[FOM] ZC vs. ZFC: a pedagogical perspective

[Jeremy Bem]

In recent threads, I've been arguing for ZC on philosophical grounds.
My argument in a nutshell: The so-called universe of ordinary
mathematics, V_{omega+omega}, can be viewed as a model of ZC in the
usual sense of first-order logic.  This is not true of the von Neumann
universe and ZFC.  For me, this outweighs the fact that replacement is
reasonably intuitive and has some interesting consequences.

More personally, a peculiarity of my education is that I was
introduced to ZFC without having independently encountered any math
that requires replacement.  As such, ZFC was under-motivated.
Moreover, the accompanying definition of V (as a union over all
ordinals) stretched my previous understanding of mathematical rigor,
yet was supposed to convince me that ZFC was consistent.  At least,
that was my understanding of its role at the time.

This experience diminished my appreciation of axiomatic set theory.
Had replacement been omitted, and the smaller universe used instead,
then not only would the axiom system have been better motivated, but
the consistency argument would have seemed rigorous to me (modulo
choice).

It seems to me that my experience is more-or-less the natural
consequence of a standard mathematical education.  Is that a fair
description?  What was your experience?  What would a better
explanation of ZFC look like?

On what basis are the consequences of replacement considered important
enough to complicate our foundations, but not important enough to
include in a standard curriculum?

(I studied logic at Berkeley, where I was required to pass exams on
analysis and algebra -- but I don't believe that any of that material
required replacement.)

-Jeremy