FOM: Axiom of Extensionality

[Dean Buckner]

What is this axiom?  Does it say that a set A is the same as set B iff they
have the same members?

But then, if a set is something apart from its members, and perhaps defined
by properties they have in common, how is that possible?  There is a set of
unicorns, which is empty.  If there were unicorns, would this be a different
set?  But then how could both be the set "of unicorns"?

Or take the set {Alice, Bob, Carol}.  Does this continue to exist even if
they cease to?  Is it that it currently has the number 3, but the number 0
if they do not exist?

Either (you would think) the set is one and the same with its members, in
which case the axiom reduces to {Alice, Bob, Carol} = {Alice, Bob, Carol},
and hardly seems necessary.

Or it is something different, which raises the possibilty of its changing
its membership, while remaining the same.  E.g. the set of people who live
in Spencer Walk, which changes slowly through the years, bit by bit, but
which (presumably) remains the same set.  In which case the axiom is clearly
false.

Or is the formulation I have used incorrent?  I have seen a number of
different ones, and they are all subtly different.  One says that "two" sets
are one and the same iff...  what?  Iff they are not two sets after all?


Dean Buckner
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London, SW15 1PL
ENGLAND

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