FOM: Conway's foundational ideas -- reply to Simpson

[Kanovei]

Date: Wed, 19 May 1999 14:25:44 -0400
From: Joe Shipman <shipman at savera.com>

>"Pure ZFC" is unrealistically
>parsimonious and we want to be able to extend it freely by defining
>notations for operations on sets satisfying given properties.
>Particular examples which are known to be OK:
>ordered pair (no metamathematical problems after Kuratowski),
>cardinal number (no metamathematical problems after Von Neumann),
>global choice function (slightly stronger than ZFC but consistent with
>ZFC)

The 3rd example is very different from the two first, 
actually a kind of antipode. Indeed, you can be a 
personal hater of the Kuratowski pair as libertarians or 
whoever are, but as a mathematician you cannot not agree 
that {{x},{x,y}} is a well-defined mathematical object. 
On the contrary 
(and despite the known conservativity 
results, of Felgner I think) 
the global choice function is not a well-defined 
mathematical object, e.g. you cannot even think of 
a question like what is G(0). 

Accepting GCF, you simply extend the "core" of 
mathematical foundations by something which does not 
belong to this "core": this is not at all unlawful 
(as soon as the consistency etc. has been established) 
but this cannot go unnoticed. 

In that GCF is similar to AC, with the only difference 
that accepting the "set-size" AC mathematicians solved 
a lot of otherwise dead questions on sets of real 
numbers. 

V.Kanovei