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Kim Lee
science forum beginner
Joined: 09 May 2006
Posts: 3
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 Posted: Tue May 09, 2006 7:12 am Post subject:
Countable sets [0,1]
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So I have a few questions on countable/uncountable sets when dealing
with the closed interval [0,1].
Let < be the usual ordering on the real line.
1. Is there a countable set X subset of [0,1], ordered by <, and order
isomorphic to w^2?
2. Is there a Y subset of [0,1], Y well ordered by <, with Y
uncountable? Prove or Disprove.
I believe this is true. Is my proof here correct?
[0,1] is equinumerous to the real line and the cardinality of the real
line is 2^(aleph_0) which is uncountable. |
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Denis Feldmann
science forum beginner
Joined: 23 Apr 2006
Posts: 5
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| Posted: Tue May 09, 2006 7:22 am Post subject:
Re: Countable sets [0,1]
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Kimmy a écrit :
| Quote: | So I have a few questions on countable/uncountable sets when dealing
with the closed interval [0,1].
Let < be the usual ordering on the real line.
1. Is there a countable set X subset of [0,1], ordered by <, and order
isomorphic to w^2?
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Yes (use the set (1/2+1/4+...1/2^m)+(1/2^m(1/3+1/9+...1/3^p))
| Quote: |
2. Is there a Y subset of [0,1], Y well ordered by <, with Y
uncountable? Prove or Disprove.
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No (proof : use the fact that a non-denumerable series with terms
strictly positive diverges)
| Quote: |
I believe this is true. Is my proof here correct?
[0,1] is equinumerous to the real line and the cardinality of the real
line is 2^(aleph_0) which is uncountable.
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No : you forgot the all-important word "well-ordered" |
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David C. Ullrich
science forum Guru
Joined: 28 Apr 2005
Posts: 2250
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| Posted: Tue May 09, 2006 11:43 am Post subject:
Re: Countable sets [0,1]
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On Tue, 09 May 2006 01:12:12 -0600, Kimmy <kimmy4life@comcast.net>
wrote:
| Quote: | So I have a few questions on countable/uncountable sets when dealing
with the closed interval [0,1].
Let < be the usual ordering on the real line.
1. Is there a countable set X subset of [0,1], ordered by <, and order
isomorphic to w^2?
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In fact this is true with any countable ordinal in place of w^w.
| Quote: | 2. Is there a Y subset of [0,1], Y well ordered by <, with Y
uncountable? Prove or Disprove.
I believe this is true. Is my proof here correct?
[0,1] is equinumerous to the real line and the cardinality of the real
line is 2^(aleph_0) which is uncountable.
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************************
David C. Ullrich |
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Dave L. Renfro
science forum Guru
Joined: 29 Apr 2005
Posts: 570
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| Posted: Tue May 09, 2006 3:33 pm Post subject:
Re: Countable sets [0,1]
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Kimmy wrote (in part):
| Quote: | 1. Is there a countable set X subset of [0,1],
ordered by <, and order isomorphic to w^2?
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David C. Ullrich wrote (in part):
| Quote: | In fact this is true with any countable ordinal
in place of w^w.
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Or even for any countable linearly ordered set with
[0,1] replaced by the collection of rational numbers
in any specified open interval. More generally, any
nonempty [countable] linear ordering that is dense
in itself and unbounded contains order isomorphic
copies of every countable linear ordering. I think
this was proved by Cantor in the 1880's, but I don't
have my notes on linearly ordered sets with me
right now to double check this.
No ulterior motive for this post, by the way,
other than there's nothing I felt like commenting
on right now in the 11:14 5K Collatz conjecture
thread, so I thought I'd poke my nose into this
thread.
Dave L. Renfro |
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The World Wide Wade
science forum Guru
Joined: 24 Mar 2005
Posts: 790
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| Posted: Tue May 09, 2006 6:00 pm Post subject:
Re: Countable sets [0,1]
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In article <RICdnfaDqfpG3f3ZRVn-tA@comcast.com>,
Kimmy <kimmy4life@comcast.net> wrote:
| Quote: | 2. Is there a Y subset of [0,1], Y well ordered by <, with Y
uncountable? Prove or Disprove.
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HInt: For each x in Y, there exists the "next" n(x) in Y. The
intervals (x,n(x)) are pairwise disjoint. |
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