Re: Cantor Confusion


the_wign@xxxxxxxxx wrote:
Cantor's proof is one of the most popular topics on this NG. It
seems that people are confused or uncomfortable with it, so
I've tried to summarize it to the simplest terms:

1. Assume there is a list containing all the reals.
2. Show that a real can be defined/constructed from that list.
3. Show why the real from step 2 is not on the list.
4. Conclude that the premise is wrong because of the contradiction.


Isn't it important to ASSUME that the original list (in #1 above) is in
a one-to-one correspondence to the integers. You need this to count
down and across to construct a real that CAN'T be one of these in the
list. You need a bigger infinity...

Similarly, if you make a horizontal and vertical list of all integers,
going to infinity in both directions, then ANY rational can be placed
uniquely on that grid, and assigned a unique integer position number.
You DON'T need a bigger infinity here...

.

Relevant Pages

  • Re: Cantor Confusion
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    ... Assume there is a list containing all the reals. ... This is hardly the simplest terms. ... enumeration of denumerable sequences of digits, we say, "Suppose f is ... I said no such thing by analogy or otherwise. ...
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  • Re: Cantor Confusion
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  • Re: Cantor Confusion
    ... I've tried to summarize it to the simplest terms: ... Assume there is a list containing all the reals. ... That is a proof by contradiction, ... One can modify it slightly to get a more direct proof: ...
    (sci.math)