FOM: infinitesimals and AST
Vaughan Pratt
pratt at cs.Stanford.EDU
Wed Nov 26 13:17:15 EST 1997
>Rick Sommer, on Nov. 11th, observed in connection with
>infinitesimals, infeasible numbers etc. that
> To me the best explanation for these objects is through
> identifying infinite integers with something akin to the
> "infeasible numbers" that have been discussed in recent
> postings. Roughly, the idea is to finitize the
> empirically relevant part of mathematical reality
> (including infinitesimal and infinite numbers).
>
>It may be of interest to some readers of FOM that Petr
>Vopenka during the 1970ies (partly in collaboration with
>A.Sochor) developed a possibly related foundational program
>which distinguished between standard and non-standard
>objects (in particular: numbers) already in the setup of
>its underlying 'alternative' set theory (AST), cf.
>Vopenka's short book
Infeasible and infinite are surely orthogonal. \omega collects all
numbers, feasible and infeasible. As a collection it is neither
feasible nor infeasible as a number, but rather among the most feasible
collections.
My position would be that \omega is a feasible ordinal, while \omega + n
and \omega.n are feasible ordinals iff n is a feasible number (finite
ordinal). For cardinals, I would take beth_0 through beth_5 to be
feasible but beth_6 not, via roughly Parikh's reasoning showing that
1024 is feasible but 2^1024 is not, as per Vladimir Sazonov's post,
more precisely via my variant making 0, 1, 2, 4, 16, 65536 feasible but
2^65536 not.
Vaughan Pratt
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