[bibtex key=”FreireHamkins:Bi-interpretation-in-weak-set-theories”]

Abstract. In contrast to the robust mutual interpretability phenomenon in set theory, Ali Enayat proved that bi-interpretation is absent: distinct theories extending ZF are never bi-interpretable and models of ZF are bi-interpretable only when they are isomorphic. Nevertheless, for natural weaker set theories, we prove, including Zermelo-Fraenkel set theory $\newcommand\ZFCm{\text{ZFC}^-}\ZFCm$ without power set and Zermelo set theory Z, there are nontrivial instances of bi-interpretation. Specifically, there are well-founded models of ZFC- that are bi-interpretable, but not isomorphic — even $\langle H_{\omega_1},\in\rangle$ and $\langle H_{\omega_2},\in\rangle$ can be bi-interpretable — and there are distinct bi-interpretable theories extending ZFC-. Similarly, using a construction of Mathias, we prove that every model of ZF is bi-interpretable with a model of Zermelo set theory in which the replacement axiom fails.

Set theory exhibits a robust mutual interpretability phenomenon: in a given model of set theory, we can define diverse other interpreted models of set theory. In any model of Zermelo-Fraenkel ZF set theory, for example, we can define an interpreted model of ZFC + GCH, via the constructible universe, as well as definable interpreted models of ZF + ¬AC, of ZFC + MA + ¬CH, of ZFC + $\mathfrak{b}<\mathfrak{d}$, and so on for hundreds of other theories. For these latter theories, set theorists often use forcing to construct outer models of the given model; but nevertheless the Boolean ultrapower method provides definable interpreted models of these theories inside the original model (explained in theorem 7). Similarly, in models of ZFC with large cardinals, one can define fine-structural canonical inner models with large cardinals and models of ZF satisfying various determinacy principles, and vice versa. In this way, set theory exhibits an abundance of natural mutually interpretable theories.

Do these instances of mutual interpretation fulfill the more vigourous conception of bi-interpretation? Two models or theories are mutually interpretable, when merely each is interpreted in the other, whereas bi-interpretation requires that the interpretations are invertible in a sense after iteration, so that if one should interpret one model or theory in the other and then re-interpret the first theory inside that, then the resulting model should be definably isomorphic to the original universe (precise definitions in sections 2 and 3). The interpretations mentioned above are not bi-interpretations, for if we start in a model of ZFC+¬CH and then go to L in order to interpret a model of ZFC+GCH, then we’ve already discarded too much set-theoretic information to expect that we could get a copy of our original model back by interpreting inside L. This problem is inherent, in light of the following theorem of Ali Enayat, showing that indeed there is no nontrivial bi-interpretation phenomenon to be found amongst the set-theoretic models and theories satisfying ZF. In interpretation, one must inevitably discard set-theoretic information.

Theorem. (Enayat 2016)

1. ZF is solid: no two models of ZF are bi-interpretable.
2. ZF is tight: no two distinct theories extending ZF are bi-interpretable.

The proofs of these theorems, provided in section 6, seem to use the full strength of ZF, and Enayat had consequently inquired whether the solidity/tightness phenomenon somehow required the strength of ZF set theory. In this paper, we shall find support for that conjecture by establishing nontrivial instances of bi-interpretation in various natural weak set theories, including Zermelo-Fraenkel theory $\ZFCm$, without the power set axiom, and Zermelo set theory Z, without the replacement axiom.

Main Theorems

1. $\ZFCm$ is not solid: there are well-founded models of $\ZFCm$ that are bi-interpretable, but not isomorphic.
2. Indeed, it is relatively consistent with ZFC that $\langle H_{\omega_1},\in\rangle$ and $\langle H_{\omega_2},\in\rangle$ are bi-interpretable.
3. $\ZFCm$ is not tight: there are distinct bi-interpretable extensions of $\ZFCm$.
4. Z is not solid: there are well-founded models of Z that are bi-interpretable, but not isomorphic.
5. Indeed, every model of ZF is bi-interpretable with a transitive inner model of Z in which the replacement axiom fails.
6. Z is not tight: there are distinct bi-interpretable extensions of Z.

   These claims are made and proved in theorems 20, 17, 21 and 22. We shall in addition prove the following theorems on this theme:

7. Well-founded models of ZF set theory are never mutually interpretable.
8. The Väänänen internal categoricity theorem does not hold for $\ZFCm$, not even for well-founded models.

These are theorems 14 and 16. Statement (8) concerns the existence of a model $\langle M,\in,\bar\in\rangle$ satisfying $\ZFCm(\in,\bar\in)$, meaning $\ZFCm$ in the common language with both predicates, using either $\in$ or $\bar\in$ as the membership relation, such that $\langle M,\in\rangle$ and $\langle M,\bar\in\rangle$ are not isomorphic.

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