What is the meaning of "true"?

Question

As I study logic, I became more confused of the meaning of "true".

Let's say $A_1, A_2, \dots, A_{n_a}$ are finite number of axioms, and $R_1, R_2, \dots, R_{n_r}$ are finite number of rules.

Is a statement $S$ said to be true if and only if it is an axiom or can be deduced in a finite sequence of statements $S_1, S_2, \dots, S_n=S$ such that $S_i$ is either an axiom $A_j$ for some $1 \leq j\leq n_a$ or a statement that can be deduced from $S_{k_1}, S_{k_2}, \dots, S_{k_{R_l}}$ for some $1\leq k_1, k_2, \dots, k_{R_l}\leq i$ by a rule $R_l$ for some $1\leq l \leq n_r$?

I know that there are some statements that are "true" yet not be able to be proven. Then what is the real meaning of "true"? (I heard that in Gödel's argument of his incompleteness theorem, "this statement is unprovable" is a statement that is a statement that is both true and unprovable. I this sense, "true" is not only something that can be achieved by a proof, right?)

What is "true"? Is it something independent from axioms made by human? or is it something made by human?

Hot Network Questions. A lot of questions with interesting names go on the list. — Soham Chowdhury, Sep 25 '14 at 10:41
@Soham: I guess that the algorithm that picks the Hot Questions looks to see if anyone left comments like yours... :P — Asaf Karagila, Sep 26 '14 at 12:00
@AsafKaragila Soham is struck with a Spell of Dumbness for two turns Over at the Code Golf site we've gotten pretty good at predicting HNQ questions :P — Soham Chowdhury, Sep 26 '14 at 13:55

score 5 · Answer 1 · answered Sep 25 '14 at 10:51

Logic is divided into two parts, syntax and semantics. Axioms and proofs are syntactic, and truth and models are semantics.

If $\cal L$ is a language, $\varphi$ is a sentence in the language $\cal L$, and $M$ is an interpretation of $\cal L$, we say that $\varphi$ is true in $M$ if it is true there, under the usual definition of truth in a structure of a language.

If $T$ is a theory in $\cal L$, namely a collection of axioms, we say that $\varphi$ is provable from $T$ if there is a finite sequence of deductions from the axioms of $T$ using the inference rules of first-order logic.

First-order logic has two properties which connect syntax and semantics, specifically provability and truth:

The definition of truth in $M$ tells us that if $M$ satisfies all the axioms of $T$, and $\varphi$ is provable from $T$, then $\varphi$ is true in $M$.
Completeness of the logic tells us, on the other hand, that if $\varphi$ is true in every $M$ which satisfies $T$, then $\varphi$ is provable from $T$.

And because of this we sometimes abuse the meaning of the word true and say things like "This is true in $T$", where we really mean that it is provable from $T$, and therefore true in any model of $T$.

There is some exceptions to that rule, for example in the case of arithmetic and Peano axioms, there is a standard model for the theory (which is unique up to isomorphism), and we say that $\varphi$ is true if it is true in that standard model, even if it is not provable from the axioms of Peano arithmetic.

Why are you talking like that, it makes it unnecessarily difficult to read what you're trying to say. — Kainui, Sep 25 '14 at 10:55

What is the meaning of "true"?

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