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Topic: [HM] A question as to Cantor's Diagonal Method.
Replies: 13   Last Post: Oct 25, 2005 6:01 AM

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 Alexander Zenkin Posts: 50 Registered: 12/3/04
Re: [HM] A question as to Cantor's Diagonal Method.
Posted: Sep 6, 2005 3:06 AM

-----Original Message-----
From: owner-historia-matematica@chasque.apc.org
[mailto:owner-historia-matematica@chasque.apc.org] On Behalf Of Roger Cooke
Sent: Tuesday, August 30, 2005 6:40 PM
To: historia-matematica@chasque.apc.org
Subject: Re: [HM] A question as to Cantor's Diagonal Method.

I think this topic has been discussed before. In his personal notebooks,
Luzin speculated that the Cantor diagonal proof shows only that the real
numbers are not effectively enumerable. But I have always had trouble
understanding what it could mean to say that " |X| =3D |N| " is true if
no
such enumeration "exists." But I think William Waterhouse's objection
misses the point: the set X *is* the real numbers, by assumption. Of
course taking Z in place of X produces a different result: so what? Or
am I missing something myself?

Roger Cooke

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Dear Roger,

Your question prompts the following quite general result.

Here Z is any set, P(Z) is a power-set for Z.

THEOREM 5a. For any set Z, If |Z| = |P(Z)|, then there is NOT a
rule/algorithm producing a 1-1-correspondence between elements of the sets,
Z and P(Z).
RAA-PROOF. Assume that |Z| = |P(Z)|, BUT there is a rule/algorithm
producing a 1-1-correspondence, say PSY, between elements of Z and P(Z).
Then the traditional Cantor counter-example Y={z from Z : z is not belonging
to PSY(z)} is constructed (see, e.g., Paul J.Cohen, Set Theory And The
Continuum Hypothesis. - Stanford University. W.A.BEJAMIN, INC., New York
1966 Amsterdam. - Chapter II, §4, Cardinal Numbers.). In such a case, the

I think the THEOREM 5a proves strictly the known Borel thesis that
there are countable, but not effectively enumerable sets (see, e.g.,
F.A.Medvedev, French School of the Theory of Functions and Sets at the
threshold of XIX-XX centuries. ­ Moscow : Science, 1976) and, in
particular, explains Luzins (and Borels) view that the set of real numbers
is countable, but not effectively enumerable one.

Alexander Zenkin