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Topic: Problem with Cantor's diagonal argument
Replies: 65   Last Post: Mar 4, 2002 1:36 PM

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Dave Seaman

Posts: 555
Registered: 12/6/04
Re: Problem with Cantor's diagonal argument
Posted: Feb 14, 2002 1:44 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply



In article <a4gvml$7ud9$1@ID-110648.news.dfncis.de>,
Herman Jurjus <h.jurjus@hetnet.nl> wrote:

>"Dave Seaman" <dseaman@seaman.cc.purdue.edu> wrote in message
>news://a4gf7t$2or@seaman.cc.purdue.edu...


>> We don't have to create the list. The list is given to us in the
>> hypothesis of the theorem. In geometry, we accept that infinitely long
>> lines exist, even though we can never finish drawing them. In analysis,
>> we accept that real numbers exist, even though we can never finish
>> writing them down.


>OK. But aren't there people outthere who can imagine 'potentially' infinite
>sets, but not 'actual' infinite sets?
>To those people, would Cantor's argument make sense?


There is more than one Cantor argument. The one that seems most
appropriate here is the power set version. Given any set X, the power
set of X, P(X), is the set of all subsets of X. A version of Cantor's
theorem says that if X is any set at all (finite or infinite), then |X| <
|P(X)|.

Proof: There is an obvious injection f: X -> P(X) given by x |-> {x}.
This establishes that |X| <= |P(X)|. In order to show that the
cardinalities are not equal, we need to show that no mapping f: X -> P(X)
is a surjection. That is, given f: X -> P(X), we need to find a member S
of P(X) (depending on f) such that S is not in the range of f.

Let S = { x in X : not(x in f(x)) }. It follows from this definition
that for every x in X, x belongs to S <==> x does not belong to f(x).
Therefore, S differs from f(x) for every x in X, which is what it means
to say S is not in the range of f.

Corollary. The powerset of the natural numbers, P(N), is uncountable.

Now, you can deny that infinite sets exist if you like, but you can't
deny that Cantor's proof applies to all sets that exist. You can deny
that there is a set N if you like, but you can't deny that *if* there is
a set N, then the set P(N) is larger than N.

--
Dave Seaman dseaman@purdue.edu
Amnesty International says Mumia Abu-Jamal decision falls short of justice.







Date Subject Author
2/13/02
Read Problem with Cantor's diagonal argument
Henry
2/13/02
Read Re: Problem with Cantor's diagonal argument
Andy Berget
2/14/02
Read Re: Problem with Cantor's diagonal argument
Mike Oliver
2/14/02
Read Re: Problem with Cantor's diagonal argument
Doug Norris
2/14/02
Read Re: Problem with Cantor's diagonal argument
Keith Keller
2/14/02
Read Re: Problem with Cantor's diagonal argument
Dudley Brooks
2/14/02
Read Re: Problem with Cantor's diagonal argument
Mike Oliver
2/14/02
Read Re: Problem with Cantor's diagonal argument
Dudley Brooks
2/14/02
Read Re: Problem with Cantor's diagonal argument
Dave Seaman
2/14/02
Read Re: Problem with Cantor's diagonal argument
Dudley Brooks
2/14/02
Read Re: Problem with Cantor's diagonal argument
Dave Seaman
2/14/02
Read Re: Problem with Cantor's diagonal argument
Dudley Brooks
2/14/02
Read Re: Problem with Cantor's diagonal argument
Bob Kolker
2/14/02
Read Re: Problem with Cantor's diagonal argument
Dave Seaman
2/14/02
Read Re: Problem with Cantor's diagonal argument
Seth Dutter
2/14/02
Read Re: Problem with Cantor's diagonal argument
mareg@mimosa.csv.warwick.ac.uk
2/14/02
Read Re: Problem with Cantor's diagonal argument
Nico Benschop
2/14/02
Read Re: Problem with Cantor's diagonal argument
mareg@mimosa.csv.warwick.ac.uk
2/14/02
Read Re: Problem with Cantor's diagonal argument
Willondon
2/14/02
Read Re: Problem with Cantor's diagonal argument
Henry
2/14/02
Read Re: Problem with Cantor's diagonal argument
magidin@math.berkeley.edu
2/15/02
Read Re: Problem with Cantor's diagonal argument
Doug Magnoli
2/14/02
Read Re: Problem with Cantor's diagonal argument
mareg@mimosa.csv.warwick.ac.uk
2/14/02
Read Re: Problem with Cantor's diagonal argument
Dudley Brooks
2/14/02
Read Re: Problem with Cantor's diagonal argument
Nico Benschop
2/15/02
Read Re: Problem with Cantor's diagonal argument (re finite case)
Nico Benschop
2/15/02
Read cancel <3C6CD566.97EA8F20@chello.nl>
Nico Benschop
2/15/02
Read Re: Problem with Cantor's diagonal argument
Nico Benschop
2/14/02
Read Re: Problem with Cantor's diagonal argument
Dave Seaman
2/14/02
Read Re: Problem with Cantor's diagonal argument
Herman Jurjus
2/14/02
Read Re: Problem with Cantor's diagonal argument
Dave Seaman
2/15/02
Read Re: Problem with Cantor's diagonal argument
Jon and Mary Frances Miller
2/15/02
Read Re: Problem with Cantor's diagonal argument
Torkel Franzen
2/15/02
Read Re: Problem with Cantor's diagonal argument
Virgil
2/15/02
Read Re: Problem with Cantor's diagonal argument
Harlan Messinger
2/15/02
Read Re: Problem with Cantor's diagonal argument
Virgil
2/15/02
Read Re: Problem with Cantor's diagonal argument
Harlan Messinger
2/15/02
Read Re: Problem with Cantor's diagonal argument
Virgil
2/18/02
Read Re: Problem with Cantor's diagonal argument
Harlan Messinger
2/18/02
Read Re: Problem with Cantor's diagonal argument
Virgil
2/19/02
Read Re: Problem with Cantor's diagonal argument
Harlan Messinger
2/19/02
Read Re: Problem with Cantor's diagonal argument
Virgil
2/19/02
Read Re: Problem with Cantor's diagonal argument
Dudley Brooks
3/4/02
Read Re: Problem with Cantor's diagonal argument
Alexey Dejneka
3/4/02
Read Re: Problem with Cantor's diagonal argument
Torkel Franzen
3/4/02
Read Re: Problem with Cantor's diagonal argument
Alan Stern
2/16/02
Read Re: Problem with Cantor's diagonal argument
Chip Eastham
2/20/02
Read Re: Problem with Cantor's diagonal argument
SRK
2/14/02
Read Re: Problem with Cantor's diagonal argument
Dale Hurliman
2/14/02
Read Re: Problem with Cantor's diagonal argument
Randy Poe
2/14/02
Read Re: Problem with Cantor's diagonal argument
Henry
2/14/02
Read Re: Problem with Cantor's diagonal argument
Randy Poe
2/14/02
Read Re: Problem with Cantor's diagonal argument
Wade Ramey
2/14/02
Read Re: Problem with Cantor's diagonal argument
nospam@auerbachatunity.ncsu.edu
2/14/02
Read Re: Problem with Cantor's diagonal argument
Dudley Brooks
2/15/02
Read Re: Problem with Cantor's diagonal argument
Chris Menzel
2/15/02
Read Re: Problem with Cantor's diagonal argument
Dudley Brooks
2/14/02
Read Re: Problem with Cantor's diagonal argument
Phil Carmody
2/14/02
Read Re: Problem with Cantor's diagonal argument
Harlan Messinger
2/14/02
Read Re: Problem with Cantor's diagonal argument
Jim Heckman
2/15/02
Read Re: Problem with Cantor's diagonal argument
Randy Poe
2/15/02
Read Re: Problem with Cantor's diagonal argument
LarryLard
2/18/02
Read Re: Problem with Cantor's diagonal argument
Harlan Messinger
2/14/02
Read Re: Problem with Cantor's diagonal argument
George Greene
2/15/02
Read Re: Problem with Cantor's diagonal argument
Duran Castore
2/18/02
Read Re: Problem with Cantor's diagonal argument
Jonathan Hoyle

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