FOM: rigor and intuition

Vladimir Sazonov V.Sazonov at
Wed Feb 13 08:48:20 EST 2002

Gordon Fisher wrote:
> Vladimir Sazonov wrote:
> > Peter Schuster wrote:
> >
> > > >Some conflict is inevitable, as it is shown by the example of quite
> > > >intuitive Axiom of Choice leading to non measurable sets and other
> > > >"paradoxes".
> > >
> > > How can you call a principle "quite intuitive" among whose consequences
> > > there are some which are commonly considered to be contra-intuitive?
> >
> > Who knows in advance which consequences some "quite intuitive"
> > axiom can have. Getting these consequences we could start think
> > more about this axiom. But, e.g. AC seems to me (and seemingly
> > to most of  mathematicians), nevertheless, sufficiently intuitive.
> >
> Intuitive with respect to countable infinities, not so intuitive with
> respect to uncountable infinities?

Yes, "we could start think more about this axiom" and open a 
way to new, more delicate intuitions around it. But, why ZFC, 
not its part ZF or even ZF + not AC, is so widely adopted by 
seemingly overwhelming majority of mathematicians 
(consciously or not)? 

I should note, that I am not a specialist in this very topic. 
I just tried to reflect some general opinions. I also wanted 
only to tell that some conflict between an intuition and 
its formalization is usually inevitable, and this is a normal 
thing which we should take into account when formalizing 
("putting in a Procrustean bed") any idea. This contradicts 
to the opinion (seemingly assumed in other postings)  
that there should be complete coherence between intuition 
and its formalization. Mathematics demonstrates just the 
converse. The discussion started with some assertions on 
some counter-intuitiveness of (expressing of) some formal 
logical rules using "arbitrary objects". My reaction is: 
1) this is understandable quite intuitively and straightforwardly 
in some reasonable terms and 2) in any case, a small discomfort, 
which still may exist, is a normal thing. Of course, if there is 
****sufficiently strong reason*** for reconsidering logical 
rules and their semantics by some developing the concept of 
"arbitrary objects", it should be done. I did not read 
the book of Kit Fine to judge. The very starting point 
(something on incomplete coherence with intuition) is not 
sufficiently convincing for me. 

> Gordon Fisher     gfisher at

Vladimir Sazonov                        V.Sazonov at 
Department of Computer Science          tel: (+44) 0151 794-6792
University of Liverpool                 fax: (+44) 0151 794 3715
Liverpool L69 7ZF, U.K.

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