Re: Axiom of Induction???

In article <56ts28F2akevpU1@xxxxxxxxxxxxxxxxxx>,
Dennis Ritchie <dmr@xxxxxxxxxxxxx> wrote:

"Herman Rubin" <hrubin@xxxxxxxxxxxxxxxxxxxx> wrote in message
news:eubcik$187a@xxxxxxxxxxxxxxxxxxxxxxx
....
With the axiom of choice, much can be done which cannot
be done otherwise. However, the negation of the axiom
of choice is consistent, and one typically wants the
weakest system which can be used....

When you (or others) say "the negation of the axiom of choice",
what exactly do you mean? Is it just (say) ZF, but not including
the AC, or an axiom that explicitly denies the existence of
certain choice sets?

Dennis

There are oodles of models which do not satisfy the axiom
of choice. One could certainly include an axiom that such
happens. This is rarely done. For example, you could
include an axiom which states the universe can be linearly
ordered, and another which states that there is a set
whose Hartogs function (always finite or an aleph) is
aleph_0. This would given an infinite set which is not
transfinite.

I only know of one "constructive" axiom which has been
proposed which negates the axiom of choice. This is
the axiom of determinateness, which is that every game
in which players put down digits altenatively, with one
player trying to get the resulting number in a set, and
the other player not, is determined. This is assuming
perfect information, and the paradox from the axiom
of choice is that there are such games which do not
have values. It has been shown that the consistency
of this axiom is equivalent to that of measurable
cardinals.


--
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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