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Topic: Every Subspace of R^N has an Orthogonal Basis?
Replies: 175   Last Post: Jul 4, 2006 9:20 AM

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 Denis Feldmann Posts: 533 Registered: 4/23/06
Re: Cardinals of fields (was, Re: Every Subspace of R^N has an Orthogonal
Basis?)

Posted: Jul 2, 2006 4:39 PM
 Plain Text Reply

Herman Rubin a écrit :
> In article <44a7eabb\$1\$fuzhry+tra\$mr2ice@news.patriot.net>,
> Shmuel (Seymour J.) Metz <spamtrap@library.lspace.org.invalid> wrote:

>> In <e80t3h\$ai5\$1@panix2.panix.com>, on 06/29/2006
>> at 11:54 AM, lrudolph@panix.com (Lee Rudolph) said:

>
>>> I am using "the set X has a cardinality" to mean "there is a
>>> bijection between X and a cardinal number", where a "cardinal number"
>>> is an initial ordinal.

>
>> While I don't like that definition, I agree that it makes your
>> statement meaningful.

>
> It does, but it makes a lot of things more complicated.
> With it, one cannot discuss cardinalities of sets which
> cannot be well-ordered.
>
> There is the rank definition of Scott if the sets of
> every given rank form a set.
>
> But the ideas of comparing the cardinalities of sets do
> not depend on any of these, and are what are used in practice.
>

>>> You appear to be using "cardinality" to mean "equivalence class
>>> under bijection". Qualms (perhaps unjustified) about proper classes
>>> keep me from talking that way these days.

>
>> I understand the qualms, but to me it appears to be throwing out the
>> baby with the bath water.

>
> Agreed. The baby does not need a name.
>

>> IAC, I'm hoping that someone has a definitive answer for Aleph_0 <
>> dimension < C.

>
> Even with the Axiom of Choice, this cannot be answered.

This? With choice, it is pretty obvious that any infinite set can be
made into a Q-vector space, which was the original question, IIRC...

> There are some alephs which C cannot be, but "most" of
> them work. Also, aleph_1 and C might be incomparable,
> but we can say that aleph_1 < 2^(2^C).

Date Subject Author
6/26/06 Hatto von Aquitanien
6/26/06 Lee Rudolph
6/26/06 Jose Carlos Santos
6/26/06 Hatto von Aquitanien
6/26/06 Jose Carlos Santos
6/26/06 Hatto von Aquitanien
6/26/06 magidin@math.berkeley.edu
6/26/06 Robert Low
6/26/06 James Dolan
6/26/06 Hatto von Aquitanien
6/26/06 magidin@math.berkeley.edu
6/26/06 Hatto von Aquitanien
6/26/06 magidin@math.berkeley.edu
6/26/06 Hatto von Aquitanien
6/26/06 magidin@math.berkeley.edu
6/27/06 Hatto von Aquitanien
6/27/06 Virgil
6/27/06 Hatto von Aquitanien
6/27/06 Gene Ward Smith
6/27/06 Hatto von Aquitanien
6/28/06 Brian Quincy Hutchings
6/27/06 Michael L. Siemon
6/27/06 Hatto von Aquitanien
6/27/06 Michael L. Siemon
6/27/06 Virgil
6/27/06 magidin@math.berkeley.edu
6/27/06 Tim Golden http://bandtech.com
6/27/06 magidin@math.berkeley.edu
6/28/06 David C. Ullrich
6/28/06 Tim Golden http://bandtech.com
6/28/06 magidin@math.berkeley.edu
6/28/06 Tim Golden http://bandtech.com
6/29/06 Robert Low
6/29/06 Tim Golden http://bandtech.com
6/29/06 David C. Ullrich
6/27/06 Dave Seaman
6/27/06 Hatto von Aquitanien
6/27/06 David C. Ullrich
6/27/06 Hatto von Aquitanien
6/27/06 David C. Ullrich
6/27/06 magidin@math.berkeley.edu
6/26/06 Virgil
6/29/06 Pinky
6/29/06 Hatto von Aquitanien
6/26/06 Gene Ward Smith
6/27/06 Hatto von Aquitanien
6/27/06 Virgil
6/27/06 Hatto von Aquitanien
6/27/06 Gene Ward Smith
6/27/06 Hatto von Aquitanien
6/27/06 Gene Ward Smith
6/27/06 Gene Ward Smith
6/27/06 Hatto von Aquitanien
6/27/06 Gene Ward Smith
6/27/06 Mariano
6/26/06 Virgil
6/27/06 Hatto von Aquitanien
6/27/06 Virgil
6/27/06 Hatto von Aquitanien
6/27/06 Virgil
6/28/06 Hatto von Aquitanien
6/28/06 David C. Ullrich
6/28/06 Hatto von Aquitanien
6/29/06 David C. Ullrich
6/29/06 Hatto von Aquitanien
6/29/06 David C. Ullrich
6/29/06 Lee Rudolph
6/29/06 K. E. Pledger
6/29/06 Hatto von Aquitanien
6/29/06 Denis Feldmann
6/29/06 Hatto von Aquitanien
6/29/06 Denis Feldmann
6/29/06 Hatto von Aquitanien
6/29/06 Virgil
6/29/06 magidin@math.berkeley.edu
6/29/06 David C. Ullrich
6/29/06 Gene Ward Smith
6/29/06 Hatto von Aquitanien
6/29/06 magidin@math.berkeley.edu
6/29/06 Hatto von Aquitanien
6/28/06 magidin@math.berkeley.edu
6/27/06 David C. Ullrich
6/26/06 Jose Carlos Santos
6/26/06 David C. Ullrich
6/26/06 Robert Low
6/26/06 David C. Ullrich
6/26/06 David C. Ullrich
6/26/06 Hatto von Aquitanien
6/26/06 Robert Low
6/26/06 magidin@math.berkeley.edu
6/26/06 Hatto von Aquitanien
6/26/06 David C. Ullrich
6/27/06 kunzmilan@atlas.cz
6/27/06 Hatto von Aquitanien
6/27/06 Robert Low
6/27/06 Lee Rudolph
6/27/06 Denis Feldmann
6/27/06 Hatto von Aquitanien
6/27/06 Robert Low
6/28/06 David C. Ullrich
6/28/06 Hatto von Aquitanien
6/29/06 David C. Ullrich
6/29/06 Hatto von Aquitanien
6/29/06 Virgil
6/30/06 Hatto von Aquitanien
6/30/06 Denis Feldmann
6/30/06 Hatto von Aquitanien
6/30/06 Denis Feldmann
6/27/06 Hatto von Aquitanien
6/28/06 Robert Low
6/27/06 Denis Feldmann
6/27/06 David C. Ullrich
6/27/06 Virgil
6/28/06 Hatto von Aquitanien
6/28/06 Robert Low
6/28/06 Hatto von Aquitanien
6/28/06 Robert Low
6/28/06 Gene Ward Smith
6/28/06 Hatto von Aquitanien
6/28/06 Hatto von Aquitanien
6/28/06 Lee Rudolph
6/28/06 Mariano
6/28/06 Mariano
6/29/06 Lee Rudolph
6/29/06 magidin@math.berkeley.edu
6/29/06 Shmuel (Seymour J.) Metz
6/29/06 Lee Rudolph
6/30/06 Bertie Reed
6/30/06 Denis Feldmann
6/30/06 Mariano
6/30/06 Denis Feldmann
6/30/06 Mariano
6/30/06 Aatu Koskensilta
7/2/06 Shmuel (Seymour J.) Metz
7/2/06 Herman Rubin
7/2/06 Denis Feldmann
6/29/06 Mariano
6/29/06 Gene Ward Smith
6/28/06 David C. Ullrich
6/29/06 Hatto von Aquitanien
6/29/06 Virgil
6/29/06 Robert Low
6/29/06 Lee Rudolph
6/29/06 Virgil
6/29/06 Lee Rudolph
6/29/06 Virgil
6/29/06 W. Dale Hall
6/30/06 Robert Low
6/30/06 Virgil
6/30/06 Hatto von Aquitanien
6/30/06 Virgil
6/30/06 Hatto von Aquitanien
6/30/06 magidin@math.berkeley.edu
7/1/06 Hatto von Aquitanien
7/1/06 magidin@math.berkeley.edu
7/2/06 Hatto von Aquitanien
7/2/06 Denis Feldmann
7/2/06 David C. Ullrich
7/2/06 Hatto von Aquitanien
7/2/06 magidin@math.berkeley.edu
7/3/06 Gerry Myerson
7/3/06 Hatto von Aquitanien
7/3/06 David C. Ullrich
7/3/06 magidin@math.berkeley.edu
7/3/06 Hatto von Aquitanien
7/3/06 magidin@math.berkeley.edu
7/3/06 Hatto von Aquitanien
7/3/06 Virgil
7/3/06 David C. Ullrich
6/30/06 Robert Low
6/29/06 Hatto von Aquitanien
6/29/06 Virgil
6/29/06 Hatto von Aquitanien
6/29/06 Virgil
7/4/06 kunzmilan@atlas.cz
7/4/06 Lee Rudolph

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