[FOM] Arithmetical soundness of ZFC (platonic)

[Timothy Y. Chow]

Panu Raatikainen wrote:

>"Timothy Y. Chow" <tchow at alum.mit.edu>:
>> We got to ZFC from ZF and to ZF from Z,
>> and the extensions were motivated neither by the desire to find the 
>> most powerful axioms nor the desire to be totally safe from error.  A 
>> large part of the motivation was to capture actual mathematical 
>> practice as simply and elegantly as possible.
>
>I am afraid this is not true - unless by "actual mathematical  
>practice" one means Zermelo's own, at the time highly controversial  
>proof of the well-ordering theorem from 1904.

Hold on, there.  Wasn't the sequence of events roughly as follows?  Frege 
tries to ground all of mathematics in logic.  Russell punctures Frege's 
system with his famous paradox.  Zermelo comes up with Z as a way to avoid 
the set-theoretic antinomies.  Replacement and Choice are added later in 
order to capture certain arguments that don't go through in Z.

If so, then I still think my sketch is roughly correct.  The motivation 
was not to find the most parsimonious system for all of mathematics, nor 
to find the broadest possible axioms.  It was to capture mathematics in a 
relatively simple set of axioms.

>I don't think there existed any mathematical practice (unless one  
>counts the Cantorian intuitive set theory as such) which required  
>anything like the full power of even Z.

Your sentence here strikes me as anachronistic, in that it presupposes our 
modern, highly refined understanding of the proof-theoretic strength of Z,
and reads it back into a historical period when such knowledge did not 
exist.  My point is precisely that Z certainly was *not* the most 
parsimonious possible system for capturing mathematics.  It was a first 
attempt, a proof of concept.  Zermelo didn't pick his axioms to be as weak 
as possible and still be able to capture mathematics, nor did he pick them 
to be as powerful as possible ("anything goes").  He picked them to 
capture set-theoretic reasoning in a (hopefully) consistent manner, during 
a period when it was becoming clear that set theory could serve as a 
foundation for all of mathematics.

Tim