ZFC without Power Set II: Reflection Strikes Back

PDF   Bibtex   Arχiv

V. Gitman and R. Matthews, “ZFC without Power Set II: Reflection Strikes Back,” Manuscript, 2022.

Many natural set-theoretic structures satisfy all the axioms of ${\rm ZFC}$ excluding the power set axiom. These include the structures $H_{\kappa^+}$ (the collection of all sets whose transitive closure has size at most $\kappa$ for a cardinal $\kappa$), forcing extensions of models of ${\rm ZFC}$ by pretame (but not tame) class forcing, and first-order structures bi-interpretable with models of the strong second-order set theory Kelley-Morse together with the choice scheme. The set theory that these structures satisfy is the theory ${\rm ZFC}^-$, whose axioms consist of the axioms of ${\rm ZFC}$ with the collection scheme in place of the replacement scheme and with the well-ordering principle (the assertion that every set can be well-ordered) in place of the axiom of choice (the assertion that every non-empty family of sets has a choice function). The reason for the particular choice of axioms comprising ${\rm ZFC}^-$ is that without the existence of power sets we lose certain equivalences between set theoretic assertions that we tend to take for granted.

Definition:

Szczepaniak showed that the axiom of choice is not equivalent to the well-ordering principle over ${\rm ZF}^-$ (see [1]) and therefore we choose to take the stronger principle when formulating the theory ${\rm ZFC}^-$. Zarach showed that the theory ${\rm ZFC}\text{-}$ does not imply the collection scheme [2]. The first author et al. showed in [3] that the theory ${\rm ZFC}\text{-}$ has many other undesirable behaviors: there are models of ${\rm ZFC}\text{-}$ in which $\omega_1$ is singular, in which every set of reals is countable but $\omega_1$ exists, and in which the Łoś Theorem fails for (class) ultrapowers.

Although the theory ${\rm ZFC}^-$ avoids these pathological behaviors, there is a number of useful properties of models of full ${\rm ZFC}$ that fail or are not known to hold in models of ${\rm ZFC}^-$, mostly as a consequence of the absence in these models of a hierarchy akin to the von Neumann hierarchy. It is known that ground model definability, the assertion that the model is definable in its set forcing extensions, can fail in models of ${\rm ZFC}^-$ [4]. The intermediate model theorem, the assertion that any intermediate model between the model and its set-forcing extension is also its set-forcing extension, can fail [5]. If there is a non-trivial elementary embedding $j\colon V_{\lambda+1}\to V_{\lambda+1}$, namely the large cardinal axiom $I_1$ holds, then it gives rise to an elementary embedding $j^+ \colon H_{\lambda^+}\to H_{\lambda^+}$ which witnesses that Kunen's Inconsistency can fail for models of ${\rm ZFC}^-$ [6]. It is an open question whether ${\rm HOD}$, the collection of all hereditarily ordinal definable sets, is definable in models of ${\rm ZFC}^-$.

One of the main themes of this article is the various ways in which the scheme version of dependent choice can fail in models of ${\rm ZFC}^-$.

Definition: The ${\rm DC}_\delta$-scheme, for an infinite cardinal $\delta$, asserts for every formula $\varphi(x,y,a)$ that if for every set $x$, there is a set $y$ such that $\varphi(x,y,a)$ holds, then there is a function $f$ on $\delta$ such that for every $\xi\lt\delta$, $\varphi(f\upharpoonright\xi,f(\xi),a)$ holds. The ${\rm DC}_{\lt\rm{Ord}}$-scheme is the scheme asserting that the ${\rm DC}_\delta$-scheme holds for every cardinal $\delta$.

In other words, the ${\rm DC}_\delta$ schemes states that we can make $\delta$-many dependent choices along any definable relation without terminal nodes. The ${\rm DC}_\delta$-scheme generalizes the dependent choice axiom ${\rm DC}_\delta$ which makes the analogous assertion for set relations. The ${\rm DC}_{\lt\rm{Ord}}$-scheme follows from ${\rm ZFC}$ by reflecting the definable relation in question to some $V_\alpha$, and then using a well-ordering of $V_\alpha$ to obtain the sequence of dependent choices. It follows that the ${\rm DC}_{\lt\rm{Ord}}$-scheme holds in every structure $H_{\kappa^+}$. Also, forcing extensions of a model of ${\rm ZFC}$ by pretame forcing satisfy the the ${\rm DC}_{\lt\rm{Ord}}$-scheme.

The ${\rm DC}_\delta$-schemes have numerous applications. Over ${\rm ZFC}^-$, the ${\rm DC}_{\omega}$-scheme is equivalent to the reflection principle which is the assertion that every formula reflects to a transitive set (missing reference). Although, there is no known reformulation of the ${\rm DC}_\delta$-scheme for uncountable $\delta$ in terms of a reflection principle, such a reformulation exists under mild existence of power set assumptions. Over ${\rm ZFC}^-$, for a regular cardinal $\delta$, the ${\rm DC}_\delta$-scheme implies that every proper class surjects onto $\delta$. It is not difficult to see that in a model of ${\rm ZFC}$ the class partial order ${\rm Add}({\rm Ord},1)$, whose conditions are partial functions on the ordinals ordered by extension, forces a global well-order without adding sets. The forcing ${\rm Add}({\rm Ord},1)$ does not add sets because it is set-distributive. In a model of ${\rm ZFC}$, the forcing ${\rm Add}({\rm Ord},1)$ is set-distributive if and only if the ${\rm DC}_{\lt\rm{Ord}}$-scheme holds. Thus, in a model of ${\rm ZFC} + {\rm DC}_{\lt\rm{Ord}}$-scheme we can force a global well-order without adding sets, and actually conversely if we can force a global well-order without adding sets using some forcing, then ${\rm Add}({\rm Ord},1)$ must be set-distributive. Thus, over models of ${\rm ZFC}^-$, the ${\rm DC}_{\lt\rm{Ord}}$-scheme is equivalent to being able to force a global well-order without adding sets. We will see another application of the ${\rm DC}_{\lt\rm{Ord}}$-scheme shortly to establishing a form of Kunen's Inconsistency for models of ${\rm ZFC}^-$.

Friedman et al. (missing reference) showed that the ${\rm DC}_\omega$-scheme can fail in a model of ${\rm ZFC^-}$. The counterexample model is the $H_{\omega_1}$ of a symmetric submodel of a forcing extension by the iteration of Jensen's forcing along the tree $\omega_1^{\lt\omega}$, in particular, $\omega$ is the largest cardinal in this model.

In this article we obtain the following failures of the various ${\rm DC}_\delta$-schemes in models of ${\rm ZFC}^-$.

Theorem 1: Suppose that $V\models{\rm ZFC}+{\rm CH}$. Then every Cohen forcing extension of $V$ has a proper class transitive submodel satisfying ${\rm ZFC}^-$ in which the ${\rm DC}_{\omega}$-scheme holds, but the ${\rm DC}_{\omega_2}$-scheme fails.

The model was constructed by Zarach in [1], but we show that the ${\rm DC}_{\omega_2}$-scheme fails there. The status of the ${\rm DC}_{\omega_1}$-scheme in this model is open. Note that this model, unlike the counterexample model of (missing reference), must have unboundedly many cardinals by virtue of being a proper class transitive submodel of a model of ${\rm ZFC}$.

Theorem 2: Suppose that $V \models {\rm ZFC} + 2^\delta = \delta^+$ for some regular cardinal $\delta$. Then every forcing extension of $V$ by the poset ${\rm Add}(\delta,1)$ has a proper class transitive submodel satisfying ${\rm ZFC}^-$ in which the ${\rm DC}_\delta$-scheme holds, but the ${\rm DC}_{\delta^{++}}$-scheme fails.

The model above is constructed by generalizing Zarach's construction, however in these models we are unable to determine the status of the ${\rm DC}_{\delta^+}$-scheme. On the other hand, we are able to remove this gap by using Jensen's forcing instead.

Theorem 3: Every forcing extension of $L$ by the $\omega$-length finite support product of Jensen's forcing has a proper class submodel satisfying ${\rm ZFC}^-$ in which the ${\rm DC}_\omega$-scheme holds, but the ${\rm DC}_{\omega_1}$-scheme fails.

Theorem 4: Suppose that $\kappa$ is inaccessible in $L$. Every forcing extension of $L$ by the $\kappa$-length bounded support product of the inaccessible Jensen's forcing has a proper class transitive submodel satisfying ${\rm ZFC}^-$ in which the ${\rm DC}_{\kappa}$-scheme holds, but the ${\rm DC}_{\kappa^+}$-scheme fails.

The inaccessible Jensen's forcing is a generalization of Jensen's forcing to a sub-poset of the $\kappa$-Sacks forcing, for an inaccessible cardinal $\kappa$, that was recently developed in (missing reference). Using this idea of union models we extend the result of (missing reference) to obtain a model of ${\rm ZFC}^-$ in which the reflection principle fails and for which there are unboundedly many cardinals.

Theorem 5: Every forcing extension of $L$ by the iteration of Jensen's forcing along the class tree ${\rm Ord}^{\lt\omega}$ has a proper class transitive submodel $N$ satisfying ${\rm ZFC}^-$, with unboundedly many cardinals, in which the ${\rm DC}_{\omega}$-scheme fails.

The results of this article were originally motivated by an open question from the work of the second author on Kunen's Inconsistency in models of ${\rm ZFC}^-$ [6]. Suppose that $W\models{\rm ZFC}^-$ and $A\subseteq W$. We will say that $W\models{\rm ZFC}^-_A$ if $W$ continues to satisfy ${\rm ZFC}^-$ in the language expanded by a predicate for $A$.

Theorem 6: Suppose that $W\models{\rm ZFC}^-$. There is no non-trivial, cofinal, $\Sigma_0$-elementary embedding $j \colon W\to W$ such that $V_{\text{crit}(j)}$ exists in $W$ and $W\models{\rm ZFC}^-_j$.

Thus, in particular, the elementary embedding $j^+ \colon H_{\lambda^+}\to H_{\lambda^+}$ resulting from an $I_1$-embedding $j \colon V_{\lambda+1}\to V_{\lambda+1}$ cannot be cofinal. Moreover, the second author showed that if the model $W$ additionally satisfies the ${\rm DC}_{\lt\rm{Ord}}$-scheme, then the existence of $V_{\text{crit}(j)}$ follows from the other assumptions [6]. Thus, we have:

Theorem 7: Suppose that $W \models {\rm ZFC}^- + {\rm DC}_{\lt\rm{Ord}}$-scheme. There is no non-trivial, cofinal, $\Sigma_0$-elementary embedding $j \colon W\to W$ such that $W\models{\rm ZFC}^-_j$.

As a step toward determining whether the existence of $V_{\text{crit}(j)}$ is truly necessary for Theorem 6, the second author asked:

Question: Suppose that $W\models{\rm ZFC}^-_j$ for a non-trivial, cofinal $j \colon W\to M$ with $M\subseteq W$. Does $V_{\text{crit}(j)}$ exist in $W$, does $\mathcal P (\omega)$ exist in $W$?

We answer the question negatively here using models of ${\rm ZFC}^-$ in which the ${\rm DC}_\delta$-scheme fails for some $\delta$.

Theorem: There is a model $W\models{\rm ZFC}^-$ in which $\mathcal P (\omega)$ does not exist and which has a definable, non-trivial, cofinal, elementary embedding $j \colon W\to M\subseteq W$.

References

  1. A. Zarach, “Unions of ${\rm ZF}^{-}$-models which are themselves ${\rm ZF}^{-}$-models,” in Logic Colloquium ’80 (Prague, 1980), vol. 108, Amsterdam: North-Holland, 1982, pp. 315–342.
  2. A. M. Zarach, “Replacement $\nrightarrow$ collection,” in Gödel ’96 (Brno, 1996), vol. 6, Berlin: Springer, 1996, pp. 307–322.
  3. V. Gitman, J. D. Hamkins, and T. A. Johnstone, “What is the theory $\mathsf {ZFC}$ without power set?,” MLQ Math. Log. Q., vol. 62, no. 4-5, pp. 391–406, 2016.
  4. V. Gitman and T. A. Johnstone, “On ground model definability,” in Infinity, Computability, and Metamathematics: Festschrift in honour of the 60th birthdays of Peter Koepke and Philip Welch, London, GB: College publications, 2014.
  5. C. Antos, S.-D. Friedman, and V. Gitman, “Boolean-valued class forcing,” Fund. Math., vol. 255, no. 3, pp. 231–254, 2021.
  6. R. Matthews, “Taking Reinhardt’s Power Away,” Manuscript, 2020.