To: |
"[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx> |
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From: |
Chris Menzel <chris.menzel@xxxxxxxxx> |

Date: |
Sun, 30 Sep 2012 00:50:08 -0500 |

Message-id: |
<CAO_JD6PFX-XGG1PjoqPQWfWQFe8yzGh4tSz0BXMPCg4tWw36zg@xxxxxxxxxxxxxx> |

On Sat, Sep 29, 2012 at 8:51 PM, John F Sowa <sowa@xxxxxxxxxxx> wrote:
Chris and Avril, I'm not sure how this addresses the claim you made to which I objected — that set theory has two operations (better, relations) and mereology one, which I took to imply that set theory has two
primitive relations. That's just not faithful to the way set theories are almost always developed.With Cantor's elementOf operator, you have an enormous amount of The membership relation has no power at all in itself. The power in set theory comes from the axioms you associate with the membership relation. For example, without the axiom of infinite you don't get an infinite set; without the Powerset axiom you don't get higher infinities.
Lesniewski considered that to be too much power. He claimed that And lacking appropriate axioms, you can't construct it in set theory either. AS
And in some set theories, too. I was a bit careless in using "set theory" above to mean Zermelo-Fraenkel set theory, which includes the axiom of foundation that rules out "a = {a}". Without foundation, you can consistently assert the existence of such "non-well-founded" sets. Indeed, there are well-developed set theories with anti-foundation axioms that entail the existence of a wide variety of non-well-founded sets.
-chris _________________________________________________________________ Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/ Config Subscr: http://ontolog.cim3.net/mailman/listinfo/ontolog-forum/ Unsubscribe: mailto:ontolog-forum-leave@xxxxxxxxxxxxxxxx Shared Files: http://ontolog.cim3.net/file/ Community Wiki: http://ontolog.cim3.net/wiki/ To join: http://ontolog.cim3.net/cgi-bin/wiki.pl?WikiHomePage#nid1J (01) |

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