FOM: infinitesimals and AST

Stephen G Simpson simpson at
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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