language (the language you care about) and the metalanguage, which is the language you use to talk about the object language.

That's literally everything I know about it. But Tarski is the guy who gets credit for defining truth.

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

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

For what it's worth, that is not the mainstream view. But I'm not familiar with the subject so I can't really say.

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.

Could you give me a link to more resources then? — guptanishank

Tarski's work is taken seriously. You are dismissing it. Clearly logicians don't dismiss it. As I say I'm not in a position to discuss the issue in detail. I'm only pointing out that Wikipedia doesn't say, "Tarski's idea is circular and not taken seriously by anyone." On the contrary, Tarski's name comes up whenever anyone discusses the definition of truth in logic. His work in that area is regarded as important.

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.

I see your point. Googling around shows many people asking if Tarski's definition is circular. You know more about this than I do.

And now you see why this "semantic" notion confuses me such.
It is not a definition as far as I know, but just a notion.
It's not well defined.

You were being sarcastic right? :P (when you said I knew more about this)

Please do not take offense.

So a sentence in the object language, which one wants to show or define as true, has to hold in the metalanguage as well.

So there is no concept of truth in the metalanguage?

I am currently reading all that I can about this. Please allow me to come back with more intelligent questions.

Sorry if you took offense, over my comments on Tarski.

We wish to prove the truth of a sentence in the object language ( L ), but we use a metalanguage (M) to do so, the definition is based around terms used in the sentence of M, but without the notion of "truth".

The whole thing seems circular. I mean not the definition of truth, but the usage of the words in the object language and the metalanguage. Because, now the words in L are dependent on truth, and truth is now dependent on the same words M.

This is the paper I had found, and thought that it was widely known that Tarski's definition is circular.
http://www.sa-logic.org/sajl-v1-i1/06-Greimann-SAJL.pdf

https://philosophy.stackexchange.com/questions/47018/how-do-you-define-truth?
Also, asking the same question here.

Hopefully, I understand it better!

Hopefully, I understand it better! — guptanishank

I don't know anything about the subject and can't respond to any of your questions. Hopefully you'll get some insight on Stackexchange.

Alright. Thank you for answering all my other questions.
It's been a ton of help really!

Tarski's definition is valid and very accurate, under the assumption of infinity.

↪fishfry Tarski's definition is valid and very accurate, under the assumption of infinity. — guptanishank

Do you mean the axiom of infinity? I would be curious to see a reference for this relationship. I know some set theory but I have never heard of any relation between the axiom of infinity and Tarski's definition of truth. That doesn't mean there isn't one, just that I wonder if you could provide some context and/or references.

I am not sure IF it's with the axiom of infinity or a weaker assumption of the same kind.

Truth in the object language depends on the metalanguage. And for truth in metalanguage, you form a bigger metalanguage and so on... at least as far as I was able to understand it.

Truth in the object language depends on the metalanguage. And for truth in metalanguage, you form a bigger metalanguage and so on... at least as far as I was able to understand it. — guptanishank

Do you have a reference for this? Or is the infinite tower of meta's something you are bringing to the discussion that's not in the primary literature?

sorry, just a conversation with someone on stackexchange, I can link the question. But if he's right, then the definition is definitely valid. So not digging more.

Tarski defines truth in terms of the notion of the satisfaction of a formula of LCC by an infinite sequence of assignments (of appropriate objects: subclasses of the universe of individuals in the case of LCC). He gives first a recursive definition and immediately indicates how to transform it into a normal or explicit definition. The recursive definition is this: an infinite sequence of classes f satisfies formula F if and only if f and F are such that

https://plato.stanford.edu/entries/tarski/

Perhaps it points to the same thing. Will give it a better read later.