Most sources I can find about Gödel's incompleteness theorems summarize the result as "there exist true arithmetical statements that have no proof."

It seems coherent to say that there exist undecidable statements for any formal system. That is, there exist statements G for which the formal system has no proof of either G or ~G. I can easily understand this part of Gödel's theorem.

The "truth" aspect doesn't seem coherent, unless we take a platonist view that some assertions "really are true" or "really are false." Let's say we don't take such a view. Then can the "truth" part of Gödel's theorem remain coherent?

Of course it seems natural to say that Gödel sentences like "This statement cannot be proven in F" are true, if we know they cannot be proven in F. But this is intuition based and relies on a platonist viewpoint that the sentence is true or false without respect to some formal system. If we accept such non-rigorous arguments, there is really no need for formal logic in the first place.

Furthermore, Gödel's completeness theorem, as I understand it, guarantees that we can add either of the undecidable propositions G or ~G (not both) to formal system F and the resulting system remains consistent. Thus it seems untenable that ~G is "false" in any formal sense. And as a final point, there exist many undecidable statements, like the continuum hypothesis, that have no agreed-on truth value absent of a formal system. Thus it seems arbitrary to believe that some undecidable statements have truth value, when it is already accepted that others do not.

  • Gödel's incompleteness theorem already says that there are undecidable statements in any formal system that extends Peano arithmetic (and is consistent and recursively axiomatizable). It can be rephrased without using "true", just that they are neither provable nor disprovable. "True" refers to the "standard model", which is a platonist idea, but in the case of the Gödel sentence for arithmetic it is simply provable in set theory, hence "true". Taking all statements to have a truth value is technically convenient and standard in the model theory, different truth values hold in different models – Conifold Jan 20 at 23:00
  • The topic confuses me but question seems a good one and important. Has it been answered? It seems to me the two answers given so far don't quite address the question. This may be a misunderstanding on my part so I'd like to ask whether the OP feels his point is being addressed. – user20253 Jan 22 at 11:35
  • @PeterJ I like the answers posted so far and am still thinking about them. I was hoping to get both philosophical and model-theoretic perspectives. The answers so far are mostly philosophical, and I’m still curious to hear model-theorists articulate their notion of “truth.” Discussions that claim Godel’s sentence is true sometimes invoke model-theoretic ideas I don’t understand. – WillG Jan 23 at 2:43

I think the title and body questions are subtly different. Here I'm going to address the title question, which I'll paraphrase for clarity as:

What sort of "mathematical truth" can a non-Platonist make sense of?

I think this is less strange than it may first appear, since there is an existing parallel: "sharp" vs. "fuzzy" referents in natural language. Roughly speaking, while Platonism does commit to the position that every mathematical statement has a definite truth value, rejecting Platonism does not require us to reject all mathematical statements as meaningless; we can have "degrees of realism" in our mathematics.

Various authors have written about the distinction(s) at play here without committing to (or even explicitly rejecting) Platonism; see e.g. Feferman, Koellner responding to Feferman, or Hamkins - or more generally the materials from Harvard's EFI.

(This old answer of mine is related.)

The situation I want to focus on in natural language is twofold:

  • In order to have a definite truth value, the "ingredients" of a sentence - including but not limited to the objects referred to - have to be "sufficiently meaningful" (I've used the term "sharp" above for this).

  • There are various levels of meaningfulness - or perhaps more palatably, there are various levels of vagueness.

For example, I think that - under some very mild ontological assumptions - we'd all agree that the sentence "the Earth orbits the Sun" is unproblematically true. However,

(And this doesn't even get into the issue of properties: how should we think about a sentence like "Steve is friendly" or "the weather outside is cold"?)

Mathematics, one can argue, is subject to a similar phenomenon - although for various reasons we can ignore it for the most part: there are different degrees of meaningfulness. Here's one particular take on what that might look like:

  • The number "3" is totally meaningful (and "3 is odd" has a truth value unproblematically) since it's feasibly realizable: I can right now instantiate it by raising the middle, index, and ring fingers of my right hand (which I happen to have).

  • The number "8394756" is almost totally meaningful: while it's not feasibly realizable we do have a high degree of confidence that it is physically realizable (e.g. that there actually exist at least that many grains of sand on the beach, so with enough time we could instantiate the number in question).

  • The number 10^50 is very meaningful: while according to our current understanding of the universe it is instantiated in some sense (there are believed to be more than that many atoms in the universe), there are serious obstacles to instantiating it (e.g. the time it would take to "collect" all those atoms "in one place" might be so long that they cease to be atoms in the first place.

  • Graham's number is pretty meaningful: there is no current understanding of the universe according to which it's even plausibly physically instantiated, but we can imagine alternate universes with more-or-less the same laws of physics in which it is. That is, its "abstract finiteness" is its saving grace.

  • But an infinite cardinal like $\aleph_0$ is pushing into the only-somewhat-meaningful territory: we'd have to accommodate an infinitely large universe in order for it to be instantiated in any sense, and it's not clear that that would even count.

  • And then things really break down when we hit stuff like $2^{2^{\aleph_0}}$ or similar.

There are, of course, a couple major questions here:

  • Is this gradation meaningful? The Platonist would say no, but a non-Platonist might find it quite interesting - even a formalist might see something to get out of it (e.g. I personally do) on the grounds that it's actually quite hard to get away from the "real-ness" of 3.

  • Is this gradation arbitrary? Even assuming that my relevant beliefs about the physical universe are accurate, the spectrum I've gestured at above is arguably quite contingent. The last sentence of your question ("it seems arbitrary to believe that some undecidable statements have truth value, when it is already accepted that others do not") strikes hard: what criteria do we use to judge meaningfulness, and in particular to what extent do we have to commit to a "meta-Platonism" ("there is a definite fact of the matter about many questions around comparative meaningfulness") - and why is that justified?

While these criticisms are quite important, I think that ultimately the approach above is valuable (and in fact it's one I adhere to).

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  • I am curious about the asymmetry of "Platonism does commit to the position that every mathematical statement has a definite truth value, rejecting Platonism does not require us to reject all mathematical statements as meaningless". It seems we can accept mathematical statements as meaningful while rejecting that they have truth values (Wittgenstein or fictionalists can be so interpreted) or at least that they have non-relative truth values (for pluralists). You later apply "meaningful" to objects rather than to statements. Should all statements about "meaningful" objects have a truth value? – Conifold Jan 21 at 21:39
  • @Conifold I tried to gloss over that issue since I really don't think I have the expertise required to treat it well in a reasonable amount of space. You're absolutely right that there's even more flexibility here - I just wanted to point out the most basic level, which is that even if we completely collapse the meaningfulness/existence/classicality/etc. distinctions we still wind up with a framework which can allow some-but-not-too-much truthositude. – Noah Schweber Jan 22 at 1:43
  • I think it is an interesting question how much the "degrees of meaningfulness" translate into expectations of a definite "truth value", and why. Hilbert's "We must know - we will know" (and he was no platonist), and widespread beliefs that "normal" mathematical questions have provable answers indicate that there are some strong intuitions among mathematicians in this regard. – Conifold Jan 22 at 1:53
  • @Conifold Oh I definitely agree - I just don't think that I personally have the background necessary to treat it well in this context. A lengthy essay on a blog? Sure ... – Noah Schweber Jan 22 at 1:53
  • I wonder if this topic tends to confuse what we can know is true with what we can prove is true within some logical system. I'm even quite sure what I mean, but if we can see that a statement is true yet undecidable within the system then there is a meta-system, and if this is true for all (relevant) systems then there is a meta-system beyond all such systems. I suppose I'm wondering whether the question is really all about logic or partly about how we know things. – user20253 Jan 24 at 18:22

The decision is about formal systems. So the problem arises only within Formalism. It is possible to deny Platonism and still not anchor mathematics in axiomatic systems.

The program that led to Goedel's completeness competes with more radical reactions to the gap in Frege's work. The first among these is the original form of Intuitionism proposed by Brouwer.

Mathematical truth for Brouwer was anchored in human intuition and creative power. So an otherwise provably undecidable statement might be accepted and elaborated upon if it had a given kind of natural appeal.

At the same time, a large number of results immediately became 'unproven' by Brouwer's standards on the basis of the fact that the Law of the Excluded Middle was seen as the most intuitive cause of Russel's paradox and abandoned. This has been followed up mostly with mathematics that backs away further into finite and approximative methods.

But some later Intuitionist (e.g. Steven Kleene) have gone the opposite direction and admitted it is just fine (if a bit cavalier?) for classical mathematics to hold onto the Law of the Excluded middle and cast about for something else to discard in order to avoid the various paradoxes, as long as they do it carefully. (By and large, they just don't, because they see the philosophical problem as an obsessive intrusion into their work.) That means other undecidable propositions should be tried out, to see if they net consequences that have intuitive appeal.

So folks have looked at the potential upside of adopting a strong negation of the Axiom of Choice known as Infinite Game Determinacy, to both get rid of Tarski's paradox and potentially find more interesting and appealing territory. Some number theorists have adopted the Continuum Hypothesis because having a given model of infinite orders makes some finite results simpler.

A large branch of the field of Ordinal Theory is about what happens when you do or do not accept various Large Cardinal Axioms, and to build a sort of map of options, in the hope that this grander view can develop the intuition and give us a good way to select our axioms in the future. This is the project of Woodin.

There remain active developments in non-Platonist mathematics outside Formalism.

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