Randall Holmes has made a quite convincing argument *against* the fact that the full axiom schema of replacement should be considered as “intuitively obvious”—even though he does believe ZFC to be probably consistent. Instead, he claims that only $\Sigma_2$ replacement axioms should be considered as obvious.

As far as I understand it, the rationale behind believing in the full axiom schema of replacement is the belief that *any* collection of sets (whether it is first-order definable or not) the same size of a set should be a set. We will call this intuitive belief “limitation of size” (LOS)—but note that it is not an axiom strictly speaking, because we are thinking in Platonist, not formal, terms.

At this point, it is important to notice that, similarly, the rationale behind believing in the axiom schema of separation is that we believe that *all* the “subsets” (in the intuitive sense) of a set are sets too! I will call this belief the “Platonist subset” (PS) “axiom”.

Now, if we believe in a Platonist world of sets in which all the axioms of ZFC are satisfied, with separation replaced by PS and replacement replaced (!) by LOS, the collection of all the ordinals of this world has the structure of a inaccessible (in the Platonist sense) cardinal. For that reason, I think that anyone who believes in the consistency of ZFC because they believe in the Platonist truth of LOS should also believe in the consistency of (ZFC + inaccessible cardinal).

But one might also believe in the existence of an inaccessible cardinal without believing in LOS! My question is, would that belief (together with the Platonist belief in PS and in $\Sigma_2$ replacement) imply that you believe in the *consistency* of ZFC? Well, this question is actually ill-stated; but it is essentially equivalent to the perfectly precise following question, which is the point of my post:

**Is (ZC + $\Sigma_2$ replacement + inaccessible cardinal) equiconsistent with (ZFC + inaccessible cardinal)?**