[FOM] Richard's Quest >> Feasible Numbers

Vladimir Sazonov V.Sazonov at csc.liv.ac.uk
Fri Jan 13 09:52:05 EST 2006

Quoting José Félix Costa <fgc at math.ist.utl.pt>:

> «And, for example, I am interested in exploring how much interesting, useful
> mathematics can be done by allowing oneself, when talking about natural
> numbers, to talk onlyabout natural numbers less than or equal to some
> unspecified, rather large natural number.»

a paper by Vladimir Sazonov (Liverpool):
> Feasible Numbers
> (donwloadable)
> Arithmetics is rethought assuming a large, unspecified, finite, limit,
> natural number.

More precisely, the number mentioned is *specified*, but it serves 
as one of possible upper bounds for feasible numbers which are closed 
under successor and therefore constitute an infinite semiset (if to use 
the terminology of Vopenka for infinite parts of formally finite sets). 
In fact a precise formalism is suggested. 

By the way, there is a formal statemet in the framework of feasible 
arithmetic that the continuum is both discrete and continuous (without 
a formal contradiction). Of course, this is not yet a full fledged 
theory (a feasible version of non-standard analysis), but at least 
a possible framework for such a theory. 

As to unspecified maximal number which is not considered as 
non-feasible, but serves rather as parameter, a formal system was 
also suggested in 1980 which is in fact a theory of polynomial time 
computability in a precise sense (of descriptive complexity theory). 
This can be also considered as a theory based on a kind of 
rejection of the abstraction of potential infinity. In a precise 
sense such kind of theories assuming the maximal natural number 
are nothing else as a form of bounded arithmetic (under a natural 
mutual interpretation with BA). Unlike the ordinary BA, it is bounded 
not because the induction axiom is such (it is formally unbounded), 
but because the world of natural numbers is postulated so. 

Vladimir Sazonov

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