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

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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