FOM: Cantor and Kronecker

William Tait wtait at
Wed Mar 17 11:19:21 EST 1999

Neil Tennant 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.


Note that Cantor's 1874 proof, to which I was referring, is not an example
of a diagonal argument: it is a nested intervals argument. I cheated in
saying that it is literaly constructive---but it is easily turned into a
constructive proof. [Let F be the 1-1 enumeration of reals and (a_0,b_0)
the given interval. Having defined (a_n,b_n), suppose that c=F(n) is in
this interval. (Since F is 1-1, this can be effectively decided.) Set a_n+1
= c and b_n+1= (c+b_n)/2. If c is not in this interval, let a_n+1 = (2a_n +
b_n)/3 and b_n+1 =(a_n+2b_n)/3. The intervals are strictly nested and |b_n
- a_n| converges to 0.]

As for what Kronecker would say, we have only the passage that Dedekind
cited in the paper on modules. In particular, in the footnote, Kronecker
explicitly rejects the notion of an arbitrary infinite sequence. As for his
reasons, that was what Dedekind was challenging him to give.

Best, Bill

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