[FOM] Set Theory and Intuition

[Dean Buckner]

There is an argument that frequently occurs in the popular literature on set
theory that I ought to mention.  It goes: one intuition, a very powerful
one, is that any set is somehow larger than any of its proper subsets.
However there is another intuition, almost as powerful, that we can match
every member of a infinite set to a unique member of one of its proper
subsets (Galileo, and numbers and their squares, usually come into the story
here).  Thus, our ordinary intutions are inconsistent.  We can only resolve
the inconsistency by dropping one of them (the first).  Thus, our ordinary
conceptual scheme leads (for all but the slow-witted or deliberately obtuse)
inexorably to & through the gates of Cantor's paradise.

Wrong.  Undoubtedly set theory is consistent, it doesn't follow that our
intuitions are not.  The whole argument rests on the assumption we have any
intuition about what the expression "infinite set" denotes.  I have no such
intuition.  Given any set of whole numbers, we can find a set of prime
number to match, agreed.  But then the set must have a number, and therefore
cannot be infinite, surely?  Every set of numbers is finite, and there are
infinitely many of them.  That's my intuition, and its perfectly consistent
with the intuition that if any set of F's that wholly contains some set of
G's, then there are more F's than G's.

If any set theorist want to claim he some contrary intution, that's fine.
But then he has to drop the intuition that is more natural (my one).

Thus I do not understand Insall's claim that the axioms of set theory are
"right, because they actually do capture a reasonable portion of our
intuition about sets".  Surely they don't.  They fail to capture the most
reasonable portion of our intuition about sets, which is that the whole is
greater than the part.  To that extent (& only to that extent, I agree) set
theory is "wrong".



Dean Buckner
London
ENGLAND

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