[FOM] Re: Arithmetic-free theory of formal systems?

Vladimir Sazonov V.Sazonov at csc.liv.ac.uk
Sun May 23 16:50:26 EDT 2004

> Is this formal theory of feasible numbers published somewhere
> (in English)?  

See "On feasible numbers" in


and the end of my recent posting Re: [FOM] A formalism for Ultrafinitism

>>Of course, these two versions of N cannot be represented as sets in
>>"the" universe for ZFC. But corresponding axiomatizations are
> In other words, you sacrifice the completeness theorem?

Why do you think that any formal system, even based on
the first order *language* (and on some weak version of first
order logic) should have a model *in* "the" universe for ZFC?

By the way, completeness theorem is, *in a sense*, evident:
if a formalism is consistent then we can imagine corresponding
"universe" and have some image in our minds. This image serves
as a model. Of course, some generalization of formally provable
completeness theorem (not in ZFC) is desirable (if possible at
all) for formalisms like that for feasible numbers.

> It was not my primary purpose to defend platonism, only to argue that the 
> inconsistency of PA would not present any *new* challenges to platonism
> that aren't already provided by known mathematical results.  You yourself
> have argued that formal set theory and formal number theory are parallel
> in many ways; the inconsistency of PA would not, as far as I can see now, 
> present new philosophical challenges that, say, Russell's paradox did not 
> already present.

Inconsistency of PA (if possible at all) would, I believe, lead to
strong doubts on uniqueness of the meaning of "and so on". So the
idea of "standard" N (unlike the vague idea on N) will vanish
(lose a lot of its defenders). Note, that these doubts exist anyway,
without any contradiction in PA. How can these doubts (which you
seemingly almost accepted?) be consistent with platonism?

> Tim


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