# Infinite descending consistency chains

What are some examples of consistent theories $$T_i$$ (extending elementary arithmetic EA) such that for $$∀i∈ℕ \,\, T_i ⊢ \mathrm{Con}(T_{i+1})$$?

Such theories exist; see for example An infinitely descending sequence of sound theories each proving the next consistent. However, in the link, the theories are given using (non-well-founded) self-reference, which in an intuitive informal sense, leaves one wondering what exactly is each theory asserting.

Thus, I am especially interested in natural or intuitive examples that do not use self-reference. Here is what I have.

If we only require $$\mathrm{PA} ⊬ (\mathrm{Con}(T_{i+1}) ⇒ \mathrm{Con}(T_i))$$, then for an example, fix a proof system that is constructively provably in $$\mathrm{EA}$$ polynomial time equivalent to sequent calculus. Let $$n$$ (provably in EA) be the number of bits in the shortest inconsistency in $$\mathrm{PA}$$ if there is any. If $$T_i = \mathrm{EA} + \mathrm{Con}(\mathrm{PA}) ∨ 2^i ∤ ⌊\log \log \log n⌋$$, then for all $$i$$, $$T_i ⊢ T_{i+1}$$ and $$\mathrm{PA} ⊬ (\mathrm{Con}(T_{i+1}) ⇒ \mathrm{Con}(T_i))$$. (To get the unpredictability of $$n$$, the proof relies on $$\mathrm{PA}$$ provability of $$\mathrm{PA}$$ bounded consistency in polynomial time, and the impossibility of a subpolynomial proof length of bounded consistency.)

To get $$T_i ⊢ \mathrm{Con}(T_{i+1})$$, I have two potential natural examples, but I do not have a proof that they work.
Update: Per Fedor Pakhomov's answer, the original constructions (given in brackets for reference) did not work; below are the modified versions. See the answers for constructions that have been shown to work; the answers complement each other.

Construction 1: Let $$S_0 = \mathrm{PA}$$ and $$S_{i+1} = \mathrm{PA} + \mathrm{Con}(S_i)$$, and let $$h_{ε_0}$$ be the $$ε_0$$ function in the Hardy hierarchy. Set $$T_i = \mathrm{PA} + ∀n \,\, (h_{ε_0 2}(i+n) \text{ exists} ⇒ \mathrm{Con}(S_n))$$ [the original used $$h_{ε_0}$$, but perhaps even $$h_{ε_0 2} = λx.h_{ε_0}(h_{ε_0}(x))$$ is too small]

Construction 2: Let $$S_0 = \mathrm{PA}$$ and $$S_{i+1} = \mathrm{PA} + \mathrm{Con}(S_i)$$. We can arrange that $$⌈\mathrm{Con}(S_i)⌉$$ is $$i^{O(1)}$$, but we only need $$2^{2^{2^{O(i)}}}$$. Let $$\mathrm{Con}_m(A)$$ mean that $$A$$ has no inconsistency shorter than $$m$$ bits.
Set $$T_i = \mathrm{PA} + ∀n \, (\mathrm{Con}_{2^{n^i}}(S_{n+i}) ⇒ \mathrm{Con}(S_n))$$ [the original used $$S_n$$ in place of $$S_{n+i}$$].
Note: If we can show in $$\mathrm{PA}$$ that each $$T_i$$ (for $$i>0$$) is consistent relative to $$S=∪S_i$$, then the construction works because if $$S$$ is inconsistent, then working in $$T_i$$, $$T_{i+1}$$ is equivalent to $$S_{i'}$$ for a low enough $$i'$$ so as to be consistent.

Also, the well-ordering of $$Π^0_1$$ ordinals imposes fundamental limits on descending consistency chains. I suspect that for sound $$T$$, $$∃i \, T_i ⊬ ∀j \, (T_j ⊢ \mathrm{Con}(T_{j+1}))$$.


Let $$\psi_i$$ be the assertion that there are $$n-i$$ many inaccessible cardinals, where $$n$$ is the size of the least proof of a contradiction from ZFC plus a measurable cardinal, if this theory is indeed inconsistent.

Let me assume that ZFC plus a measurable cardinal is consistent. In this case, $$\ZFC+\neg\Con(\ZFC+\exists\text{ measurable})$$ is also consistent, and in any model of ZFC plus $$\neg\Con(\ZFC+\exists\text{ measurable})$$, the proof of a contradiction from that theory will be necessarily nonstandard. So $$n$$ will be nonstandard when it exists. Since $$i$$ is standard, however, the subtraction $$n-i$$ is sensible and we will never hit zero.

Furthermore, in any model of $$\neg\Con(\ZFC+\exists\text{ measurable})$$, each $$\psi_i$$ directly implies the truth and indeed the consistency of $$\psi_{i+1}$$, since with $$n-i$$ inaccessible cardinals we will have an extra one, and can therefore make a model with only $$n-(i+1)$$ inaccessible cardinals, as desired, simply by chopping off at the biggest one.

Thus, the theories $$\ZFC+\neg\Con(\ZFC+\exists\text{ measurable})+\psi_i$$ are descending in consistency strength.

Of course, you can use other large cardinals, if you like, and form additional similar examples.

• I believe that one can also use the ideas of the universal algorithm (see jdh.hamkins.org/…), using essentially the assertions that the algorithm has $i$ successful stages. The nature of the extension property is that every model in which the algorithm has exactly $i$ stages thinks it is consistent (over a nonstandard fragment of its $\text{PA}$) that it has exactly $i+1$ stages. So this is an infinite descent. – Joel David Hamkins Jan 13 at 16:09
• This is a good example. Note that ZFC + $ψ_i$ also works (since provably in ZFC, if a measurable is consistent, then so are ZFC + $ψ_i$); an optional change is to have $ω$ inaccessibles if a measurable is consistent. Also, the unsound $T_i$ = ZFC + $ψ_i$ + ¬Con(ZFC+measurable) proves not only consistency but $Π^V_2$ soundness of $T_{i+1}$, which (per the link in Fedor Pakhomov's answer) would not be possible for even $Π^0_3$ soundness for $Π^0_3$ sound $T_i$. – Dmytro Taranovsky Jan 14 at 8:38
• (correction to my comment) $Π^V_2$-soundness should have been $Σ^V_2$-soundness (though there are variations for $Π^V_n$-soundness). – Dmytro Taranovsky Jan 14 at 10:39

I know two constructions of the chains of this sort that aren't based on explicit diagonalization.

In a recent work by James Walsh and me https://arxiv.org/abs/1805.02095 we gave an example (Theorem 3.8) of a chain of theories $$T_0,T_1,\ldots$$ that have even stronger property $$T_i\vdash \mathsf{RFN}_{\Sigma_1}(T_{i+1})$$. The theory $$T_i$$ is defined to be $$\mathsf{EA}+\mbox{there is a proof p of a false \Sigma_1 sentence in \mathsf{I\Sigma}_2 s.t. \mathsf{RFN}^{p\mathop{\dot -}i}_{\Sigma_1}(\mathsf{EA}) holds''}.$$

In the construction above one could replace $$\mathsf{EA}$$ with $$\mathsf{I}\Sigma_1$$ and sustain the same property. But in the case of $$\mathsf{I}\Sigma_1$$ the $$T_i$$'s could be defined in an alternative way that makes them very similar to your Construction 1. Let $$R_{\mathsf{I\Sigma_2}}$$ be the following fast-growing function \begin{aligned} R_{\mathsf{I\Sigma_2}}\colon x \longmapsto \sup \{ y\mid & y\mbox{ is the least witness for a \Sigma_1 sentence provable in } \\ & \mbox{\mathsf{I\Sigma_2} by a proof with the Gödel number \le x}\}\end{aligned}. Alternatively $$T_i$$'s could be defined as $$T_i\colon \mathsf{I}\Sigma_1+\forall x\; (R_{\mathsf{I}\Sigma_2}(x+i)\mbox{ is defined } \Rightarrow \mathsf{RFN}^x_{\Sigma_1}(\mathsf{I}\Sigma_1))+R_{\mathsf{I}\Sigma_2}\mbox{ isn't total}.$$

Although, the similarity, your Construction 1 seems not to work. Using some manipulations with models of arithmetic (injecting inconsistencies, and cuts in large non-standard intervals) if I am not missing anything, it is possible to construct a model $$\mathfrak{M}$$ of $$T_0$$ such that $$\mathfrak{M}\not\models \mathsf{Con}(S_a)$$, for some non-standard $$a\in \mathfrak{M}$$. But it is easy to observe that $$\mathfrak{M}\not\models \mathsf{Con}(T_1)$$. Assume for a contradiction that $$\mathfrak{M}\models \mathsf{Con}(T_1)$$. Since $$\mathsf{PA}\vdash \forall x\;\mathsf{Prv}_{\mathsf{PA}}(h_{\varepsilon_0}(x)\mbox{ is defined})$$, we would have $$\mathfrak{M}\models \mathsf{Con}(T_1+h_{\varepsilon_0}(a+1)\mbox{ is defined})$$. But the latter would imply that $$\mathfrak{M}\models \mathsf{Con}(\mathsf{PA}+\mathsf{Con}(S_a))$$ and hence $$\mathfrak{M}\models \mathsf{Con}(S_a)$$, contradiction.

I don't see how you thought to achieve the desired effect with your Construction 2. Let us consider a model $$\mathfrak{M}$$ of $$T_1$$ where we have $$\mathfrak{M}\not\models \mathsf{Con}(S_a)$$, for some non-standard $$a$$ (such a model could be constructed using injecting inconsistencies theorem). Without loss of generality we could assume that $$a$$ is the least non-standard number with this property. Assume for a contradiction that $$\mathfrak{M}\models \mathsf{Con}(T_2)$$. It is easy to see that for any $$b we have $$\mathfrak{M}\models \mathsf{Prv}_{\mathsf{PA}}(\mathsf{Con}_{2^{b^2}}(S_b))$$. Hence we have $$\mathfrak{M}\models \mathsf{Con}(T_2+\mathsf{Con}_{2^{(a-1)^2}}(S_{a-1}))$$. Thus we have $$\mathfrak{M}\models \mathsf{Con}(\mathsf{PA}+\mathsf{Con}(S_{a-1}))$$ and therefore $$\mathfrak{M}\models \mathsf{Con}(S_a)$$, contradiction.

A more natural example of a descending chain that I know is based on the notion of slow consistency. For a computable function $$f$$ let $$\mathsf{PA}\upharpoonright f =\{\mathsf{I}\Sigma_x\mid f(x) \mbox{ is defined}\}.$$ In the case when $$f$$ is total but not provably total in some theory $$T$$ the theory $$\mathsf{PA}\upharpoonright f$$ from the external point of view just coincide with $$\mathsf{PA}$$, but $$T$$ might not be able to prove that $$\mathsf{PA}$$ and $$\mathsf{PA}\upharpoonright f$$ coincide. The sentences $$\mathsf{Con}(\mathsf{PA}\upharpoonright f)$$ are known as slow consistency sentences and were introduced by Friedman, Rathjen, and Weiermann. For rationals $$q>0$$ let me consider the functions $$f_{q}(x)=\mathsf{h}_{\varepsilon_0}([qx])$$. Let $$T_q=\mathsf{I}\Sigma_1+\mathsf{Con}(\mathsf{PA}\upharpoonright f_q)$$. The claim is that for $$p>q>0$$ we have $$T_q\vdash \mathsf{Con}(T_p)$$. Let me reason in $$T_q$$. Actually I will present a model-theoretic argument that isn't directly available in an extension of $$\mathsf{I\Sigma}_1$$, but with some additional care it could be formalized in $$\mathsf{WKL}_0+\mathsf{Con}(\mathsf{PA}\upharpoonright f_q)$$ and then transfered to $$T_q$$ by the arithmetical conservativity of $$\mathsf{WKL}_0$$ over $$\mathsf{I}\Sigma_1$$. First assume that $$f_q$$ is total, this actually is equivalent to $$\Sigma_1$$-soundness of $$\mathsf{PA}$$, and it is fairly easy to prove consistency of $$T_p$$ in this case. Now we assume that there is the number $$a$$ s.t. $$f_q(a)$$ is defined but $$f_q(a+1)$$ is not. Since we externally know that $$f_q$$ is total, for any given $$N$$ we could prove in $$T_q$$ that $$a>N$$. By taking $$N$$ to be large enough we prove in $$T_q$$ that there is a number $$b$$ s.t. $$f_p(b)$$ is undefined and $$[bp]<[ap]$$. Clearly we have $$\mathsf{PA}\upharpoonright f_q=\mathsf{I}\Sigma_a$$. We construct a model $$\mathfrak{M}$$ of $$\mathsf{I}\Sigma_a$$. If $$f_p(a)$$ is undefined in $$\mathfrak{M}$$, then we are done since $$\mathfrak{M}\models \mathsf{PA}\upharpoonright f_p\subseteq \mathsf{I}\Sigma_{a-1}$$, $$\mathsf{I}\Sigma_a\vdash \mathsf{Con}(\mathsf{I}\Sigma_{a-1})$$ and hence $$\mathfrak{M}\models \mathsf{Con}( \mathsf{PA}\upharpoonright f_p)$$. Otherwise, in $$\mathfrak{M}$$ we have a non-standard interval $$[f_p(b),f_p(a)]$$. Using the fact that $$[pb]<[pa]$$ it is easy to show that $$\mathfrak{M}\models A(f_p(b)), where $$A(x)$$ is the diagonal of Ackermann's function. From Sommer's results it follow that there is a cut $$\mathfrak{J}\models \mathsf{I}\Sigma_1$$ of $$\mathfrak{M}$$ such that $$f_p(b)<\mathfrak{J}. Clearly we have $$\mathfrak{J}\models \mathsf{Con}(\mathsf{I}\Sigma_{a-1})\;\;\;\mbox{ and }\;\;\;\mathfrak{J}\models f_p(a)\mbox{ is undefined}$$ and hence $$\mathfrak{J}\models \mathsf{Con}(\mathsf{PA}\upharpoonright f_p)$$.

As for your last conjecture. It is known that for any sound arithmetical theory $$U$$ and a recursive chain $$T_i$$ of r.e. extensions of $$U$$ it couldn't be the case that $$U\vdash \forall x\;(\mathsf{Prv}_{T_x}(\mathsf{Con}(T_{x+1})))$$ (this is due to Friedman, Smorynski, and Solovay; in my paper with Walsh that I mentioned earlier we give a simple proof of this (Theorem 3.6); you could find reference to the earlier works about it there just before Theorem 3.6).

Through this answer I mentioned several times construction of models of arithmetic as cuts of non-standard intervals and construction of non-standard models of arithmetic by injecting inconsistencies in other non-standard models of arithmetic. Here are the relevant references:

R. Sommer. Transfinite induction within Peano arithmetic. Annals of Pure and Applied Logic, 76(3):231 – 289, 1995.

J. Krajíček and P. Pudlák. On the structure of initial segments of models of arithmetic. Archive for Mathematical Logic, 28(2):91–98, 1989.

• Thank you for the detailed answer. With its help, I added a new answer; let me know if there are gaps in the proof. Also, in the first $T_i$ you give, does "equivalently given as ..." work as EA does not appear to have enough induction to get the least counterexample to the totality of $R_{\mathrm{IΣ}_2}$? Also, near the end of the answer (about the conjecture) "some $T_i$" should be "all $T_i$" (since we can make $T_0$ arbitrarily strong). – Dmytro Taranovsky Jan 14 at 9:11
• @DmytroTaranovsky Indeed, you are right the alternative definition of $T_i$'s in the first example, shouldn't be equivalent to the initial one; one would need to use $\mathsf{I}\Sigma_1$ as the base theory to overcome this problem. – Fedor Pakhomov Jan 14 at 9:51
• @DmytroTaranovsky And you are right about the verification of the fact that a chain of theories is $<_{\mathsf{Con}}$-descending. The theorem in the paper by Walsh and me was that this couldn't be verified in $\mathsf{EA}$ (I don't remember the precise formulations Smorynski and Solovay used). This could be generalize to the result that for any sound r.e. arithmetical $U\supseteq \mathsf{EA}$ and recursive chain $T_i$ of extensions of $U$ it couldn't be the case that $U\vdash \forall x\;\mathsf{Prv}_{T_x}(\mathsf{Con}(T_{x+1}))$. But I don't know how to prove your conjecture. – Fedor Pakhomov Jan 14 at 10:02

Informed by Fedor Pakhomov's excellent answer, and using "slow" iterated $$Σ_1$$-soundness, here is an infinite sequence of sound theories $$T$$ such that $$T_i$$ = PA + 1-Con($$T_{i+1}$$).

Set $$T_i$$ = PA + "$$h'_{g(n)∸i}(n)$$ is total"
using fast-growing or Hardy hierarchy $$h$$ (either one works), with $$h'_α$$ being $$h_{ε_α}$$, and $$g(n) = {h'_{ω+1}}^{-1}(n)$$, where $$f^{-1}(n) = \max(m: ∀m' (or any reasonable variation on this). Also, $$a∸b = \max(a-b,0)$$.

Alternatively, almost any reasonable monotonic total recursive $$g$$ that is sufficiently slow-growing but tending to infinity works, but note that $$T$$ depends on $$g$$. The construction also generalizes to other theories extending $$Σ^0_1$$-PA (the proof uses $$Σ^0_1$$ induction) by replacing $$h'_m(n)$$ with a function corresponding to $$1+m$$ iterations of $$Σ_1$$-soundness.

Provably in PA, if $$g$$ is unbounded (equivalently, $$h'_{ω+1}$$ is total), then $$T_i$$ (and even PA + "$$h'_ω$$ is total") is $$Σ_1$$-sound. Thus, it suffices to prove that for each $$i$$, PA + "$$g$$ is bounded" proves "$$h'_{g(n)∸i}(n)$$ is total" $$⇔$$ 1-Con($$T_{i+1}$$). (However, the quantification over $$i$$ will be unprovable even in $$T_0$$).

Working in PA + "$$g$$ is bounded", let $$k$$ be the the maximum value of $$g$$. Since $$k$$ is (externally) nonstandard, $$k>i$$. Because 1-Con is unaffected by true $$Π^0_1$$ statements, in 1-Con(...), we can freely assume/assert that $$k$$ is the maximum value of $$g$$. Thus, 1-Con($$T_{i+1}$$) $$⇔$$ 1-Con(PA + "$$h'_{k-(i+1)}(n)$$ is total") $$⇔$$ "$$h'_{k-i}(n)$$ is total" (the latter follows from standard results in ordinal analysis), as required.

Addendum ($$Σ^1_1$$-soundness): Per the linked paper in Fedor Pakhomov's answer, the $$Σ^0_1$$ soundness cannot be improved to $$Σ^0_2$$ (equivalently $$Π^0_3$$) for c.e. (in $$i$$) $$T$$. However, if $$T$$ is not required to be computable enumerable in $$i$$, there is an infinite sequence $$T_i$$ of sound c.e. extensions of Z$$_2$$ such that each $$T_i$$ proves $$Σ^1_1$$-soundness of $$T_{i+1}$$. $$T_i$$ will be satisfied by every transitive model of Z$$_2$$ (and analogously for theories other than Z$$_2$$ extending $$Π^1_1$$-CA$$_0$$).
To see this, given a recursive ordering $$X$$, let $$T_a$$ state Z$$_2$$ plus existence of a real $$M$$ such that for all $$b ≤_X a$$, $$M_b$$ is a (countably coded) $$ω$$ model of Z$$_2$$, and for all $$c <_X b$$, $$M_c$$ is an element of (the model coded by) $$M_b$$. Such $$M$$ exists for all $$a$$ in the well-founded part of $$X$$, so if $$X$$ is a recursive pseudowellordering, then such $$M$$ must also exist for some $$a'$$ outside of the well-founded part of $$X$$. If $$a_i$$ with $$a_0=a'$$ is an infinite $$X$$-descending sequence, then $$T_{a_i}$$ proves $$Σ^1_1$$-soundness $$T_{a_{i+1}}$$. Also, I suspect that there are sound finitely axiomatizable $$T_i = Π^1_1\text{-CA}_0 + \mathrm{RFN}_{Σ^1_1}(T_{i+1})$$, and similarly with many $$Σ^1_1$$ strengthenings of $$\mathrm{RFN}_{Σ^1_1}$$.

• It is a nice example. And I don't see any gaps in the construction. – Fedor Pakhomov Jan 14 at 10:18