[math-fun] Infinite epsilon-chains (set theory)
On of the Zermelo-Frankel axioms for set theory (the Axiom of Regularity, aka the Axiom of Foundation) implies the nonexistence of left-infinite epsilon chains. Let eps denote set membership. Then Regularity implies there is no function f with domain Z+ such that for all n, f(n+1) eps f(n). I'm not at all convinced this is a necessary condition for set theory. If you think of a set as a matryoshka (babushka) doll, there seems no problem in conceiving of an inwardly nested sequence of dolls indexed by Z+. E.g., the spheres of radius 1 + 1/n about 0 in R^3 for n = 1,2,3,.... The objection to such a situation seems to be that a set S with a left-infinite epsilon-chain is not ultimately "founded" on anything -- which I agree is a bit troubling. But such a set S seems analogous to the concept that time has no beginning, that it might locally correspond to all real numbers less than "now". This too is a bit troubling, but imo by no means beyond the realm of possibility. ------------------------------------------------------------------------ Another consequence of Regularity is the nonexistence of (finite) circular epsilon chains: A_1 eps A_2 eps ... eps A_n eps A_1. This I find even less palatable than a left-infinite epsilon-chain -- but not enough to want to ban it. Because, topologically it's easy to conceive of a finite set of circularly-nested matryoshka dolls: (Let S_k be the sphere of radius k about 0 in R^3. Now let X be all the space between S_1 and S_(n+1), inclusive. Now identify S_(n+1) with S_1 by central projection to get a quotint spac Y of X. Then Y has a finite circular sequence of spheres S_1,...,S_n, each "contained" within the next.) ----------------------------------------------------------------------------- What do others think of possibly tossing out Regularity as a set theory axiom? --Dan
Well, Dan, how would you do induction? Or can you do without it? --ms Daniel Asimov wrote:
On of the Zermelo-Frankel axioms for set theory (the Axiom of Regularity, aka the Axiom of Foundation) implies the nonexistence of left-infinite epsilon chains.
Let eps denote set membership.
Then Regularity implies there is no function f with domain Z+ such that for all n, f(n+1) eps f(n).
I'm not at all convinced this is a necessary condition for set theory.
If you think of a set as a matryoshka (babushka) doll, there seems no problem in conceiving of an inwardly nested sequence of dolls indexed by Z+. E.g., the spheres of radius 1 + 1/n about 0 in R^3 for n = 1,2,3,....
The objection to such a situation seems to be that a set S with a left-infinite epsilon-chain is not ultimately "founded" on anything -- which I agree is a bit troubling.
But such a set S seems analogous to the concept that time has no beginning, that it might locally correspond to all real numbers less than "now".
This too is a bit troubling, but imo by no means beyond the realm of possibility. ------------------------------------------------------------------------
Another consequence of Regularity is the nonexistence of (finite) circular epsilon chains:
A_1 eps A_2 eps ... eps A_n eps A_1.
This I find even less palatable than a left-infinite epsilon-chain -- but not enough to want to ban it. Because, topologically it's easy to conceive of a finite set of circularly-nested matryoshka dolls:
(Let S_k be the sphere of radius k about 0 in R^3. Now let X be all the space between S_1 and S_(n+1), inclusive. Now identify S_(n+1) with S_1 by central projection to get a quotint spac Y of X. Then Y has a finite circular sequence of spheres S_1,...,S_n, each "contained" within the next.) -----------------------------------------------------------------------------
What do others think of possibly tossing out Regularity as a set theory axiom?
--Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On Wednesday 20 April 2005 23:57, Daniel Asimov wrote:
On of the Zermelo-Frankel axioms for set theory (the Axiom of Regularity, aka the Axiom of Foundation) implies the nonexistence of left-infinite epsilon chains. ... What do others think of possibly tossing out Regularity as a set theory axiom?
Well, e.g., Peter Aczel thought it was a good idea, so he wrote a book about it. And then Barwise and Moss thought so too, and wrote another. I expect there are several more :-). Aczel, "Non-well-founded sets", CSLI Lecture Notes #14. Barwise & Moss, "Vicious circles", CSLI Lecture Notes #60. If you ask Google about "antifoundation axioms", the first thing you find is a review of Barwise&Moss by Thomas Forster, who knows much about such things. -- g
participants (3)
-
Daniel Asimov -
Gareth McCaughan -
Mike Speciner