FOM: infinitesimals and AST
Stephen G Simpson
simpson at math.psu.edu
Thu Nov 27 16:51:23 EST 1997
Walter Felscher writes:
> 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
> Mathematics in the Alternative Set Theory
> Leipzig, Teubner 1979 , 120 pp .
Yes, I looked at this book in the 1970s. As I recall it, AST is an
interesting approach to both feasible/infeasible and
standard/nonstandard integers. If I recall correctly, a typical model
of AST is a countably saturated ultrapower of the hereditarily finite
sets, together with a predicate for the standard (feasible) integers.
AST has one advantage over the Robinson setup in ZFC. Namely, under
AST, there is a canonical set of nonstandard integers. Tell me, is it
provable in AST that there exists a definable infinitesimal?
I have my doubts about whether AST provides a decent foundation for
all of mathematics, as ZFC does.
-- Steve Simpson
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