FOM: The Axiom of Choice -- positive and negative versions

[Joe Shipman]

Simpson:
>>
My feeling is that, on the contrary and perhaps somewhat
paradoxically, the full axiom of choice (= AC) is *more* plausible
than its restriction to sets of reals.  First, the naive idea of
choosing elements from nonempty sets seems completely general and
intuitive, almost a "law of thought"; to restrict it to sets of reals
strikes me as an artificial or arbitrary restriction.  Second, full AC
is equivalent to Zorn's lemma, which is traditionally regarded as a
nice principle with many nice consequences, while AC for sets of reals
leads to many pathological counterexamples, e.g. a non-measurable set,
or a set of reals X such that neither X nor R \ X contains a perfect
set.
<<

I agree that AC is more natural and intuitive in its general form (even
more so when you express it in the equivalent form "a product of
nonempty sets is nonempty"). Any "law of thought" ought to be stated in
terms of arbitrary collections rather than mathematical ones.  But you
raise an interesting issue regarding "positive" and "negative" versions
of AC.  The provable equivalence of Zorn's Lemma or similar versions
(which are used to prove the "good" general theorems in ordinary
mathematics") with the Well-Ordering Theorem (which is used to get the
"bad" or "pathological" counterexamples) seems so straightforward to me
that I don't see the existence of a nonmeasurable set or a set neither
containing nor disjoint from a perfect set as counterintuitive.

It's not clear why the reals "ought" to have a countably additive
translation invariant measure.  There does exist a finitely additive
translation invariant measure (which you prove using one of the "good"
versions of AC) and this is as far as my intuition goes, since the
identification of the mathematical reals with the intuitive continuum is
not at all perfect, and it is not clear the quasi-physical intuitive
continuum allows for the kind of infinite subdivision relevant to
countable additivity.

What I *do* find somewhat surprising and counterintuitive is the
Banach-Tarski theorem which applies in dimension >=3 (there is no
*finitely* additive rigid-motion-invariant measure) -- but rotational
invariance is a lot more complicated than translational invariance
because there is a free nonabelian group of rotations.  (I strongly
recommend Stan Wagon's book "The Banach-Tarski paradox".)

In the same way I find the "strong Fubini theorems" of the form "any
iterated integrals which exist are equal" intuitively appealing because
the only kind of invariance involved is under coordinate permutations,
not under arbitrary rotations, and this comes from axioms of symmetry of
the type discussed by Freiling (JSL 1986) and Riis (FOM 1998).  In my
thesis (T.A.M.S. 10/90) I show that the strong Fubini theorems follow
from the existence of a countably additive measure defined on all
subsets of R.  We already know such a measure can't be
translation-invariant but
that's not implausible--we should expect a finite subdivision to
be compatible with translations but the intuition is not clear for
infinite subdivisions.

-- Joe Shipman