[FOM] Cantor's argument

[Dean Buckner]

I'm a philosopher, not a mathematician, so unable to comment on the
mathematical application of Cantor's Theorem.  But consider the following,
non-mathematical application of his argument, to prove that we cannot count
or list out all concepts or ideas.

(A)   Let the expression  "does not satisfy the object it counts" denote the
concept c.  Assume some object x counts c.  Then if x satisfies c, it does
not satisfy c.  And if it does not satisfy c, it satisfies c.
Contradiction, therefore our assumption that some object counts the concept
c is false.  Therefore concepts as a whole are uncountable: any list of
concepts will omit some concept.

Is this valid?  Or do we question the assumption that we can "let" the
expression "does not satisfy the object it counts" stand for a concept at
all?  If it did, it would be a noun phrase, but grammatically it is not a
noun phrase, since it implicitly  includes the verb "is".

We can remove the verb from the expression, but then we don't get the
desired result.  The expression "does not satisfy the object it counts" is
equivalent to "is whatever it does not count", which turns into the noun
phrase "whatever it does not count" when to take out "is".  Then, if we
re-write Cantor's argument with this in
mind, we get the trivial result that there can be no x such that:

      x counts whatever x does not count

otherwise if x counted itself, x would not count itself, and if it did not
count itself, it would count itself.  This does not yield the (for
Cantoreans) desired conclusion, since it does not prove the existence of
anything that cannot be counted.

This also explains why there is no x such that

      x satisfies "does not satisfy itself"

since it means (once re-analysed as above) that there is no x such that

      x is whatever it is not

It is difficult even to state Russell's paradox outside set theory.  Of
course this proves very little, since it rests on a merely grammatical
consideration, but I mention the point in case it is of interest to FOM
readers.


                           Martin Davis
                    Visiting Scholar UC Berkeley
                      Professor Emeritus, NYU
                          martin at eipye.com
                          (Add 1 and get 0)
                        http://www.eipye.com