Re: cantor's theorem - the pieces of the puzzle

On Nov 15, 5:26 pm, Ken Quirici <ken.quir...@xxxxxxxxxx> wrote:
AP says:

the set of all possible digit arrangements in (0,1), that is
the reals in that interval, is countable

Cantor says:

OK, so we can make a list of 'em. I can construct
by valid arithmetic/mathematical methods, from that
list, a number in that interval that is NOT in the
list.

AP says:

but I said it was the set of all possible digit arrangements,
so that digit arrangement you found HAS to be in the
set of all possible digit arrangements

At that point, there's no basis for an intelligent discussion. It's
already over. The AP character's remark shows only that he does not
understand the basic logic of universal generalization.

Cantor says:

but I've constructed a digit arrangement that isn't
in the list you gave me, and you CANNOT
fault my construction EXCEPT by continuing
to insist on your assumption.

No only that. But our Cantor character here doesn't even need to
submit this rebuttal. The proof is correct and AP's rejoinder does not
make the proof incorrect, and there's no need even to address AP's
rejoinder, which shows only that AP does not understand the basic loic
of universal generalization.

I say:

OK gents, drinks on the house. What we have here
is called in mathematics a CONTRADICTION -
two opposing claims, one derived by valid
arithmetic/mathematical means (Cantor's) from
the other (AP's).

There's no need even to mention that a theorem contradicts the
negation of a theorem. Cantor's proof is correct and AP's objection is
simply ignorant of the basic logic of universal generalization.

I hope AP, the self-proclaimed expert on proofs
by contradiction notices that what indeed we have here
is a CONTRADICTION. GIVEN the assumption of
the countablility of the reals, we found a
CONTRADICTION. THEREFORE THE COUNTABILITY
OF THE REALS LEADS TO A CONTRADICTION AND
CAN"T BE TRUE.

No need for this "backup" argument. Cantor's proof is not proof by
contradiction even. Rather, by proof (that is not a proof by
contradiction), we show that there is no surjection from the set of
naturals onto the set of denumerable binary sequences. There's no need
to even grant AP a rejoinder at all.

This has nothing to do with non-Euclidean geometry or
AP-adics or any other damn fool thing. It's fairly
simple math. I grant you that I feel uncomfortable
with the notion of an infinity greater than that of
the integers, but my mind accepts the proof(s).

The proof exists as a correct application of basic logic to ordinary
mathematical premises. If AP doesn't understand that, then he's
hopeless.

MoeBlee

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