[FOM] Is CH vague?

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

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) 
questions they've already answered?)

I spoke to Sol at the talk and mentioned that his view of CH as vague 
was currently being discussed on FOM.  It would be great if he could 
weigh in.

Vaughan Pratt

More information about the FOM mailing list