[FOM] geometry and non-well-founded sets
allenph at unimelb.edu.au
Fri Sep 14 04:38:38 EDT 2007
Robert Black asks
>Someone once said to me (and it sounds true) that using a
>non-well-founded set theory you could so axiomatize projective
>geometry that *both* a line is identified with the set of points
>lying on it *and* a point is identified with the pencil of lines
>passing through it. Can anyone give me a reference for somewhere this
is actually done?
I know of (but have not carefully studied, have not even looked at
for years, and can't usefully comment on) one published attempt:
J. Kuper, "An application of non-wellfounded sets to the foundation
of geometry," in "Zeitschrift f. Masthematische Logik u. Grundlagen d.
Mathematik," vol. 37 (1991), pp. 257-264.
Since not ALL sets of points are (on) lines and not ALL sets of lines
are convergent on points, the cardinality problems expected from
letting Xs be sets of sets of Xs don't arise. The particular
non-well-founded set theory needed will not, however have any of the
strong alternatives to foundation (AFA, SAFA...) considered in
Aczel's monograph on n.w.f. set theory.
University of Melbourne
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