[FOM] Constructivism and physics

Alasdair Urquhart urquhart at cs.toronto.edu
Fri Feb 10 11:01:14 EST 2006

Nonstandard analysis can be done in a formal system
that is a conservative extension of standard set
theory ZFC.  For example, Hrbacek set theory HST,
which is a powerful version of nonstandard analysis,
is equiconsistent with ZFC, and is conservative with
respect to standard theorems of ZFC (see
"Nonstandard Analysis, Axiomatically" by Kanovei
and Reeken, p. 20).  Similar theorems can be
proved for other theories such as Nelson's Internal
Set Theory IST.

It follows from this that to look for a theorem that
you can only prove by nonstandard methods, but not
by standard methods, is to go on a wild-goose chase.
The strength of infinitesimal methods lies in their heuristic
power, as Leibniz said several centuries ago.

Incidentally,  Dana Scott pointed out
how expensive the book by Kanovei and Reeken is, as well
as another book I mentioned earlier.
However, Dover has recently
re-issued three excellent books on nonstandard analysis in
their admirable mathematics catalogue:

"Infinitesimal Calculus" by Henle and Kleinberg -- an
introductory calculus text using the hyperreals;

"Nonstandard Analysis" by Alain Robert -- an
introductory text using Nelson's IST, but including
some more advanced material such as the Bernstein-
Robinson theorem on invariant subspaces;

"Applied Nonstandard Analysis" by Martin Davis --
includes some advanced material on Hilbert space
and real analysis.

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