FOM: Cantor and Kronecker

Neil Tennant neilt at
Tue Mar 16 14:01:09 EST 1999

Bill Tait writes:

 > ... The statement that Cantor's 1874 proof that a one-to-one enumeration of
 > reals does not contain evry real in a given interval is constructive does
 > not of course mean that Kronecker would have accepted it. It is
 > constructive in the sense of Bishop's conception of constructive
 > mathematics. But Kronecker held to a strict finitist conception, which
 > excludes even the conception of an arbitrary one-to-one enumeration of
 > reals.

This is interesting. What objection, exactly, would Kronecker have had to
the straightforwardly constructive `diagonal' proof of absurdity from the 
assumption (for reductio) that

	for every set y of F-things there is exactly one F-thing x
	such that R(x,y)  ?

Here R is a placeholder for any relation whatsoever, be it (constructively)
definable or not.  The reader of the reductio proof need not conceive of R
as a finished totality of ordered pairs whose first member is an F-thing
and whose second member is a set of F-things. 

Likewise, F is a placeholder for any property or predicate that *might* be
used to define the set of all Fs; and then again, might not be, depending
on one's philosophical preferences concerning sets.  One might be reluctant
to countenance the set of all F-things (for certain F), while yet claiming
a grasp of the notion "y is a set all of whose members are F-things".

The *logical* status of the diagonal reductio seems to be invariant 
across different philosophical conceptions of what R might in general be,
and whether the set of all Fs exists.

We are so used to stating the Cantor-type result in terms of sets (e.g.
`there is no 1-1 mapping of the set of reals onto the set of all sets of
reals'; or, more generally: `every set has strictly more subsets than
members') that we tend to lose sight of the fact that the impossibility
that Cantor's reasoning established is something that confronts us pretty
much regardless of our philosophical convictions concerning how sets
should be formed, which ones really exist, etc.

Neil Tennant

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