From Tarski to G\"odel. Or, how to derive the Second Incompleteness Theorem from the Undefinability of Truth without Self-reference

In this paper, we provide a fairly general self-reference-free proof of the Second Incompleteness Theorem from Tarski's Theorem of the Undefinability of Truth.


Prelude
Self-reference is a great and beautiful thing, but it may be interesting to see what one can do without it. In this paper, we provide a self-reference-free proof of the Second Incompleteness Theorem from Tarski's Theorem of the Undefinability of Truth. Thus, we at least reduce the self-referential arguments needed for the Second Incompleteness Theorem and Tarski's Theorem of the Undefinability of Truth taken together.
We take self-reference, in this paper, as broadly as possible. There are all kind of subtle discussions that we wish to side-step: are Yablo-style constructions selfreference? does the Grelling paradox involve true self-reference? etcetera. We want to avoid even the slightest smell of diagonalization, self-reference, the recursion theorem, circularity or majorization.
1.1. Motivation. Why is a proof of Gödel's Second Incompleteness Theorem from Tarski's Theorem interesting? Our result is a step in a program to find selfreference-free proofs both of the Second Incompleteness Theorem and of Tarski's Theorem. I do not think there are, at present, convincing self-reference-free proofs -at least, not in the very broad sense of self-reference discussed above-of either the Second Incompleteness Theorem or of Tarski's Theorem. There is a selfreference-free proof, due to Kotlarski, of Tarski's Theorem for the case of Uniform Biconditionals. See [Kot04]. However, Kotlarski's attempted proof for non-uniform biconditionals fails. Moreover, Kotlarski's proof uses majorization which could be viewed as a form of diagonalization.
Why is the quest for a self-reference-free proof of these two great theorems important? Well, first there is the 'because the question is there' argument. It is always annoying when we have in essence just one kind of proof of a great theorem. Secondly, a different proof may lead to new generalizations and new insights.
For example, this research could help us find a solution of Jan Krajíček's problem to find a proof of the non-interpretability of the extension PC(A) of a consistent finitely axiomatized sequential theory A with predicative comprehension in A itself that does not run via the Second Incompleteness Theorem.
1.2. About the Proof. Our proof is roughly the result of combining a proof plan due to Kreisel (see [Smo77]) with a proof plan due to Adamowic and Bigorajska (see [AB01]). I give two variants of the proof. The first avoids the use of the third Löb condition to wit that provability implies provability of provability. The second does use the third Löb condition. Since the two variants share a lot of text, I will represent the first variant as the main line and I will give the additions for the second line in typewriter font.
Remark 1.1. The ideal would be to study a new proof in a setting that is so general that we do not have the resources to even prove things like the Gödel Fixed Point Lemma. We do not aspire to this ideal here. The theories studied all have the resources to do diagonalization.
1.3. Prerequisites. The full proof presupposes some knowledge of relevant materials from the text books [HP93] and [Kay91]. For the benefit of the reader who is not acquainted with weak theories we sketch the proof for the case of Peano Arithmetic in Section 2. This special case has the advantage that almost all technical complications disappear, while the essence of the proof-idea is still visible.
We will employ, for the full proof, the notations and conventions and elementary facts of [Vis17a] to which the present paper is a sequel. We will use the Interpretation Existence Lemma. For a careful exposition of this last result, see [Vis17b]. Finally, we will make use of a result from [Vis14].

The Second Incompleteness Theorem for Peano Arithmetic
We prove Second Incompleteness Theorem for Peano Arithmetic PA. Let the standard axiomatization of PA be π. We write ✷ π A for the arithmetization of the provability of A from π and ✸ π A := ¬✷ π ¬A for the consistency of (the theory axiomatized by) π extended with A.
We will use Tarski's Theorem of the Undefinability of Truth for the case of PA.

Theorem 2.2. For any arithmetical formula
Proof of Theorem 2.1 from Theorem 2.2. We assume Theorem 2.2. Suppose, in order to arrive at a contradiction, that PA ⊢ ✸ π ⊤.
Let S be a maximal set of Σ 0 1 -sentences such that PA + S is consistent. Let M be a model of PA + S.
We apply the Henkin Construction as described in e.g. [Fef60] (see also [Vis17b]), to construct an internal Henkin model H(M) of PA based on ✸ π ⊤. We claim that M and H(M) satisfy the same Σ 0 1 -sentences, to wit the Σ 0 1 -sentences of S.
We may conclude that H(M) |= S. On the other hand, by the maximality of S, the models M and H(M) cannot satisfy more Σ 0 1 -sentences than those in S. Since, H(M) again will contain ✸ π ⊤ we can repeat the construction H to obtain HH(M).
We claim that H(M) and HH(M) are elementary equivalent. The reason is as follows. The Henkin construction is based on yes-no-decisions that depend on the truth or falsity of Σ 0 1 -questions. On the standard level it, thus, depends on the truth or falsity of Σ 0 1 -sentences. Since M and H(M) satisfy the same Σ 0 1 -sentences we find that, on standard levels, the same choices are made in the Henkin construction and hence that H(M) and HH(M) are elementary equivalent.
The Henkin construction provides an internal truth-predicate H such that we have PA ⊢ H( B ) ↔ B H . So we find: But this contradicts Theorem 2.2 on the undefinability of truth. ✷ We note that our construction effectively yields a ∆ 0 2 -truth-predicate H such that is an arithmetical sentence} is consistent, in case PA proves its own consistency.
We also note that the proof works on the weaker assumption that PA + S ⊢ ✸ π ⊤. So, it follows that ✷ π ⊥ is in any maximal set of Σ 0 1 -sentences S such that PA + S is consistent.

Statement of Two Theorems
In this section, we state both Second Incompleteness Theorem and Tarski's Theorem of the Undefinability of Truth in the general forms we consider in the present paper.
3.1. Our Version of the Second Incompleteness Theorem. We work with S 1 2 as our basic basic weak arithmetic. See [Bus86] or [HP93]. We use ✷ σ for the arithmetization of provability from axiom set σ and ✸ σ for ¬✷ σ ¬.
We will prove the following version of the Second Incompleteness Theorem.
1 -formula ] that defines an axiom set of U in the standard model. Suppose that N : U ✄ (S 1 2 + ✸ σ ⊤). Then U is inconsistent. The basic idea of a version of the Second Incompleteness Theorem using interpretations is due to Feferman [Fef60]. Of course, Feferman, around 1960, was thinking of PA as base theory rather than S 1 2 . The variant where σ is Σ 0 1 is stronger than the one where σ is Σ b 1 . However, there are easy arguments to reduce the Second Incompleteness Theorem for Σ 0 1axiomatizations to the Second Incompleteness Theorem for Σ b 1 -axiomatizations. See [Vis17a].

Our Version of Tarski's Theorem.
We employ the following version of Tarski's Theorem. Let Θ be a signature. Suppose N : U ✄ R, where R is the very weak arithmetic introduced in [TMR53]. We take TB N,A Θ to be the set of all Tarski biconditionals A( B ) ↔ B, for B a Θ-sentence and B an N -numeral.
Theorem 3.2. Let U be a theory of signature Θ. Suppose N : U ✄ R and, for some U -formula A, we have U ⊢ TB N,A Θ . Then, U is inconsistent. We can reduce our version of Tarski's Theorem to the following special case. This reduction uses the recursion theorem, so since we want to exclude anything that even smells of self-reference we, probably, have to exclude this reduction from our main line of argument. This is a pity since it would have been nice to reduce all instances of the Second Incompleteness Theorem to one single application of self-reference. Here is our argument for the reduction of Theorem 3.2 to Theorem 3.3. Suppose N : U ✄ R and U ⊢ TB N,A Θ . Let ν be the translation associated with N . We interpret R + TB ID,T A in U via the translation ν ⋆ which is ν on the arithmetical vocabulary and which is A(tr ν ⋆ (v)) on T. Here, tr ν ⋆ is the arithmetization of the function B → B ν ⋆ . We evidently need the recursion theorem to make our definition work.

The Henkin Construction
The main tool of our proof will be the Interpretation Existence Lemma. In this section we state this lemma and collect the relevant facts around it.
For a proof of this result see [Vis17b] which also discusses the history of the result.

The Proof
Theorem 4.3 tells us that, in order to to prove Theorem 3.1, it is sufficient to For the first line, we use the following fact.
We now take S a maximal set of Σ 0 1,1 -sentences such that U + S K is consistent. Let M be a model of U + S K . Let J be a K-internally definable cut between K(M) and KHK(M), where H := H[σ]. We have: Step (2) holds since K(M) |= W and the definition of J is K-internal. We have Step (3), since Σ 0 1,1 -sentences are upwards preserved from cuts.
By the maximality of S, it follows that S ∈ S iff K(M) |= S and S ∈ S iff KHK(M) |= S. We may conclude that K(M) and KHK(M) are Σ 0 1,1 -elementary equivalent. Thus, we have our desired result with K in the role of N .
[ We treat the second line. Suppose σ is a Σ b 1 -formula that represents the axioms of U in the standard model. Suppose N : U ✄ (S 1 2 + ✸ σ ⊤). Let S be a maximal set of ∃Σ b 1 -sentences such that U + S N is consistent. Let H := H[σ]. We have, by verifiable ∃Σ b 1 -completeness in S 1 2 , for S ∈ S, that: Now let M be a model of U +S N . It follows that HN (M) |= S N . By maximality, it follows that, for any S ∈ ∃Σ b 1 , we have N (M) |= S ⇔ s ∈ S and N HN (M) |= S ⇔ s ∈ S.

So we are done. ]
We note that the second line looks somewhat more efficient. However, the first line avoids the more refined syntactic analysis that is the basis of the second line.
Open Question 5.2. Our argument is presented as a model-theoretic argument. So, it is itself not obviously formalizable in a weak theory. However, it seems to me that the models can be eliminated from the argument. They mainly function as a heuristic tool. So, the question is how much resources do we need to internalize our argument in a theory. Is S 1 2 sufficient? We note that, since Löb's Principle follows, in the classical case, from the Second Incompleteness Theorem and since the Third Löb Condition follows from Löb's Principle, an internalized version of the first variant of our proof implies the Third Löb Condition.
A closely related question is whether our argument can be made constructive. This seems, at first sight, rather hopeless because of the radically non-constructive character of the Henkin construction. However, one can reduce the question the Second Incompleteness Theorem for constructive theories to the Second Incompleteness Theorem for classical theories. Now if we can make the argument completely theory-internal we would be there.