[FOM] Cantor's argument

[Vladik Kreinovich]

There actually is a well known rephrasing of Russell's paradox: a barber is 
assigned to a miliary unit with the task of shaving everyone who does not shave 
themselves. The question is: does this barber shave himslef or not? If he does, 
he violates his instruction because he then shaves a person who shaves homself; 
similarly, if he does not shave himself he also violates his instructions.

The conclusion here, of course, is not that the number of soldiers is 
uncountable but that the instructions issued to the barber are, if taken 
literally, inconsistent. 

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> From: Dean Buckner <Dean.Buckner at btopenworld.com>
> Subject: [FOM] Cantor's argument
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> 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
> 
> 
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