# ZFC without Power Set II: Reflection Strikes Back

V. Gitman and R. Matthews, “ZFC without Power Set II: Reflection Strikes Back,” *To appear in Fundamenta Mathematicae*, 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:**

- Let ${\rm ZF}\text{-}$ be the theory ${\rm ZF}$ with the power set axiom removed. That is, ${\rm ZF}\text{-}$ consists of the axioms: extensionality, empty set, pairing, unions, infinity, the foundation scheme, the separation scheme and the replacement scheme.
- Let ${\rm ZFC}\text{-}$ denote the theory ${\rm ZF}\text{-}$ plus the well-ordering principle.
- Let $\rm{ZF(C)}^-$ denote the theory $\rm{ZF(C)}^-$ plus the collection scheme.

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 $H_{\kappa^+}$. It is not known whether pretame class forcing over models of ${\rm ZFC}$ preserves the ${\rm DC}_{\delta}$-schemes, unless the forcing has no proper class-sized antichains.

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 [7]. 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. This is because, using ${\rm AC}$, any set can be coded as a subset of an ordinal and, by genericity, this subset will appear somewhere in the generic class function from ${\rm Ord}$ into $2$. We can then well-order the sets by comparing the least location in the generic function where a code appears. In a model of ${\rm ZFC}^-$, the forcing ${\rm Add}({\rm Ord}, 1)$ is pretame if and only if the ${\rm DC}_{\rm Ord}$-scheme holds. Thus, in a model of ${\rm ZFC}^- + {\rm DC}_{\rm Ord}$-scheme, we can force a global well-order without adding sets, and conversely if we can force a global well-order without adding sets using some forcing, then ${\rm Add}({\rm Ord}, 1)$ must be pretame. We will see another application of the ${\rm DC}_{\rm Ord}$-scheme shortly to establishing a form of Kunen's Inconsistency for models of ${\rm ZFC}^-$.

Friedman et al. [7] showed that the ${\rm DC}_\omega$-scheme can fail in a model of ${\rm ZFC^-}$. Moreover, this failure is witnessed by a $\Pi^1_2$ formula, which turns out to be the simplest complexity for which such a failure can occur (see [7] and Theorem VII.9.2 of [8] for more details). 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. The symmetric submodel in question satisfies ${\rm AC}_\omega$, but has a $\Pi^1_2$-definable failure of ${\rm DC}$, which translates to its $H_{\omega_1}$ having the requisite properties.

There are two principle difficulties in constructing such consistency results; having second-order definable failures of ${\rm DC}$ and satisfying the full axiom of choice. For example, it is an old result of Jensen that it is possible to produce models of ${\rm ZF}$ in which the axiom of choice for families of size at most $\delta$ holds (where $\delta$ is an arbitrary regular cardinal), but ${\rm DC}_\omega$ already fails. Furthermore, by Pincus, for any regular cardinal $\delta$ there is a model of ${\rm ZF} + {\rm DC}_\delta + \neg {\rm DC}_{\delta^+}$. We refer the reader to Chapter 8 of Jech's book on the axiom of choice, [9], for more details.

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. If we assume further that $V = L$ holds then the ${\rm DC}_{\omega_1}$-scheme additionally fails.

The model was constructed by Zarach in [1], and the failure of ${\rm DC}_{\omega_1}$ follows by a result of Blass on the Cohen forcing ${\rm Add}(\omega,1)$ . 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 ${\rm ZFC}$. The second model is constructed by generalizing Zarach's construction as well as generalizing Blass's result to the forcing ${\rm Add}(\delta,1)$.

**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. If we assume further that $V = L$ holds then the ${\rm DC}_{\delta^+}$-scheme also fails.

Using the idea of union models we extend the result of [7] to obtain a model of ${\rm ZFC}^-$ in which the reflection principle fails and for which there are unboundedly many cardinals.

**Theorem 3:**
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 4:** (Matthews [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 5:**
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 natural next step, the second author asked whether the existence of $V_{\text{crit}(j)}$ is truly necessary for Theorem 5.

**Question:**
Is the following situation consistent: There is a non-trivial, cofinal, elementary embedding $j : W \rightarrow W$ such that $W \models {\rm ZFC}^-_j$?

In private communications with the second author, Yair Hayut has shown that the above situation is inconsistent, that is there are no non-trivial, cofinal, elementary embeddings $j : W \rightarrow W$ for which $W \models {\rm ZFC}^-_j$. However, as an initial attempt to answering this question, the second author asked:

**Question**: Suppose that $W\models{\rm ZFC}^-_j$ for a non-trivial, cofinal, elementary embedding $j: W\to M$ 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 $j : W\to M\subseteq W$.

## References

- 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. - A. M. Zarach, “Replacement $\nrightarrow$ collection,” in
*Gödel ’96 (Brno, 1996)*, vol. 6, Berlin: Springer, 1996, pp. 307–322. - 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. - 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. - C. Antos, S.-D. Friedman, and V. Gitman, “Boolean-valued class forcing,”
*Fund. Math.*, vol. 255, no. 3, pp. 231–254, 2021. - R. Matthews, “Taking Reinhardt’s power away,”
*J. Symb. Log.*, vol. 87, no. 4, pp. 1643–1662, 2022. Available at: https://doi.org/10.1017/jsl.2022.9 - S.-D. Friedman, V. Gitman, and V. Kanovei, “A model of second-order arithmetic satisfying AC but not
DC,”
*J. Math. Log.*, vol. 19, no. 1, pp. 1850013, 39, 2019. - S. G. Simpson,
*Subsystems of second order arithmetic*, Second. Cambridge University Press, Cambridge; Association for Symbolic Logic, Poughkeepsie, NY, 2009, p. xvi+444. - T. J. Jech,
*The axiom of choice*. Amsterdam: North-Holland Publishing Co., 1973, p. xi+202.