Re: How much Choice is needed

In article <rvud539te2iufdeu9pp6vaub9g2goub2o4@xxxxxxx>,
Gonalo Rodrigues <nospam@xxxxxxxxxxxx> wrote:
Hi all,

This has probably been hashed and rehashed many a time - but recently
I was in a thread where a related issue came up, so I'd like the
experts input on this.

First, let me say that for me the Axiom of Choice is intuitively
obvious and it is the fuss about it that is a little bit
incomprehensible. It is intuitively clear to me that the product of
any family of sets exists and is non empty or that any surjection has
a section - even if no effective procedure for building such a section
is available - such a thingy exists somewhere out there in outer space
(as per a Sun Ra musical piece).

I have read many a time that for many results you do not need the full
strength of the axiom of choice and something weaker is enough. Here
go some examples:

1. Let me start with the Tychonoff theorem. This is probably the
single most important theorem in general topology since many spaces
arise as closed subspaces of a product of compact ones. Examples
include the Banach-Alaoglu theorem or the Gelfand-Naimark equivalence
between the category of C*-algebras and compact Hausdorff spaces. I am
not scholarly enough to know if you can do the proofs of these without
the Tychonoff theorem, but these two examples alone are already very
important pieces of mathematics.

As far as I know, Tychonoff theorem is equivalent to the full axiom of
choice (AC),

Correct.

but if you add up Hausdorffness then the axiom of
dependent choice (ADC) is enough. Can anyone confirm this?

No. It is equivalent to the prime ideal theorem. For
other equivalences, see the abstract of Rubin and Scott,
I believe in 1954.

2. Baire category theorem. Most of the more spectacular theorems with
complete metric spaces use this one in one form or other. This
includes the open mapping theorem and the principle of uniform
boundedness.

How much choice is needed? Is countable choice (ACC) enough? Not
really having thought about it, it sure seems so.

I do not think so, but dependent choice certainly is.

3. The Hahn-Banach theorem.

I have read that this one is strictly weaker than AC and that topos
theorists have managed to prove it in some topoi, but restricting
ourselves to set theory, how much choice is needed for this one?

MUCH less. I do not know how much.

4. Stone's theorem on extension of ideals in Boolean algebras. This
creeps in analysis via the Stone space functor.

I have read somewhere that this is also strictly weaker than AC but it
is enough to prove the existence of non-measurable sets. How much
Choice is needed for this one?

This is the prime ideal theorem.

These are examples mainly from topology and analysis. One from
algebra:

5. Every field has an algebraic closure. Same question for this one.

I suspect this is equivalent to choice, but I am not sure.

Feel free to clarify on the above examples or add more. Also, can
someone tell me if there are mathematicians that feel that ACC or ADC
is intuitively more obvious than full AC and why is it so. Having
devoted almost no thought to it, they all seem equally
obvious/non-obvious to me.

Best regards,
G. Rodrigues

For a large collection of equivalents and consequents of the
axiom of choice, with proofs and references, see the book
by Paul Howard and Jean Rubin, _Consequences of the Axiom of Choice_.
--
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
.

Relevant Pages

  • Re: Henkin Semantics
    ... It is equivalent to the Axiom of Choice. ... See Rubin and Rubin, ... are those of the Statistics Department or of Purdue University. ... Herman Rubin, Department of Statistics, Purdue University ...
    (sci.math.research)
  • Re: axiom of choice and equivalent statements
    ... Well-ordering Principle: ... copy of "Equivalents of the Axiom of Choice". ... First edition is Rubin & ... Herman Rubin, Department of Statistics, Purdue University ...
    (sci.math)
  • Re: Cardinal arithmetic question
    ... had proved that the Axiom of Choice implied the ... Well-Ordering Theorem, and after Frankel showed ... See H. Rubin and J. Rubin, ... if maxis always defined for infinite cardinals k and u ...
    (sci.math)
  • Re: Henkin Semantics
    ... for second order logic with Henkin semantics? ... It is equivalent to the Axiom of Choice. ...  See Rubin and Rubin, ... constants satisfying each statement for each substitution, ...
    (sci.math.research)
  • Re: axiom of choice and equivalent statements
    ... Well-ordering Principle: ... A is a nonempty set, then A^a is nonempty for any ordinal number a. ... 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. ...
    (sci.math)