[FOM] ZC vs. ZFC: a pedagogical perspective

[Timothy Y. Chow]

On Sun, 31 Jan 2010, Jeremy Bem wrote:
> 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.

To be clear about one point: There are a few "ordinary" facts that don't 
quite fit into ACA_0.  "Every vector space has a basis" is a simple 
example.  ACA_0 is fundamentally a system for doing mathematics with 
*countable* sets and can't talk about arbitrary sets.  So if you work in 
ACA_0 you must tacitly agree to restrict yourself to "countable" math.  In 
practice this turns out to be much less of a restriction than one might 
think, but it is a restriction.

> 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.

The trouble with proposing ZC is that ZC may seem shockingly non-rigorous 
to people who were brought up in a different religion.  Nowadays it is not 
uncommon to run into people who feel that only finitistic math or 
computable math is rigorous, and that all set theory is bunk.  Picking ZC 
will be better than picking ZFC for people whose psychological profile 
happens to match yours, but otherwise I don't think it will accomplish 
much.

Tim