FOM: Existence of Axioms and Existence of Collections ofAxioms

V. Sazonov V.Sazonov at
Wed Nov 8 10:40:31 EST 2000

Matt Insall wrote:
> Professor Wilson:
> >Isn't "the theory PA" itself an infinite set?!
> Matt Insall:
> That is the way I see it.  However, if professors Kanovei and Sazonov are
> correct,
> then we cannot draw this conclusion.  For they appear to assume the existence
> of theories with infinitely many axioms, but argue that no infinite set
> actually exists.

I do not mix the syntactic reality [of really existing objects 
such as (usually) small finite sets of formal (schemes) of 
rules and (schemes) of axiom written physically in a sheet 
of paper] with the abstract infinity concept investigated by 
mathematicians. [Alternatively, we could speak on finite computer 
program generating the axioms and rules. However, the first 
way is more reasonable and practical for working mathematicians.] 
Moreover, here is no assumption on the infinity of the "set" of 
axioms, inference rules, really written inferences and theorems 
deduced in such theories. My life and the life of the population 
of the Earth is probably bounded. Even the whole World is 
probably bounded (finite?). This absolutely does not matter for 
considering such theories on infinity as ZFC and IMAGINING if 
we like that we investigate, e.g. some infinite cardinals as if 
they really existed. 

It would be very interesting to me to know, is anybody here who 
is able to really exist in this imaginary world, to prove 
mathematical theorem in that world as he/she also eat, sleep, 
travel for holidays, etc. in the real world? 

It seems, Professor Davis is right by saying: 

> Can I be the only fom-er getting tired of this discussion going round and
> round in circles? 

I naively would like to hope, that it is rather dialectical 
spiral. However, I am also somewhat tired from this still 
existing misunderstanding concerning trivialities. 

Vladimir Sazonov

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