[FOM] Foreman's preface to HST

joeshipman@aol.com joeshipman at aol.com
Tue Apr 27 03:16:27 EDT 2010

If you only care about solving the mathematical problems normally 
considered important, and not about philosophy, "first order logic + 
ZFC" appears to provide a good enough foundation that it is sufficient 
to search for additional set-theoretical axioms and not to go outside 
that framework. For example, the important results that were obtained 
using category theory can be easily obtained within ZFC if you add the 
"Grothendieck Universes" axiom (inaccessibles are unbounded among the 

This is merely a practical observation; if you know of any serious 
mathematical investigation of questions which can be stated in the 
language of set theory that cannot easily be conducted in the framework 
of first-order-logic + ZFC, I'd like to hear about it.

-- JS

-----Original Message-----
From: Marc Alcobé <malcobe at gmail.com>

Let me explain one sense in which I feel that the Preface is biased.
The idea it seems to convey is that after first-order logic and ZFC,
no foundational effort other than searching for new axioms settling
mathematical problems proven independent of ZFC is necessary. I guess
not much people outside set theory would agree with that...

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