FOM: ultrafinitism; objective vs. subjective

[Vladimir Sazonov]

Todd Wilson wrote:
> 
> Vladimir Sazonov writes:
> > Then, we are using some ideas, abstract notions like the natural
> > numbers.  These ideas are also subjective in the sense that they
> > are our creations (with the roots in reality, of course) and
> > often vague. What is not vague, is any concrete tool we use (a
> > formal system with explicitly fixed rules and axioms).
> 
> But isn't a "concrete" formal system with axioms and rules at least as
> vague as the natural numbers?  The formulas being manipulated by such
> formal systems, as well as the proofs themselves, are defined by
> structural induction and can grow to arbitrary size, just as the
> natural numbers can.

Yeas, you are right. But when we prove a concrete theorem, 
everything is OK. 

Vladimir Sazonov
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