Does $\omega$-consistency depend on the encoding?

Question

A given theory $T$ can interpret the language of arithmetic in different ways, and therefore it seems to me like it is possible that in one of these ways $T$ would be $\omega$-consistent but not in the other.

Assuming PA is $\omega$-consistent, there seems to be an easy way to construct such a theory: Let $T$ have two sorts and put the axioms of PA (Peano Arithmetic) for one of the sorts and PA+not(Con(PA)) for the other. Then depending on which sort we use to interpret arithmetic we would either get something $\omega$-consistent or not.

Is this correct? If so, $\omega$-consistency seems like an ill-defined concept. How do we deal with this issue? A similar question can arise if we have some model of some theory and want to know whether $Con(T)$ for some $T$ is true for that model, as the answer depends on the encoding.

Answer 1

This is a great question! In general, "reasonably strong" theories will always have distinguished implementations (the jargon is actually interpretations, but meh) of the natural numbers, and it is with respect to these that we make sense of things like $\omega$-consistency.

Let's look at $\mathsf{ZFC}$ for concreteness. We can prove in $\mathsf{ZFC}$ that there is a smallest model of (say) Robinson's arithmetic $\mathsf{Q}$. (Here "smallest" is meant in terms of initial segment embeddability.) In particular, if $\mathcal{M}$ is a model of $\mathsf{ZFC}$, $\mathbb{N}^\mathcal{M}$ is the finite ordinals of $\mathcal{M}$ construed as a model of arithmetic in the usual way, and $X$ is some other "implementation" of a model of $\mathsf{Q}$ inside $\mathcal{M}$, we have in $\mathcal{M}$ that there is a unique initial segment embedding $\mathbb{N}^\mathcal{M}\rightarrow X$. Moreover, we have - again $\mathsf{ZFC}$-provably - that any two such implementations are isomorphic.

So there is a very precise sense in which the "obvious" way of implementing the naturals inside $\mathsf{ZFC}$ is the best possible.