Re: Cantor Confusion

In article <1159417542.425540.214160@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
cbrown@xxxxxxxxxxxxxxxxx wrote:

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.

That is a proof by contradiction, which many constructionists object to.

One can modify it slightly to get a more direct proof:
1. Assume one is given any list of reals (i.e., an arbitrary function
with domain N and codomain R, the only condition being that there is one
real for each natural number)
2. Show that a real can be defined/constructed from that list in such a
way as not to be a member of that list.
3. Conclude that every list must omit at least one real, so no list is
complete.
.

Relevant Pages

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