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Isn't a statement only true if you can prove it?

Edit: To elaborate, when reading about foundations of math, there seems to be concepts of completeness and decidability that seems to suggest they are proving and deciding if true are different...

csp2018
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  • If a statement is taken to be true then it is known as an axiom. Showing the consequences of these axioms through logical arguments is the art of proving statements. – WaveX Oct 05 '19 at 16:47
  • The concept of provability is concrete, but I feel the truth of a sentence is vague. We can define what a truth of a sentence of a given structure, but it seems not trivial to answer whether a given sentence (e.g. continuum hypothesis) is true or not. – Hanul Jeon Oct 05 '19 at 17:02
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    In mathematical logic one has to distinguish between those two concepts: Provability is purely syntactical and depends on the choosen axiom system. Truthness is semantical and is usually dependent on some specific interpretation/structure: In a given structure, a sentence $\phi$ is either true or false – Andrei Kh Oct 05 '19 at 17:10

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What the authors are getting at is that provability and truth are fundamentally different things. Provability means that we have some rigorous argument for it from some assumptions that we believe are true. Truth means... well... that it is true. The latter can in principle hold without the former: there can be a statement that is true and yet there is no proof of it. (The former holding without the latter shouldn't happen provided our methodologies for proving things are only capable of proving true statements, which is something we should demand.)

The fact that we may not be able to prove every true statement is called incompleteness, and as you have no doubt heard, this really happens in mathematics. What further cements the distinction between truth and provability further is that there are statements that are unprovable in a certain system, and yet we can prove them true in a more powerful system. For instance we can't prove the consistency of arithmetic using arithmetical methods, and yet we can prove it using more powerful set-theoretical methods.

But then there are statements like the continuum hypothesis (CH) that we know for a fact that we can't prove true or false by any currently accepted mathematical methods/assumptions, even in principle. This presents a problem since the only way of 'deciding a statement is true' is giving a proof by some accepted method/assumptions.$^*$

What does this mean for the CH? Well, there are (at least) two ways forward. One is to search for more powerful method that are still "self-evidently valid" but can decide things like the CH. This program is being pursued in the form of finding stronger axioms of set theory, but even if it is eventually successful in the eyes of the pursuers, it seems unlikely to generate broad consensus, at least not quickly.

The other way is to give up on the idea of the CH, or mathematical statements more generally, always having a well-defined truth value. As others have remarked, truth (which I could only define circularly above) is something that only makes sense relative to a structure. So giving up on a well-defined truth value means giving up on the idea of there only being one unique mathematical universe we are describing about with the axioms of set theory. But then we are challenged to justify how for a seemingly concrete question about the seemingly concrete real numbers, there isn't one right answer.

$^*$ Note there is another distinction to be made here. It's one thing to have a proof, it's another to believe it is a valid argument from valid assumptions. So even give that proof is the only accepted way of generating definitive mathematical knowledge, there is some wiggle room if some people have different precepts as to what are valid inferences and assumptions. One framing is that we have many different formal deductive systems out there (a zoo of acronyms like PA, PRA, HA, ATR$_0$, ZFC, MK, etc.) and different mathematicians will sometimes differ on which systems' theorems they accept as true.

spaceisdarkgreen
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  • I'll add that for readability's sake I've been very vague and sloppy about the technical points. "Provable" and "true" are technical terms in logic and to fully see what's going on, you need to study syntax and semantics rigorously, as per Andrei's comment below the question. – spaceisdarkgreen Oct 05 '19 at 17:53
  • Thanks, but could you elaborate more on how decidability fits into this picture? – csp2018 Oct 05 '19 at 20:20
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    @csp2018 undecidability in this context means that a particular statement can't be proven or disproven in a given system. So for instance, related to my third paragraph, we would say CH is undecidable in ZFC. "Undecidable" can mean other things in related contexts for instance that there is no algorithm for establishing if an arbitrary sentence is provable in a given theory or not. For instance ZFC is an undecidable theory in this second sense. Does that help or were you looking for something else? – spaceisdarkgreen Oct 05 '19 at 20:34
  • Thanks, that helps. But then how is undecidability of a system different than incompleteness of the system? Is it related to your second sense, that an undecidable theory doesn't have an algorithm to know if a sentence is provable, whereas an incomplete theory doesn't have a proof of some sentence whether or not there is an algorithm to know this? – csp2018 Oct 05 '19 at 20:50
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    @csp2018 In the first usage, we call the sentence undecidable and we call the theory incomplete. So an incomplete theory is one with an undecidable sentence. (Note it's also common to say 'independent' rather than 'undecidable', which is a better terminology to my mind.) There is an slight relationship with the second sense. If a theory is complete and there's an algorithm to decide what the axioms are, then it is decidable in the second sense, since we can systematically search through every proof until we find either the proof of the sentence or its negation. – spaceisdarkgreen Oct 05 '19 at 21:00
  • how can a theory be complete yet undecidable? https://math.stackexchange.com/questions/3336583/decidability-vs-completeness – csp2018 Oct 06 '19 at 23:16
  • @csp2018 Remember how I had to stipulate how the theory's axioms were decidable in the argument above? An example of a theory that is complete but undecidable is $Th((\mathbb N,+,\cdot)),$ i.e. the theory of all true first order statements about the natural numbers involving addition and multiplication. This is manifestly complete, but if it were decidable this would run contrary to Godel's theorem. – spaceisdarkgreen Oct 06 '19 at 23:31
  • I think I was confused about one detail. https://philosophy.stackexchange.com/questions/67582/how-can-a-undecidable-theory-be-complete/67583#67583 – csp2018 Oct 06 '19 at 23:36
  • @csp2018 makes sense. To clarify my last remark, it violates Gödel because Gödel says any theory with decidable axioms that extends PA (or even Q) is incomplete. This may actually be a more natural place to deploy Tarski’s theorem, which says the theory is not even arithmetical. – spaceisdarkgreen Oct 07 '19 at 00:57
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My opinion is that there is sometimes theorems that work everytime but we still don't have a specific reasonable proof for (and they are quite rare). Also there is something called axioms that are agreed on to be true with no proof, just by convention (I know this is a little it bit different than "deciding" if it is true or not, but I am just giving some ideas that may help)

Fareed Abi Farraj
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