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From this page: https://www.reddit.com/r/math/comments/56bbd3/what_does_it_mean_for_something_to_be_true_but/

I have the following quote:

When we say true, we mean true of a particular structure. When we say provable, we mean provable from some axioms.

My questions are:

(1) what is the meaning of particular structure?

I guess it is a specific one.

(2) what is the meaning of some axioms?

I guess they are all depend on the set of positive integers.

Safwane
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  • I don't know how correct this is (which is why it's a comment), but I think about the difference as whether you are arguing within the structure (using the axioms actively to prove statements), or you are arguing about the structure from the "outside". Inside the structure there is only the notion of provable (or falsifiable). However, from the outside, you can see results that must be true in the system, for instance because no counterexample can exist, but also that it is impossible to formulate a proof within the system. – Arthur Aug 30 '19 at 10:15
  • You can try to read my answer here : https://math.stackexchange.com/questions/2598863/how-to-prove-a-non-provable-statement-that-is-weird/2598893#2598893 perhaps it will make some things clearer – Maxime Ramzi Aug 30 '19 at 10:18
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    Perhaps this (hypothetical situation) will give a helpful analogy to what @Arthur is saying: I know John started the fire, because I saw him do it, but I can't independently prove this in a way that is sufficient for a court of law. The "prove in a court of law" is a proof within the system, and my seeing him start the fire is a truth that exists outside the constraints of the court of law. (The analogy breaks down a bit in that I can't imagine how one might actually prove that one can't prove something within a court of law. Maybe there's overlooked evidence or something.) – Dave L. Renfro Aug 30 '19 at 10:36
  • A "structure" is a collection of mathematical objects with relations and operations between them. Example : the "usual" natural numbers (whose structure is denoted with $\mathbb N$) with sum, product and order ($<$). The formula $0<1$ is TRUE in $\mathbb N$. – Mauro ALLEGRANZA Aug 30 '19 at 10:57
  • Provable means derivable by rules of logic from the relevant axioms of some theory. Related to natural numbers, we may consider arithmetic with Peano's axioms. $0 < 1$ is provable from Peano axioms, i.e. is a theorem of arithmetic. – Mauro ALLEGRANZA Aug 30 '19 at 10:59
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    Have you read the many many answers on the topic that were posted here over the 9 years of this site's activity? – Asaf Karagila Aug 31 '19 at 22:14

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Raymond Smullyan likes to frame this problem as an accurate and consistent logician who doesn't know that he is accurate or consistent visiting one of those islands where some of the people always tell the truth and the rest always lie. Imagine a native coming up to the logician and saying "You will never be able to prove that I am a truth-teller".

You and I looking at this problem can tell that the native is a truth-teller -- if he were a liar, then the logician would be able to prove that he was a truth-teller, which violates his accuracy. But from the logician's perspective, he will remain undecided about the native's reliability, because he doesn't have the perspective to know his own accuracy.

So, as far as provability goes, the reasoner's logic is like the axioms of a particular logical system, and our logic is like the axioms of a meta-system that is able to prove things about the simple system that it can not prove about itself.

And truth.... In a way, I'm glad I didn't formally study logic far enough to get to Tarski's theorem. Godel was head shaking enough, but (to coin a phrase) we can't handle the truth about Truth.

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This question sums up the distinction between syntax and semantics. Proof is a syntactic notion, truth is semantic. Syntax concerns formal theories, semantics concerns structures.

In the simplest cases, a formal theory consists of a finite set of symbols (the vocabulary), plus rules specifying when a string of symbols is syntactically correct (a so-called formula), which formulas are axioms, and when a formula follows from other formulas (rules of inference). Key point: all this should be purely mechanical, and in principle programmable. Example: $$\forall x\exists y(x\cdot y=1)$$ is an axiom in the formal theory of groups.

A formula in a formal theory is provable if there is a finite list of formulas, such that every formula in the list is either an axiom, or follows by a rule of inference from earlier formulas on the list.

To define the notion of structure, we need a bit of set-theory. A structure for a theory consists of a set called the domain (or universe) of the structure, and enough relations, functions, and individuals in this domain to give meaning to the formulas of the theory. For example, a structure for the formal theory of a group consists of a set $G$ and a function $G\times G\rightarrow G$ (an 'operation') that interprets the symbol '$\cdot$' of the theory; also an individual element of $G$ that interprets '1'.

Tarski gave a definition of 'truth' (or 'satisfaction') for a class of theories known as first-order theories. If $T$ is a first-order theory, and $S$ is a structure for it, then Tarski defined the notion of "$\varphi$ is true in $S$", where $\varphi$ is a formula of $T$ (strictly speaking, a so-called closed formula of $T$).

If all the axioms of $T$ are true in a structure $S$, we say $S$ is a model of $T$.

Tarski's definition is inductive, i.e., truth for longer formulas is defined in terms of truth for shorter formulas. For example $\varphi\&\psi$ is defined to be true in $S$ if and only if both $\varphi$ and $\psi$ are true in $S$.

I'm leaving out gobs of details, which can be found easily in a zillion textbooks (or in my notes Basics of First-order Logic at diagonalargument.com). But I should add a few more generalities.

First, it's not possible to "get off the ground" without relying on an informal level of understanding. For example, Tarski's formal definition of the meaning of '$\varphi\&\psi$' assumes you understand the meaning of the word 'and'. Likewise, a certain amount of informal set theory must be taken for granted. (Set theory itself can be formalized as a first-order theory, but that doesn't erase the issue, just pushes it one level back.)

Second, the most famous example of a "true but unprovable" statement is the so-called Gödel formula in Gödel's first incompleteness theorem. The theory here is something called Peano arithmetic (PA for short). It's a set of axioms for the natural numbers. The so-called standard model for PA is just the usual natural numbers with the usual operations of addition and multiplication and the usual individual elements 0 and 1.

The Gödel formula cannot be proved in PA (if PA is a consistent theory, which most mathematicians believe). But you can give a convincing argument that the formula is true in the standard model. This proof of this argument uses notions from set theory, and cannot be formalized in PA. It can be formalized in other formal theories, however.

Michael Weiss
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There are mathematical conjectures of the form: "there is no natural number with property x" that where shown to be undecidable and hence unproveable within standard axiomatic models of maths. But knowing that such a statement is unproveable immediatly implies that it is true because if it where false one could easily prove so by giving a counter example.

quarague
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  • Is it really that easy? All you can tell a priori is that you can't describe a counterexample within your theory, not that a counterexample can't exist. Right? – Arthur Aug 30 '19 at 11:55
  • @Arthur All you need is that every single natural number either does or does not have property x and that this can in principle be checked. Then if the claim 'there is no natural number with property x' is false, it has a smallest counter example and hence one can prove that it is false by showing that the smallest counter example does satisfy property x. – quarague Aug 30 '19 at 12:52
  • "knowing that such a statement is unproveable immediatly implies that it is true because if it where false one could easily prove so by giving a counter example." That's not true in general - the property in question has to be reasonably simple. For instance, "there is no natural number which is greater than every twin prime" does not obviously have this property, since there's no obvious way to check whether the property "there are no twin primes greater than me" holds. This only works to prove $\Pi^0_1$ statements. – Noah Schweber Aug 31 '19 at 22:54
  • @Arthur It's important that we're looking at natural numbers here: every natural number is indeed definable, so - modulo the (very big) issue in my previous comment - things work out. By contrast, if we looked at statements quantifying over (say) real numbers, we'd potentially run into trouble even for fairly simple properties. – Noah Schweber Aug 31 '19 at 22:55