Non-formal provability

A simplified version of Goedel’s first incompleteness theorem (it’s really just a special case of Tarski’s indefinability of truth) goes like this:

• Given a sound semidecidable system of proof that is sufficiently rich for arithmetic, there is a true sentence g that is not provable.

Here:

• sound: if s is provable, s is true

• semidecidable: there is an algorithm that given any provable sentence verifies in a finite number of steps that it is provable.

The idea is that we start with a precisely defined ‘formal’ notion of proof that yields semidecidability of provably, and show that this concept of proof is incomplete—there are truths that can’t be proved.

But I am thinking there is another way of thinking about this stuff. Suppose that instead of working with a precisely defined concept of proof, we have something more like a non-formal or intuitive notion of proof, which itself is governed by some plausible axioms—if you can prove this, you can prove that, etc. That’s kind of how intuitionists think, but we don’t need to be intuitionists to find this approach attractive.

Note that I am not explicitly distinguishing axioms.

The idea is going to be this. The predicate P is not formally defined, but it still satisfies some formal constraints or axioms. These can be formulated in a formal language (Brouwer wouldn’t like this) that has a way of talking about strings of symbols and their concatenation and allows one to define a quotation function that given a string of symbols returns a string of symbols that refers to the first string.

One way to do this is to have a symbol ′α′ for any symbol α in the original language which refers to α, and a concatenation operator +, so one can then quote αβγ as ′α′ + ′β′ + ′γ′. I assume the language is rich enough to define a quotation function Q such that Q(x) is the quotation of a string x.

To formulate my axioms, I will employ some sloppy quotation mark shorthand, partly to compensate for the difficulty of dealing with corner quotes on the web. Thus, ′αβγ′ is shorthand for ′α′ + ′β′ + ′γ′, and as needed I will allow substitution inside the quotation marks. If there are nested quotation marks, the inner substitutions are resolved first.

1. For all sentences ϕ and ψ, if P(′ϕ↔︎∼ψ′) and P(′ϕ′), then P(′∼ϕ′).

2. For all sentences ϕ and ψ, if P(′ϕ↔︎∼ψ′) and P(′ψ→ϕ′), then P(′ϕ′).

3. For all sentences ϕ, we have P(′P(′ϕ′)→ϕ′).

4. If ϕ has a formal intuitionistic proof from sufficiently rich axioms of concatenation theory, then P(′ϕ′).

Here, (1) and (2) embody a little bit of facts about proof, both of which facts are intuitionistically and classically acceptable. Assumption (3) is the philosophically heaviest one, but it follows from its being axiom that if ϕ is provable, then ϕ, together with the fact that all axioms count as provable. That a formal intuitionistic proof is sufficient for provability is uncontroversial.

Using similar methods to those used to prove Goedel’s first incompleteness theorem, I think we should now be able to construct a sentence g and the prove, in a formal intuitionistic proof in a sufficiently rich concatenation theory, that:

5. g ↔︎  ∼ P(′g′).

But these facts imply a contradiction. Since 5 can be proved in our formal way, we have:

6. P(′g↔︎∼P(′g′)′). By 4.

7. P(′P(′g′)→g′). By 3.

8. P(′g′). By 6, 7 and 2.

9. P(′∼g′). By 6, 8 and 1.

Hence the system P is inconsistent in the sense that it makes both g and  ∼ g are provable.

This seems to me to be quite a paradox. I gave four very plausible assumptions about a provability property, and got the unacceptable conclusion that the provability property allows contradictions to be proved.

I expect the problem lies with 3: it lets one ‘cross levels’.

The lesson, I think, is that just as truth is itself something where we have to be very careful with the meta- and object-language distinction, the same is true of proof if we have something other than a formal notion.

Provability and truth

The most common argument that mathematical truth is not provability uses Tarski’s indefinability of truth theorem or Goedel’s first incompleteness theorem. But while this is a powerful argument, it won’t convince an intuitionist who rejects the law of excluded middle. Plus it’s interesting to see if a different argument can be constructed.

Here is one. It’s much less conclusive than the Tarski-Goedel approach. But it does seem to have at least a little bit of force. Sometimes we have experimental evidence (at least of the computer-based kind) for a mathematical claim. For instance, perhaps, you have defined some probabilistic setup, and you wonder what the expected value of some quantity Q is. You now set up an apparatus that implements the probabilistic setup, and you calculate the average value of your observations of Q. After a billion runs, the average value is 3.141597. It’s very reasonable to conclude that the last digit is a random deviation, and that the mathematically expected value of Q is actually π.

But is it reasonable to conclude that it’s likely provable that the expected value of Q is π? I don’t see why it would be. Or, at least, we should be much less confident that it’s provable than that the expected value is π. Hence, provability is not truth.

Incompleteness

For years in my logic classes I’ve been giving a rough but fairly accessible sketch of the fact that there are unprovable arithmetical truths (a special case of Tarski’s indefinability of truth), using an explicit Goedel sentence using concatenation of strings of symbols rather than Goedel encoding and the diagonal lemma.

I’ve finally revised the sketch to give the full First Incompleteness theorem, using Rosser’s trick. Here is a draft.

Logicism and Goedel

Famously, Goedel’s incompleteness theorems refuted (naive) logicism, the view that mathematical truth is just provability.

But one doesn’t need all of the technical machinery of the incompleteness theorems to refute that. All one needs is Goedel’s simple but powerful insight that proofs are themselves mathematical objects—sequence of symbols (an insight emphasized by Goedel numbering). For once we see that, then the logicist view is that what makes a mathematical proposition true is that a certain kind of mathematical object—a proof—exists. But the latter claim is itself a mathematical claim, and so we are off on a vicious regress.

Provability from finite and infinite theories

Let #s be the Goedel number of s. The following fact is useful for thinking about the foundations of mathematics:

Proposition. There is a finite fragment A of Peano Arithmetic such that if T is a recursively axiomatizable theory, then there is an arithmetical formula P_T(n) such that for all arithmetical sentences s, A → P_T(#s) is a theorem of FOL if and only if T proves s.

The Proposition allows us to replace the provability of a sentence from an infinite recursive theory by the provability of a sentence from a finite theory.

Sketch of Proof of Proposition. Let M be a Turing machine that given a sentence as an input goes through all possible proofs from T and halts if it arrives at one that is a proof of the given sentence.

We can encode a history of a halting (and hence finite) run of M as a natural number such that there will be a predicate H_M(m, n) and a finite fragment A of Peano Arithmetic independent of M (I expect that Robinson arithmetic will suffice) such that (a) m is a history of a halting run of M with input m if and only if H_M(m, n) and (b) for all m and n, A proves whether H_M(m, n).

Now, let P_T(n) be ∃mH_M(m, n). Then A proves P_T(#s) if and only if there is an m₀ such that A proves H_M(m₀, n). (If A proves P_T(#s), then because A is true, there is an m such that H_M(m, #s), and then A will prove H_M(m₀, #s). Conversely, if A proves H_M(m₀, #s), then it proves ∃mH_M(m, #s).) And so A proves P_T(#s) if and only if T proves s.

A bad idea in the foundations of mathematics

The relativity of FOL-validity is the fact that whether a sentence ϕ of First Order Logic is valid (equivalently, provable from no axioms beyond any axioms of FOL itself) sometimes depends on the axioms of set theory, once we encode validity arithmetically as per Goedel.

More concretely, if Zermelo-Fraenkel-Choice (ZFC) set theory is consistent, then there is an FOL formula ϕ that is FOL-provable according to some but not other models of ZFC. So which model of ZFC should real provability be relativized to?

Here is a putative solution that occurred to me today:

• Say that ϕ is really provable if and only if there is a model M of ZFC such that according to M, ϕ has a proof.

If this solution works, then the relativity of proof is quite innocent: it doesn’t matter in which model of ZFC our proofs live, because proofs in any ZFC model do the job for us.

It follows from incompleteness (cf. the link above) that real provability is strictly weaker than provability, assuming ZFC is true and consistent. Therefore, some really provable ϕ will fail to be valid, and hence there will be models of the falsity of ϕ. The idea that one can really prove a ϕ such that there is a model of the falsity of ϕ seems to me to show that my proposed notion of “really provable” is really confused.

Post-Goedelian mathematics as an empirical inquiry

Once one absorbs the lessons of the Goedel incompleteness theorems, a formalist view of mathematics as just about logical relationships such as provability becomes unsupportable (for me the strongest indication of this is the independence of logical validity). Platonism thereby becomes more plausible (but even Platonism is not unproblematic, because mathematical Platonism tends towards plenitude, and given plenitude it is difficult to identify which natural numbers we mean).

But there is another way to see post-Goedelian mathematics, as an empirical and even experimental inquiry into the question of what can be proved by beings like us. While the abstract notion of provability is subject to Goedelian concerns, the notion of provability by beings like us does not seem to be, because it is not mathematically formalizable.

We can mathematically formalize a necessary condition for something to be proved by us which we can call “stepwise validity”: each non-axiomatic step follows from the preceding steps by such-and-such formal rules. To say that something can be proved by beings like us, then, would be to say that beings like us can produce (in speech or writing or some other relevantly similar medium) a stepwise valid sequence of steps that starts with the axioms and ends with the conclusion. This is a question about our causal powers of linguistic production, and hence can be seen as empirical.

Perhaps the surest way to settle the question of provability by beings like us is for us to actually produce the stepwise valid sequence of steps, and check its stepwise validity. But in practice mathematicians usually don’t: they skip obvious steps in the sequence. In doing so, they are producing a meta-argument that makes it plausible that beings like us could produce the stepwise valid sequence if they really wanted to.

This might seem to lead to a non-realist view of mathematics. Whether it does so depends, however, on our epistemology. If in fact provability by beings like us tracks metaphysical necessity—i.e., if B is provable by beings like us from A₁, ..., A_n, then it is not possible to have A₁, ..., A_n without B—then by means of provability by beings like us we discover metaphysical necessities.

The centrality of the natural numbers

The more I think about the foundations of mathematics, the more wisdom I see in Kronecker’s famous saying: “God made the natural numbers; all else is the work of man.” There is something foundationally deep about the natural numbers. We see this in the way theories of natural numbers is equivalent (e.g., via Goedel encoding) to the theories of strings of symbols that are central to logic, and in the way that when we fix our model of natural numbers, we fix the foundational notion of provability.

Independence of FOL-validity

A sentence ϕ of a dialect of First Order Logic is FOL-valid if and only if ϕ is true in every non-empty model under every interpretation. By the Goedel Completeness Theorem, ϕ is valid if and only if ϕ is a theorem of FOL (i.e., has a proof from no axioms beyond any axioms of FOL). (Note: This does not use the Axiom of Choice since we are dealing with a single sentence.)

Here is a meta-logic fact that I think is not as widely known as it should be.

Proposition: Let T be any consistent recursive theory extending Zermelo-Fraenkel set theory. Then there is a sentence ϕ of a dialect of First Order Logic such that according to some models of T, ϕ is FOL-valid (and hence a theorem of FOL) and according to other models of T, ϕ is not FOL-valid (and hence not a theorem of FOL).

Note: The claim that ϕ is FOL-valid according to a model M is shorthand for the claim that a certain complex arithmetical claim involving the Goedel encoding of ϕ is true according to M.

The Proposition is yet another nail in the coffins of formalism and positivism. It tells us that the mere notion of FOL-theoremhood has Platonic commitments, in that it is only relative to a fixed family of universes of sets (or at least a fixed model of the natural numbers or a fixed non-recursive axiomatization) does it make unambiguous sense to predicate FOL-theoremhood and its lack. Likewise, the very notion of valid consequence, even of a finite axiom set, carries such Platonic commitments.

Proof of Proposition: Let G be a Rosser-tweaked Goedel sentence for T with G being Σ₁ (cf. remarks in Section 51.3 here). Then G is independent of T. In ZF, and hence in T, we can prove that there is a Turing machine Q that halts if and only if G holds. (Just make Q iterate over all natural numbers, halting if the number witnesses the existential quantifier at the front of the Σ₁ sentence G.) But one can construct an FOL-sentence ϕ such that one can prove in ZF that ϕ is FOL-valid if and only if Q halts (one can do this for any Turing machine Q, not just the one above). Hence, one can prove in T that ϕ is FOL-valid if and only if I holds.

Thus, in T it is provable that ϕ is FOL-valid if and only if G holds. But T is a consistent theory (otherwise one could formalize in T the proof of its inconsistency). Since G is independent of T, it follows that the FOL-validity of ϕ is as well.

Logical closure accounts of necessity

A family of views of necessity (e.g., Peacocke, Sider, Swinburne, and maybe Chalmers) identifies a family F of special true statements that get counted as necessary—say, statements giving the facts about the constitution of natural kinds, the axioms of mathematics, etc.—and then says that a statement is necessary if and only if it can be proved from F. Call these “logical closure accounts of necessity”. There are two importantly different variants: on one “F” is a definite description of the family and on the other “F” is a name for the family.

Here is a problem. Consider:

1. Statement (1) cannot be proved from F.

If you are worried about the explicit self-reference in (1), I should be able to get rid of it by a technique similar to the diagonal lemma in Goedel’s incompleteness theorem. Now, either (1) is true or it’s false. If it’s false, then it can be proved from F. Since F is a family of truths, it follows that a falsehood can be proved from truths, and that would be the end of the world. So it’s true. Thus it cannot be proved from F. But if it cannot be proved from F, then it is contingently true.

Thus (1) is true but there is a possible world w where (1) is false. In that world, (1) can be proved from F, and hence in that world (1) is necessary. Hence, (1) is false but possibly necessary, in violation of the Brouwer Axiom of modal logic (and hence of S5). Thus:

2. Logical closure accounts of necessity require the denial of the Brouwer Axiom and S5.

But things get even worse for logical closure accounts. For an account of necessity had better itself not be a contingent truth. Thus, a logical closure account of necessity if true in the actual world will also be true in w. Now in w run the earlier argument showing that (1) is true. Thus, (1) is true in w. But (1) was false in w. Contradiction! So:

3. Logical closure accounts of necessity can at best be contingently true.

Objection: This is basically the Liar Paradox.

Response: This is indeed my main worry about the argument. I am hoping, however, that it is more like Goedel’s Incompleteness Theorems than like the Liar Paradox.

Here's how I think the hope can be satisfied. The Liar Paradox and its relatives arise from unbounded application of semantic predicates like “is (not) true”. By “unbounded”, I mean that one is free to apply the semantic predicates to any sentence one wishes. Now, if F is a name for a family of statements, then it seems that (1) (or its definite description variant akin to that produced by the diagonal lemma) has no semantic vocabulary in it at all. If F is a description of a family of statements, there might be some semantic predicates there. For instance, it could be that F is explicitly said to include “all true mathematical claims” (Chalmers will do that). But then it seems that the semantic predicates are bounded—they need only be applied in the special kinds of cases that come up within F. It is a central feature of logical closure accounts of necessity that the statements in F be a limited class of statements.

Well, not quite. There is still a possible hitch. It may be that there is semantic vocabulary built into “proved”. Perhaps there are rules of proof that involve semantic vocabulary, such as Tarski’s T-schema, and perhaps these rules involve unbounded application of a semantic predicate. But if so, then the notion of “proof” involved in the account is a pretty problematic one and liable to license Liar Paradoxes.

One might also worry that my argument that (1) is true explicitly used semantic vocabulary. Yes: but that argument is in the metalanguage.

Necessity is not provability

A plausible account of necessity is that p is necessary provided that p can be proved in the correct logical system K and p is possible provided that its negation cannot be proved. Assuming K is axiomatizable and proves enough of the axioms of arithmetic, this account can be shown to be incorrect.

Fix any sentence s in K. It follows from Goedel's Second Incompleteness theorem that there is no K-proof of s's being K-unprovable (for if there were such a proof, then it would follow that there is a K-proof of K's consistency, since if K is inconsistent, then every sentence, including s, can be proved in K). But on the account of modality under consideration, this means that it is possible that s is K-provable, i.e., it is possible that s is necessary.

In other words, this account of modality implies that every sentence is possibly necessary. But it is absurd to think that 0=1 is possibly necessary!

I think much the same reasoning can be used to disprove Swinburne's account of necessity, since where we are not dealing with directly referential rigid designators, Swinburne's account agrees with the provability account.

I am skirting over distinctions between s and its Goedel number, but I think that's a mere technicality to work out in greater precision.

This makes for a nice way to see a relationship between the two incompleteness theorems. The first one tells us that not everything true is provable. From the second we learn that not even everything necessary is provable.