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Article 2516 of comp.ai.philosophy:
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>From: hook@wilbur.nas.nasa.gov (Edward C. Hook)
Newsgroups: sci.philosophy.tech,sci.logic,comp.ai.philosophy,sci.math
Subject: Re: Grasping concepts... is it polite?
Summary: NOT the Continuum Hypothesis
Message-ID: <1992Jan6.210258.9631@nas.nasa.gov>
Date: 6 Jan 92 21:02:58 GMT
References: <1991Dec30.172852.3305@csc.canterbury.ac.nz> <1992Jan2.131048.18412@news.stolaf.edu> <1992Jan6.092440.25451@etl.go.jp> <1992Jan6.130008.16471@news.stolaf.edu> <1992Jan6.172637.468@nas.nasa.gov> <1992Jan06.194634.64671@cs.cmu.edu>
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In article <1992Jan06.194634.64671@cs.cmu.edu> nickh+@CS.CMU.EDU (Nick Haines) writes:
>In article <1992Jan6.172637.468@nas.nasa.gov>, hook@nasa.nas.gov (Edward C. Hook) writes:
>|> >
>|> -- No, aleph_0^aleph_0 = 2^aleph_0 = c
>
>Isn't that the continuum hypothesis?

-- No, the Continuum Hypothesis is that c = aleph_1, i.e., that there is NO
   cardinal number which lies strictly between aleph_0 and c.

   The fact that 2^aleph_0 = c should be well-known and was probably first
   proved by Cantor himself. One way to see this is to use the fact that each
   real number in the open interval (0,1) has a unique non-terminating dyadic
   expansion - one can view such an expansion as listing the characteristic
   function of a subset of the positive integers. One doesn't get ALL subsets
   of Z+ in this way, only those whose characteristic function has infinitely
   many nonzero values, but this DOES give an injection (0,1) --> P(Z+), where
   P() is the power-set construction. To go in the other direction, an element
   of P(Z+) is a subset of Z+ with a well-defined characteristic function. If
   this function is nonzero infinitely often, it arises from a unique real
   number in (0,1) by the above construction, so send the subset to that real
   number. In the other case, the subset is finite, say = { a1,a2,...,ar } -
   send it to 1 + 2^(-a1) + 2^(-a2) + ... + 2^(-ar): this gives an injection
   from the finite subsets of Z+ into [1,2), so together these combine to
   produce an injection P(Z+) --> (0,2). Since card(P(Z+)) = 2^aleph_0 and
   card((0,1)) = card((0,2)) = c, it follows that 2^aleph_0 = c, as promised.


>
>	Nick 
>	nickh@cs.cmu.edu

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