X-Message-Number: 26695 Date: Mon, 25 Jul 2005 21:27:54 -0700 From: Mike Perry <> Subject: Once More on Identity vs Equivalence References: <> I respond to Thomas Donaldson's comments: >You say that 2 sets are identical if >their members are the same, and thus give a circular definition. Only informally did I say that (or that's how it should be taken at any rate). My actual definition is that X and Y are identical iff, for all z, z is a member of X iff z is a member of Y. So I really say nothing directly (formally) about the elements of X and Y being "the same". >Nothing prevents us from considering a set of sets which are >"identical" in the sense that they differ only on a set of measure >0, where "measure" is the Lebesgue measure on the real line. >Well, you might say, such sets do not have the same elements. Yes, and that's why I'd call them equivalent rather than identical, the notion of equivalence *in this case* meaning "symmetric difference (set difference) has Lebesgue measure zero". For mathematical "identity" your options are narrower. Mike Perry Rate This Message: http://www.cryonet.org/cgi-bin/rate.cgi?msg=26695