TITLE: The difference between KP+Inf+Pow and Z
QUESTION [5 upvotes]: I'd like to clarify some details about the theories Z (= Zermelo set theory) and KP+=KP+Infinity+Powerset (KP is Kripke-Platek set theory).
In this paper (M1), Mathias claims that Z+KP is consistent relative to Z; in particular, this implies that KP+ does not prove Con(Z). Moreover, by explicitly working in Z+KP, Mathias implies that Z does not contain KP. For the former fact he refers to this other paper (M2); however, I can't find that fact in (M2). But I could be missing something obvious - there are a ton of subtheories of ZFC discussed in (M2), and the claim could easily be a trivial consequence of one of the theorems Mathias proves, even if it's not stated explicitly.
I'd like to first double-check these claims:

Question 1: Is it true that Z+KP is consistent relative to Z, and that Z does not contain KP? (If the answer to the first point is "yes", is this proved in (M2)? If not, where?)

And then ask a couple related questions, comparing the strengths of Z and KP.
First proof-theoretically:

Question 2: Does Z prove Con(KP+)?

And then set-theoretically:

Question 3: Is the smallest height of a transitive model of KP+ strictly smaller than the smallest height of a transitive model of Z?

I suspect the answers to all three questions are "yes": that is, Z is not stronger than KP+, but it is "morally" stronger. However, I would like to check this.

REPLY [4 votes]: Questions 2 and 3 both have a negative answer. 
For 3, $V_{\omega + \omega}$ is a model of Z but KP + Inf proves the existence of all computable ordinals, so the height of a transitive model of KP+ must be much taller than $\omega + \omega$.
For 2, we in fact have that KP+ proves the consistency of Z. This is because from Powerset, $\Sigma_1$-Replacement, and the existence of $V_\omega$ we can prove the existence of $V_{\omega + \omega}$.