[FOM] RE: 195:The axiom of choice

[kremer@uchicago.edu]

Unless I am too tired to think straight -- this is not a version of the axiom 
of choice.  It is rather a direct consequence of the union axiom.

(Proof if its needed:  let x be any set.  Let y be union(x).  Assume z in x and 
assume that for some w, w in z.  Then w in y (by def of union(x)) and so for 
some w, w in z and w in y.)

--Michael Kremer

Quoting Matt Insall <montez at fidnet.com>:

> Hervey asks about equivalents of the axiom of choice with fewer than 6
> quantifiers.  I would like to point out that in the GBN theory of classes,
> (one version of) the axiom of global choice has 4 quantifiers:
> 
> (GC)  There is a class whose intersection with every nonempty set is
> nonempty.
> 
> The formalized version of (GC) is as follows:
> 
> (\exists A)(\forall x)[(\exists y)(y\in x) implies (\exists y)(y\in x and y
> \in A)].
> 
> Now, if we relativize this to ZF so that we get a comparable statement of
> the axiom of choice, we get
> 
> (AC) To every set x there corresponds a set whose interesections with the
> nonempty members of x are nonempty.
> 
> This formalizes in the language of ZF as follows:
> 
> (\forall x)(\exists y)(\forall z)[(z\in x and (\exists w)[w\in z])
> implies (\exists w)(w\in z and w\in y)].
> 
> If I have made no mistakes, then it seems this is a statement of AC with 5
> quantifiers.
> 
> 
> 
> Dr. Matt Insall
> Associate Professor of Mathematics
> Department of Mathematics and Statistics
> University of Missouri - Rolla
> Rolla MO 65409-0020
> 
> insall at umr.edu
> montez at fidnet.com
> (573)341-4901
> 
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