FOM: Re: Transfinite Logic

Dean Buckner Dean.Buckner at
Mon Jun 3 13:48:46 EDT 2002

Karl Cooper refers me "elementary textbooks on set theory" that discuss such
examples as there being a parent set (the integers), a subset (the even
integers), and a remainder (the odd integers).  "The existence of the
remainder does not make it somehow contradictory or mysterious that the even
integers are equinumerous with the integers".

But this is where I started: with a textbook that defined a proper subset as
one with "fewer" members than its parent.  This was clearly wrong (does
Cooper disagree?).  On there being a remainder, why is this not
contradictory or
mysterious?  If we define equinumerosity as one-one correspondence, there
cannot be a remainder.  Doesn't the diagonal proof depend on there being one
(the diagonal itself)?

It's clear you can correlate the even numbers with themselves perfectly, as

    1,2,3,4,5, ...
    *,2,*,4,*, ...

This way, there is a whacking great remainder, as indicated by the
asterixes.  If equinumerosity is1-1 correlation without remainder, the even
numbers are not equinumerous with the integers.  If on the other hand
remainders are OK, what does the diagonal proof prove?

Sorry to burden this predominantly mathematical group with the doubtless
elementary worries of a philosopher, but I've consulted a number of
textbooks now, none of them illuminate this great mystery.

Dean Buckner
4 Spencer Walk
London, SW15 1PL

Work 020 7676 1750
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