|Nov26-03, 06:10 AM||#1|
C(reals) = C(P(naturals))??
Could someone help me please.
I am studying Cantor's set theory at present, but am a little confused as to why he concludes that the cardinality of the set of real numbers is equal to the cardinal number of the power set of naturals (2^aleph0).
|Nov26-03, 11:43 AM||#2|
The elements of 2N can be treated as sequences of 1's and 0's where a sequence corresponding to a subset A of N has a 1 in the n th position if n is in A.
You should have no trouble seeing that this mapping is bijective.
Now, if you look at the real numbers on [0,1] base 2, you get numbers like:
which are also sequences of ones and zeros. So there's a natural mapping.
Unfortunately there is a problem because
in the reals.
But that only occurs a countable number of (N) times. So we can certainly construct a bijection to [0,1] + N.
So we have |2N| = |[0,1] + N|
but you should already know that |[0,1] + N|=|[0,1]|=|R|
(Cantor certainly did)
So by substitution we get:
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