Re: Diagonalization theorem
- From: Patricia Shanahan <pats@xxxxxxx>
- Date: Mon, 22 Jan 2007 16:25:59 GMT
iAGENT wrote:
....
And the bigger doubt is the following:
Power Set is collection of all subsets of N.
Can anybody tell me how Power Set is different from collection of
finite subsets of N?
Easy. It includes all the infinite subsets of N, as well as the finite
subsets. :-)
Think about your grouping and dovetailing approach to the finite
subsets, and see what goes wrong when you try to apply it to the
infinite subsets.
Also, think about what goes wrong with the diagonal argument if you try
to apply it to the set of finite subsets of N. The anti-diagonal is a
subset of N, but there is no way to ensure it is a finite subset of N. A
list can include all finite subsets of N without including its own
anti-diagonal.
The proofs that the set of finite subsets of N is countable do not work
for the power set of N. The proofs that the power set of N is
uncountable do not work for the set of all finite subsets of N.
Patricia
.
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- Diagonalization theorem
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