[FOM] set theory in the rest of mathematics

[Vladimir Sazonov]

Adrian-Richard-David Mathias wrote:
> 
> On Tue, 25 Mar 2003, Vladimir Sazonov wrote:
> 
> > After appearing and formalising, set theory became a part
> > of mathematics, but its role for the rest of mathematics
> > was always foundational, conceptual. Now, even if it is
> > also considered as a branch of mathematics, trying to do
> > something on large cardinals is still rather internal
> > business of set theory.
> 
> For counterexamples to this last statement, see numerous papers
> of Harvey Friedman, or see my expository piece "Strong Statements of
> Analysis", which has been published in the Bulletin of the London
> Mathematical Society 32 (2000) 513-526.

Thanks for this note and the references. Being a reader of FOM 
list, it was known to me that Harvey Friedman is working towards 
relating large cardinals to the rest of mathematics. 

Of course, any general statement of that kind (note also the 
word "rather" to slightly mild the statement) may have 
counterexamples. But the most of the work on large cardinals 
SEEMS to me is really an internal business of set theory. 
I am not competent enough to make any evaluations and 
categorical assertions. But I wanted to say that there 
may be other alternatives. These alternatives are very unusual 
today and, I believe, are worth to be, at least, mentioned. 
But something like this was quite usual at older times. 
I think, this should be a historical lesson for us. Or we 
decided to put even the future mathematics in a Procrustean 
bed of ZFC? This bed is very soft and very large, but, 
nevertheless, what about other possibilities which will 
not fit for this "bed"? 


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> 
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-- 
Vladimir Sazonov                        V.Sazonov at csc.liv.ac.uk 
Department of Computer Science          tel: (+44) 0151 794-6792
University of Liverpool                 fax: (+44) 0151 794 3715
Liverpool L69 7ZF, U.K.       http://www.csc.liv.ac.uk/~sazonov