Re: What is a proof, exactly?
From: J.E. (troubled6man_at_yahoo.com)
Date: 12/09/04
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Date: 9 Dec 2004 09:10:51 -0800
If ("y is a set" and "x belongs to y" ) implies "x is a set" then it
seems that equality is redundant is the sentance "(x belongs to y) or
it is not the case that (x belongs to y)" for every set x and every set
y since any expression "x=y" can be replaced with "Az (zey <=> zex)"
for some free variable z. Equality is redudant for sets, just
notational, unless the domain of discourse is irregular or contains
things other than sets.
As for the proper "relation" of equality, it's simply not neccissary
for most mathematics. If you considered the graph of the equality
relation it bears a striking resemblence to the class of all sets, in
fact it would likely be the class of all "singleton subsets". If
however one had multiple sets x and y with the same elements, and no
equality relation, there would be no way to distinguish between a set
containing x and a set containing y.
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