[FOM] geometry and non-well-founded sets

Allen Hazen 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.


Allen Hazen
Philosophy Department
University of Melbourne

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