[FOM] RE: 195:The axiom of choice
Matt Insall
montez at fidnet.com
Tue Nov 4 08:13:01 EST 2003
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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