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
1) Every statement has a definite truth value (under every model of the system). This is a semantic notion. — guptanishank
2) Provability is syntactical. We have a set of assumptions assigned as true, some operators defined on some symbols, — guptanishank
and say that every statement within it can be proved or not proved? — guptanishank
3) Under Godel's theorems not all true statements within an axiomatic system can be proven, — guptanishank
but we know they are true by going "outside" the system, and imposing a larger meta framework, and proving them in there? — guptanishank
Have I understood it correctly so far? — guptanishank
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
↪fishfry The axioms are more like definitions? — guptanishank
Does the notion of truth then pertain to consistency? — guptanishank
(I guess that's what you meant when you said "Every") — guptanishank
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
What's at the end of the recursion? — guptanishank
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
I think I understood you, but can't explain it in words yet. Have a diagram in my head. — guptanishank
The thing is I have a very different platonic notion of "Truth". — guptanishank
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
This semantic vs syntactical is still a a little confusing.
Is there a definition for truth as well? Non-circular — guptanishank
I am well aware of Tarski's definition.
It is circular. — guptanishank
Could you give me a link to more resources then? — guptanishank
Hopefully, I understand it better! — guptanishank
↪fishfry Tarski's definition is valid and very accurate, under the assumption of infinity. — guptanishank
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
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