Consistency and consistency strength of certain special cuts in $I\Delta_0$

Recall that $$I\Delta_0$$ is the theory in the language of arithmetic that consists of the axioms of $$\mathsf{PA}$$ with induction restricted to $$\Delta_0$$ formulas (i.e., formulas where all quantifiers are bounded).

It is not too difficult to build a model $$M$$ of $$I\Delta_0$$ with a proper cut $$N \subset M$$ such that $$M \equiv N$$. Moreover, we can ensure that $$2^n$$ exists for every $$n \in N$$ and that $$2^x$$ is not a total function on $$N$$ (and therefore not a total function on $$M$$ either). It is also a standard fact that the set $$\{ x \in M : (\exists y \in N) x < 2^y\}$$ is a cut that models $$I\Delta_0$$ (this follows from the fact that it is downwards closed and closed under addition and multiplication). My question is about this last cut actually being all of $$M$$.

Let $$C$$ be a unary predicate symbol, and let $$T$$ be the theory in the language of arithmetic augmented with $$C$$ that contains $$I\Delta_0$$ and says that

• $$C$$ is downwards closed,

• $$C$$ is not all of the structure,

• $$2^x$$ is not a total function,

• $$2^x$$ is total on $$C$$,

• the induced structure on $$C$$ has the same theory as the full structure (in particular, it is a model of $$I\Delta_0$$), and

• for every $$x$$, there exists a $$y \in C$$ such that $$x < 2^y$$.

Obviously, the fifth bullet point must be stated as an axiom scheme.

Question 1. Is $$T$$ consistent?

Question 2. Does $$I\Delta_0$$ interpret $$T$$?

Question 3. Does $$T$$ interpret $$I\Delta_0 + \mathrm{Exp}$$ (where $$\mathrm{Exp}$$ is the statement that $$2^x$$ is a total function)?

Note that since $$I\Delta_0$$ does not interpret $$I\Delta_0 + \mathrm{Exp}$$, these last two questions cannot both have a positive answer.

• In the introduction, you required that $2^x$ exists in $M$ for every $x\in N$. This condition is missing in $T$ (and this will make a lot of difference). Is this omission intentional? Mar 31 at 7:22
• No that was a mistake. Mar 31 at 8:01

$$T$$ is inconsistent. The argument below is due to Robert Solovay (it is attributed to a letter from Solovay to Nelson in Visser’s Peano Basso and Peano Corto, see Lemma 3.7).
Let $$2^x_n$$ denote the iterated exponential function $$2^x_0=x$$, $$2^x_{n+1}=2^{2^x_n}$$. It is well known that the graph of $$2^x_n$$ has a well-behaved $$\Delta_0$$ definition in $$I\Delta_0$$.
Lemma (Solovay): $$I\Delta_0+\neg\mathrm{Exp}$$ proves that there exists a unique number $$n$$ such that $$2^0_n$$ exists, but $$2^0_{n+1}$$ does not.
Proof: Take $$x$$ such that $$2^x$$ does not exist, and let $$n$$ be maximal such that $$2^0_n\le x$$. Then $$2^0_{n+1}$$ either does not exist, or satisfies $$2^0_{n+1}>x$$, hence $$2^0_{n+2}$$ does not exist. Thus, $$n$$ or $$n+1$$ satisfies the conclusion of the Lemma. QED
Let us call the $$n$$ from the Lemma as Solovay’s number.
Now, observe that the axioms of $$T$$ imply that $$C=\{x:2^x\text{ exists}\}$$. Thus, working in $$T$$ (which includes $$I\Delta_0+\neg\mathrm{Exp}$$), if $$n$$ is Solovay’s number, then $$n-1$$ is Solovay’s number in $$C$$. In particular, the universe and $$C$$ disagree on the truth of the sentence “Solovay’s number is even”, contradicting the elementary equivalence axiom.