James Dennett said unto the world upon 26/06/2005 03:51: > Steven D'Aprano wrote: > > >>On Thu, 16 Jun 2005 21:21:50 +0300, Konstantin Veretennicov wrote: >> >> >> >>>On 6/16/05, Vibha Tripathi <[EMAIL PROTECTED]> wrote: >>> >>> >>>>I need sets as sets in mathematics: >>> >>>That's tough. First of all, mathematical sets can be infinite. It's >>>just too much memory :) >>>Software implementations can't fully match mathematical abstractions. >> >> >>:-) >> >>But lists can be as long as you like, if you have enough memory. > > > But you never have enough memory to store, for example, > a list of all the prime integers (not using a regular list, > anyway).
An even better example is the set of reals in the interval (0, 1). Even an idealized Turing machine with (countably) infinite memory will choke on that :-) <snip> >>Standard Set Theory disallows various constructions, otherwise you get >>paradoxes. >> >>For example, Russell's Paradox: the set S of all sets that are not an >>element of themselves. Then S should be a set. If S is an element of >>itself, then it belongs in set S. But if it is in set S, then it is an >>element of itself and it is not an element of S. Contradiction. >> >>The price mathematicians pay to avoid paradoxes like that is that some >>sets do not exist. For instance, there exists no universal set (the set >>of all sets), no set of all cardinal numbers, etc. >> >>So even in mathematics, it is not true that sets can contain anything. > > > See "Set Theory With a Universal Set" by T. Forster, which covers > some set theories in which there *is* a set of all things, and > in which Russell's paradox is avoided in other ways (such as by > restricting the comprehension axioms). > > (Sorry for drifting offtopic, I happen to find non-standard > set theories interesting and thought that some others here > might too.) > > -- James So do I :-) Do you know of non-well-founded set theory (non-standard set theory which allows sets A, such that A is in A)? Not really on point for any of the above, but being on topic is in the rear view mirror, anyway :-) <http://cslipublications.stanford.edu/site/1575860082.html> <http://cslipublications.stanford.edu/site/0937073229.html> Best, Brian vdB -- http://mail.python.org/mailman/listinfo/python-list
