[FOM] Axioms that imply AC
John Baldwin
jbaldwin at uic.edu
Sat Feb 4 10:55:10 EST 2006
Standard references include
Equivalents of the Axiom of Choice (Jean Rubin, H. Rubin), North
Holland, 1963 (reprinted 1970).
Consequences of the Axiom of Choice (Jean Rubin, P. Howard),
Models of ZF set theory (Ulrich Felgner) SLN 223
Here is a link to a number of modern sources:
http://www.math.vanderbilt.edu/~schectex/ccc/choice.html
On Fri, 3 Feb 2006, William Tait wrote:
> The statement that there is a well-ordering of the universe (i.e. a
> universal choice function) is an example. For example, this is true
> in L. It implies that set theory can be coded as a theory of
> ordinals. Ronald Jensen [BSL 1 No. 4 (1995)] referred to this
> possibility as the 'Pythagorean' conception of set theory:
> 'Everything is number'.
>
> Bill Tait
>
> On Feb 3, 2006, at 1:33 AM, Andrej Bauer wrote:
>
>> I student of mine is writing an undergraduate thesis on theorems in
>> mathematics which are equivalent to the axiom of choice, which got me
>> thinking: are there (relatively well-known) statements in
>> mathematics, or
>> specifically set theory, which are _stronger_ than the axiom of
>> choice?
>> Some form of choice for classes perhaps? What use does it have?
>> There are of
>> course many weak forms of choice.
>>
>> Andrej Bauer
>> University of Ljubljana
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John T. Baldwin
Director, Office of Mathematics Education
Department of Mathematics, Statistics,
and Computer Science M/C 249
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