[FOM] Arithmetical soundness of ZFC (platonic)

[praatika@mappi.helsinki.fi]

Lainaus "Timothy Y. Chow" <tchow at alum.mit.edu>:

> 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.

More or less... (various paradoxes were discovered independently by  
Cantor, Hilbert, and Zermelo himself, already before Russell. And I  
don't know how much Zermelo even knew about Frege, or Russell; he was  
mainily interested in Cantorian set theory. But that is not really  
important here.)

> Replacement and Choice are added later in
> order to capture certain arguments that don't go through in Z.

Set theoretical arguments - not arguments in "actual mathematical  
practice". The reason I press this is that Cantorian set theory was,  
at the time, extremely controversial, and I just feel it is misleading  
to count it, at the time, as "actual mathematical practice". But this  
is probably just a verbal issue...

> 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.

OK, but see above (set theory vs. actual mathematical practice)

> 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.

Set-theoretic reasoning, yes. (But I don't think my point was just  
anachronistic. The great majority of the mathematicians just did not  
accept set theoretical reasoning back then.)

But I think our apparent disagreement turns out to be largely verbal.

All the Best,

Panu



Panu Raatikainen

Ph.D., Academy Research Fellow,
Docent in Theoretical Philosophy

Department of Philosophy
University of Helsinki
Finland


E-mail: panu.raatikainen at helsinki.fi

http://www.mv.helsinki.fi/home/praatika/