In these slides (see especially slide 26), Steel emphasizes the phenomenon that for all known "natural" extensions of ZFC, the ordering by consistency strength agrees with the ordering by containment of arithmetic consequences. That is, if $T$ and $T'$ are natural extensions of ZFC, then $T \leq T'$ with respect to consistency strength if and only if $T'$ proves all the arithmetic consequences of $T$. Indeed, Steel points out that as long as $T$ and $T'$ have high enough consistency strength, the same could be said of their analytical consequences, not just their arithmetical consequences.

But I find it remarkable that the first-order theories of the strongest consistency strength known (which are not known to be inconsistent) are most naturally formulated in ZF without the axiom of choice. Namely, if you look at the theories listed at Cantor's attic, the highest consistency strength theories are ZF + Reinhardt cardinals or ZF + Berkeley cardinals (I'm being vague about how many of these cardinals are stated to exist). Of course you can get stronger theories by taking Con(ZF + Reinhardt) etc. in some language, but these are the basis of things.

My question is, basically: does the phenomenon Steel discusses continue to hold for natural extensions of ZF, rather than ZFC? In particular, does ZF + Reinhardt cardinals imply all the arithmetical consequences of ZFC + $I_0$ (the strongest large cardinal principle not known to be inconsistent with choice)? How about ZF + Berkeley cardinals?

**EDIT**
Steel says a little more about "naturalness" on p. 6, footnote 10 of the related paper. I'll state it using some notation:

- If $A$ is a set of statements in the language of set theory, let $\overline{A}$ denote the deductive closure of $A$
- If $A$ is a set of statements in the language of set theory, let $\mathrm{Con}^\mathrm{Fin}(A)$ denote the set of statements $\{\mathrm{Con}(\phi) \mid \phi \in A\}$ where $\mathrm{Con}(\phi)$ is the statement that $\phi$ is consistent.
- If $S,T$ are extensions of ZFC, write $S \leq_{\mathrm{Con}}^{\ast} T$ if $\mathrm{Con}^\mathrm{Fin}(S) \subseteq T$. Note by compactness that $S\leq_\mathrm{Con}^\ast T$ implies $S \leq_\mathrm{Con} T$, where $\leq_\mathrm{Con}$ is the usual consistency strength preorder.
- If $T$ is an extension of ZFC, let $(\Pi^0_1)_T$ denote the $\Pi^0_1$ consequences of $T$.

According to Steel, the reflection principle implies for any $T$ extending ZFC that $(\Pi^0_1)_T = \overline{\mathrm{Con}^\mathrm{Fin}(T)}$. This implies that

- $(\Pi^0_1)_S \subseteq (\Pi^0_1)_T \Leftrightarrow S \leq_\mathrm{Con}^\ast T$

That is, the ordering on extensions of ZFC by $\Pi^0_1$ consequences can be recast in terms of consistency statements. Steel says that for the "natural" extensions he's talking about, it turns out that $S \leq_\mathrm{Con} T \Leftrightarrow S \leq_\mathrm{Con}^\ast T$. In some sense this is just a re-statement of another version of the phenomenon, but I think it sheds some light.

I'm not sure whether all of this should go through with ZF in place of ZFC, but if it does, it would be interesting to know whether $ZFC +I_0 \leq_\mathrm{Con}^\ast ZF + \mathrm{Reinhardt}$ for example. Of course, this is still quite a ways from talking about containment of *all* arithmetic statements. I would be happy to say something about the $\Sigma^0_1$ statements -- I'm particularly keen (for kind of frivolous reasons) to know in $ZF + \mathrm{Reinhardt}$ that a given Turing machine will halt so long as $ZFC + I_0$ thinks it will halt.

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