[FOM] ZC vs. ZFC: a pedagogical perspective

[Jeremy Bem]

Replying to Panu and Tim together:

I can't really "recall" that all my analysis and algebra can be done
in ACA_0, because I'm not familiar with it.  Thanks for the
information.

I didn't really mean to say that "the proper motivation for an
axiomatic system for set theory is that it should be the most
parsimonious system that suffices to formalize the mathematics that
you've already encountered".  I see that that's a reasonable
interpretation of what I said, which is too bad.  I was aiming to be
more casual and narrative in this thread.  I'm saying that the
definition of V seemed shockingly non-rigorous to me, in taking a
"union over all ordinals".  It felt like being told (at far too late
an age) to accept a new religion.  To realize in retrospect that it
could have been avoided, and I'd have learned a system perfectly
capable of formalizing all the math I'd ever known -- with a
consistency proof meeting my previous standards for rigor -- makes it
even more shocking to me.

As for the multiplicity of systems: I'm currently interested in
arguing that *if* there is to be a "standard" foundation for
mathematics, ZC is a better choice than ZFC.  I think this is true and
important.

I'm open to Tim's proposal that we shouldn't bless any system as
"standard", but maybe we can leave that aside for now?  It's already
making me nervous that I started this more open-ended thread at all.

-Jeremy