[FOM] Arithmetical soundness of ZFC (platonic)

Timothy Y. Chow tchow at alum.mit.edu
Wed Jul 29 10:54:56 EDT 2009

In http://www.cs.nyu.edu/pipermail/fom/2009-May/013770.html I wrote:

>Nik Weaver wrote:
>> From what I've read on the FOM list, I get the impression that
>> people basically fall into two camps.  Some want to carefully
>> build foundations up from the bottom, starting with principles
>> in which we have complete confidence and demanding a thorough
>> justification of any proposed extension.
>> The other group takes an "anything goes" attitude, the idea being
>> that we should look for the most powerful axioms we can find, and
>> if something turns out to be inconsistent we give it up.  That's
>> fine, but my feeling is that if that is your approach then don't
>> claim that what you're doing is arithmetically sound, because if
>> your axioms weren't sound you wouldn't have any reliable way of
>> knowing this.
>Perhaps someone who is more familiar with the history of ZFC can correct 
>me if I'm wrong, but my impression is that the genesis of ZFC falls into 
>neither of these alleged camps.  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.

Panu Raatikainen suggested that my version of history here was inaccurate, 
and that Zermelo's primary motivation was a defence of his well-ordering 
theorem.  I just checked out Ebbinghaus's biography of Zermelo from the 
library and have found that while the defence of the well-ordering theorem 
was certainly one of Zermelo's motivations, my version of history is not 
all that far from the truth, at least if we take Zermelo's own words at 
face value.  Ebbinghaus quotes from van Heijenoort's "From Frege to 
Goedel: A Source Book in Mathematical Logic, 1879-1931," which has an 
English translation of Zermelo's 1908 Mathematische Annalen paper 
"Untersuchungen ueber die Grundlagen der Mengenlehre. I":

  Set theory is that branch of mathematics whose task is to
  investigate mathematically the fundamental notions of number,
  order, and function in their pristine simplicity, and to develop
  thereby the logical foundations of all of arithmetic and analysis.

That is, Zermelo was not just studying set theory for its own sake, or for 
the sake of justifying his well-ordering theorem; he regarded set theory 
as a foundation for most if not all of mathematics.  Regarding the 
relevance of the antinomies:

  At present, however, the very existence of [set theory] seems to
  be threatened by certain contradictions, or "antinomies," that
  can be derived from its principles---principles necessarily
  governing our thinking, it seems---and to which no entirely
  satisfactory solution has yet been found.  [...]  Under these
  circumstances there is at this point nothing left for us to do
  but to proceed in the opposite direction and, starting from set
  theory as it is historically given, to seek out the principles
  required for establishing the foundations of this mathematical

As for Weaver's suggested dichotomy between "anything goes" and "be 
absolutely safe," Zermelo (as I suggested) did not fall into either of 
these camps:

  In solving the problem we must, on the one hand, restrict these
  principles far enough to exclude all contradictions and, on the
  other hand, leave them sufficiently wide to retain all that is
  valuable in this discipline.


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