[FOM] "Mathematician in the street" on AC
pratt at cs.stanford.edu
Tue Aug 18 02:12:44 EDT 2009
Aha, the "counted" terminology is just what I wanted, thanks Thomas.
Too bad it's not in wider use. What's a citation for the term that no
Wikipedia editor would complain about, in case it seems worth mentioning
in the article on countable sets?
There are many analogous situations where everyone is in agreement that
this sort of distinction makes a difference. One familiar example is
the distinction between R^n and an n-dimensional vector space over the
reals for finite n. It is a theorem of ZF that a basis exists for the
latter, whereas the former amounts to the same thing (up to isomorphism)
but equipped with a specific basis, which determines an inner product
and hence notions of angle and length. In consequence one can define
"n-dimensional Euclidean space" as R^n but not as an n-dimensional
vector space over R.
An example that gives computability students grief is the distinction
between a program for computing a partial recursive function and the
partial recursive function it computes. The latter is defined as one
for which a program exists. The confusion arises when one writes phi_x
to denote a partial recursive function; the student then has to
determine from context exactly who has access to the index x: the
program, the student, the instructor, an oracle... If the student is
still struggling with this when confronted with the recursion theorem,
that every partial recursive function can be implemented in a way that
gives it access to its own index, the theorem may seem either vacuously
true, meaningless, false, or (the intended impression) miraculous.
Yet another example is the category of sets and functions as usually
understood. This is cartesian closed, and bountifully so, namely in a
proper class of ways, but no one would want to appeal to its being so
closed in practice without first fixing a specific tensor product
(invariably cartesian product) and right adjoint A=>B thereof
(invariably the set of functions from A to B). More generally one
assumes coherence conditions that avoid the sort of craziness that
results from relying on uncorrelated existentials.
If "counted" were to become a generally recognized notion, the natural
theorem in ZF would be "Every counted union of counted sets is counted."
It is natural in the sense that it surely formalizes the intuitions of
those who consider it pretty much as obvious as the countability of N^2.
The version that is independent of ZF would be the less obvious variant,
"Every counted union of countable sets is countable" when defining
countable union as an omega-ary operation, and "Every countable union of
countable sets is countable" when defining countable union as a unary
operation mapping P^w(P^w(X)) to PX (where P^w(X) denotes the set of
countable subsets of X, and with the theorem being nonvacuous in this
last case only for uncountable X since P^w(X) = P(X) when X is countable).
Unfortunately when the only thing that's at stake is dependence on
something even weaker than AC, junior faculty just coming up for tenure
aren't likely to take that sort of subtlety terribly seriously.
Part of the problem here is that AC itself is too abstract to be
concretely meaningful to most "working mathematicians," and therefore
not something they feel comfortable passing value judgments on.
Accepting AC as true has something of the flavor of Pascal's Wager: if
AC is true then so are the theorems that appeal to it, but if AC is
neither true nor false then no harm has been done. (In making his wager
Pascal seems not to have reckoned with the possibility of an Antigod who
would consign him to eternal torment for worshipping God, the
counterpart of AC being false. Goedel showed that AC can't be false,
but who convinced Pascal that there is no Antigod?)
If I wanted to motivate someone to draw the distinction between a
countable set and a counted set, I would instead point out to them that
the former notion when applied to countably infinite sets entails
quantifying over uncountably many functions. I would then ask them
whether they were comfortable with the notion of countability depending
on quantifying over a domain that not only was uncountable but that
contained insane functions that, relative to each other, were wildly
intractable, going all the way up the arithmetic hierarchy and far
beyond. When put that concretely they might think twice about it.
But I would also ask them whether they thought Erdos would judge their
non-ZF proof (of the countability of countable unions of countable sets)
as "straight from The Book." This is not to imply that The Book must
outlaw every non-theorem of ZF ("every vector space is free" being a
great candidate), only those which can be moved into ZF by a better
choice of definition of some concept without losing their basic import.
Thomas Forster wrote:
> Dear Vaughan,
> Thank you for this nice exposition. The notion you are using is
> the one that Conway coined the phrase *counted set* for. However, i
> think you are being far too charitable in your description of normal
> mathematical practice. I think you are crediting these good people
> with using the notion of counted set *consistently* (in which case,
> as you observe, no AC is needed). Sadly what i think is going on is
> that the mathematician-in-the-trailer-park is engaged in a fallacy of
> equivocation between the two notions. Worse, they do not take kindly
> to having this equivocation pointed out to them.
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