Re: Cantor Confusion


Dave L. Renfro schrieb:

Peter Webb wrote (in part):

This is a complete red herring. There is no question that
the Real generated by Cantor's proof is computable (r. e,)
if the original list is, [...]

mueckenh@xxxxxxxxxxxxxxxxx wrote (in part):

Of course. That's why the diagonal proof only proves the
existence of numbers which belong to a countable set i.e. the
set of constructible reals. This proof proves in essence that
the countable set of constructible real numbers is uncountable.
A fine result of set theory.

You're overlooking Peter Webb's hypothesis "if the original
list is". You need to have a list (x_1, x_2, x_3, ...) such
that the function given by n --> x_n is computable. Thus,
before you can conclude what you're saying (which sounds like
a metalogic "proof by contradiction" to me, but no matter),
you need to come up with a computable listing of the computable
numbers (or at least, show that such a listing exists).

One cannot compute a list of all computable numbers. By this
definition,
(1) the computable numbers are uncountable.
(2) There is no question, that the computable numbers form a countable
set.
This is a contradiction. It is not necessary to come up with a list of
all computable numbers.

Regards, WM

.

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