Re: axiom of choice and equivalent statements
- From: hrubin@xxxxxxxxxxxxxxxxxxxx (Herman Rubin)
- Date: 6 Feb 2006 13:19:47 -0500
In article <1hpFf.10242$fM1.71@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Norm Dresner <ndrez@xxxxxxx> wrote:
"Li Yi" <liyi.cn@xxxxxxxxx> wrote in message
news:1139145297.879035.68570@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Prove the following statements are equivalent:
(1) axiom of choice;
(2) for any sets A and B, it holds either A <= B or B <= A. (X <= Y
means that there is an injection from X to Y)
(3) Zorn's lemma
(4) Well-ordering Principle : every set can be well-ordered.
(5) Every binary relation can be single-valued: Given a binary relation
R, there exists a function F subset R such that dom F = dom R
(6) A is a nonempty set, then A^a is nonempty for any ordinal number a.
I know that (1)(3)(4) are equivalent and have a proof of (3)=>(2).
The simplest solution to your problem is to get from your school library a
copy of "Equivalents of the Axiom of Choice". First edition is Rubin &
Rubin. Second addition added someone else whose name I can't recall.
The second edition is largely a rewriting and expansion by
the same authors; it is self-contained. There is a book by
J. E. Rubin and Paul Howard, _Consequences of the Axiom of Choice_.
Norm
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@xxxxxxxxxxxxxxx Phone: (765)494-6054 FAX: (765)494-0558
.
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