[FOM] FOM: Infinite sets and plural reference

[Dean Buckner]

As it happens, set theory assumes a set is not identical with its elements.
{Alice} is not the same thing as Alice, for instance.  {} has no members to
be identical with.  And from the mere existence of Alice and Bob, we cannot
infer the existence of {Alice, Bob}.  Some of the other axioms of set theory
would also be unnecessary if we could decompose all statements about sets,
into statements about their elements.

Is this just an accident?  Could we still get the essentials of set theory,
in particular Cantor's Theorem, on the assumption that we could decompose in
this way?

Is the idea that sets "are identical with their elements" coherent at all?
(One of my non-FOM correspondents, who is eminent by any standard, claims
not).

My hunch is that we need set theory as it stands, i.e. ~ {Alice} = Alice, to
get to the transfinite.  I have arguments for this, but interested to hear
what others think in the first place.


Dean Buckner
London
ENGLAND

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