# [FOM] Is CH vague?

Vaughan Pratt pratt at cs.stanford.edu
Fri Feb 1 00:39:53 EST 2008

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Martin Davis wrote:
> Here is one of the many equivalent form of CH:
>
> For every uncountable set of real numbers, there is a function that
> maps it *onto* the set of all real numbers.
>
> If this statement is vague it must be because there is something
> unclear about one of the concepts with which it deals. These concepts are:
> 1. real number
> 2. set of real numbers
> 3. uncountable set
> 4. function whose domain and range are sets of real numbers

Surely at least 1, 2 and 4 are vague (or indefinite if we're being
picky---Sol's comment on this when I asked him today which he had meant
was that "vague" and "indefinite" are vague and indefinite concepts,
and he didn't say "respectively" either).  This is because

5.  set of natural numbers (equivalently for the present purpose,
function whose domain is N and range is {0,1})

is vague.  1 depends on 5 because R = [0,1) x Z and [0,1) can be
identified with the non-cofinite sets of natural numbers (writing the
members of [0,1) in binary, the cofinite sets correspond to the bit
sequences that converge to 1), which must be vague if 5 is vague because
"cofinite set of natural numbers" is (surely) not vague.  (Z however as
a *particular* set is not vague.)  2 and 4 obviously depend on 1, and 3
arguably does too.

How can we pretend we know what a set of natural numbers is when we
can't even enumerate (i.e. well-order) them without resorting to Choice?

We grow comfortable with certain sets of numbers by long exposure.  We
come to understand the finite and cofinite sets early on, also the set
of natural numbers congruent to m mod n for any given m and n, and
finite unions thereof, also pairs of numbers and some effective
quotients thereof.  After working with progressively more sophisticated
examples for a long time, familiarity breeds comfort and we feel we
could handle any set of natural numbers.

So we gamely launch into the definition of real as a Dedekind cut,
meaning an order ideal in the rational line having no greatest element,
which looks very tame as sets of rational numbers go: how could an order
ideal possibly be hard to pin down exactly?  Not only is it countable,
but effectively so: just enumerate the rationals weeding out those
greater than every rational we want in our order ideal, namely those
above or equal to where we want the cut to go in the rational line.

Unfortunately that argument is circular because it presupposes we know
where the cut is.  Worse, there seems to be no good fix that removes
that circularity without accepting 5 as well defined.  If we could say
what a real is exactly, we could say what a set of natural numbers is
exactly, by the argument above.

All the evidence is that we haven't really pinned down what we're
talking about when we refer to 2^N.  We do know it's uncountable, but we
don't even know its immediate neighborhood, since what we're prepared to
commit to, specifically ZFC, fails even to determine whether the
cardinal immediately below it is countable.

Fortunately mathematical progress to date has been largely unimpeded by
this level of ignorance of its subject matter.  However exactly why that
has been the case is a very nice question not just for philosophers but
also for mathematicians.

A more practical question might be, does this ignorance hinder
mathematics in other ways than those understood to date such as not
knowing what cardinals are adjacent to the continuum?  A lunchtime talk
Grisha Mints gave at CSLI today on Terry Tao's recent writings on hard
vs. soft analysis seemed to be addressing a related question.  Soft
analysis views sets of natural numbers and real numbers as acceptable
objects of study and in general is satisfied with infinitistic notions,
hard analysis shifts the emphasis to effective sequences and insists on
keeping everything finitistic.

(I should add that I'm not at all clear as to the relationship between
Tao's program and descriptive set theory.  In particular are the
questions he's asking properly understood as within the scope of DST,
and if so do the descriptive set theorists regard them as new questions,
old questions they haven't been able to answer to date, or (unlikely)