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Re: [HM] A question as to Cantor's Diagonal Method.
Posted:
Sep 6, 2005 3:06 AM
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-----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
existence of Y leads to a contradiction. Q.E.D.
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 Luzins (and Borels) view that the set of real numbers
is countable, but not effectively enumerable one.
Alexander Zenkin
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