[FOM] Arithmetical soundness of ZFC (platonic)

[Timothy Y. Chow]

On Wed, 3 Jun 2009, Calvin Jongsma wrote:
> Panu Raatikainen's claim, which I believe to be correct, was that 
> Zermelo's axioms were proposed to systematize his reasoning behind the 
> Well Ordering Theorem in response to his earlier critics rather than to 
> systematize set theory as a whole or mathematical practice in general.  
> Nor was it advanced first of all to handle paradoxes.  I believe Greg 
> Moore's book Zermelo's Axiom of Choice takes these things up, though I 
> don't have a copy at hand to check chapter and verse.

O.K., I'll look into this more carefully...it sounds interesting.

Still, I think that (as often happens on FOM!) my main point is being lost 
because of a correct but largely tangential objection.  The main point was 
that Zermelo's set theory was formulated with neither of the two attitudes 
listed by Weaver, namely maximal caution and maximal incaution.  Whether 
Zermelo was solely motivated by trying to justify his well-ordering 
theorem or whether he was also interested in the general problem of 
axiomatizing set theory in an antinomy-free manner, it is still the case 
that he was taking a first crack at formalizing a pre-existing piece of 
mathematics.  Thus we should not be surprised that his system wasn't the 
weakest possible system (even for formalizing the argument he was most 
interested in), nor should we dismiss his effort as "probably unsound" 
because he was throwing caution to the wind.

Tim