Re: Gentzen's "constructivism"
- From: "T.H. Ray" <thray123@xxxxxxx>
- Date: Mon, 29 Jan 2007 19:14:41 EST
In Gentzen's wonderfully lucid and accessible lectureI haven't read the lecture, but I am going to take
"The concept of
infinity in mathematics" there is an interesting --
and from a modern
perspective -- somewhat puzzling slide. He urges on
us a hierarchical
picture of the universe of sets, where "new sets may
be formed only
constructively on the basis of already constructed
sets". But he then
immediately links this to a more general
"constructivist view" to the
effect that "something infinite must never be
regarded as completed".
Any suggestions for illuminating articles on Gentzen
(or about what's
going on at the time) which would help explain why
someone so smart
should make such an unremarked slide between two
apparently different
ideas?
a guess that it comes from Weyl's concept of the
continuum. (The Continuum: a Critical Examination of
the Foundation of Analysis. Dover books republication,
1994.) One of my favorites.
"...one might say that our construction of analysis
contains a _theory of the continuum_ which must
establish its own reasonableness (beyond its mere logical
consistency) in the same way as a physical theory."
(emphasis and parentheses in original. p. 93)
As a physical theory cannot accept a completed infinity,
neither can a truly scientific theory of numbers.
Tom
.
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