[FOM] Arithmetical soundness of ZFC (platonic)

[praatika@mappi.helsinki.fi]

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

> 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.

Here I can only whole-heartedly agree. And never intended to object. I  
am sorry if I ever gave that impression.

> But now you've got me curious in the historical question of how Zermelo
> set theory and its variants arrived at its current status.  When and how
> did it become widely accepted that all of mathematics could be based on
> set theory?  Obviously Bourbaki was an important influence, but surely
> there were other currents?  For example, Bourbaki didn't use ZFC in its
> current form.

That is a difficult historical question which is really beyond my  
competence. Probably different developments in different countries...  
and the change must have been gradual. But I have the impression that  
not before the second world war (except, perhaps, in Poland?).

BTW, I don't think that "ordinary" mathematicians really use ZFC, or  
even know it - rather, they use intuitive set theory amounting to more  
or less the same, most importantly, infinite sets, power sets and  
choice are freely used. Or am I wrong?

(Here in Finland, set theory was even taught from the first class in  
the elementary school when I was a kid... a funny experiment... they  
did not continue it.)


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/