[FOM] Intuitionism, predicativism, and ill-defined domains

[Nik Weaver]

Arnon Avron disputes my assertion that the law of excluded
middle may fail when the domain of discourse is indefinite.
What makes the issue subtle is the question: what does it
mean --- indeed, can it have any precise meaning --- for a
domain of discourse to be indefinite?

I am specifically thinking of a situation where I am interested
in reasoning about some type of object, such that I am able to
recognize whether any given object is of the desired type but I
have no conception of there being a well-defined collection of
all objects of the given type.

For instance, finitists might adopt this attitude toward the
natural numbers: individual numbers are not problematic but
the set of all numbers is.  Predicativists in the tradition
of Russell and Weyl would take the same attitude toward the
power set of omega.  Probably many people, without thinking
about the question too deeply, would take this attitude toward
the universe of sets.  But is it actually a coherent position?
Is it possible to reason logically on such a basis?

I am proposing that it is, *provided one uses intuitionistic
logic*.  This is what I find so remarkable about intuitionistic
logic, because the concept of an indefinite domain of discourse
at first seems to be too vague to deal with in any meaningful
way.  Yet intuitionistic logic, I claim, is precisely suited
to this situation.

I suspect that many people would not accept that every large
cardinal axiom has a definite truth-value.  Yet they might
not think about the fact that this means they are denying
the law of excluded middle and must therefore reason
intuitionistically.

To give another example, predicativists have some difficulty
with assertions of the form "such-and-such a recursive ordering
of omega is a well-ordering" (meaning, to be precise, there are
no proper progressive subsets --- we must be specific because
there are several classically equivalent versions of the concept
of well-ordering which are not predicatively equivalent).  The
problem is that assertions of this form involve a quantification
over the power set of omega, which is not predicatively recognized
as a legitimate entity.  In fact this has been a source of a great
deal of confusion in the predicativist literature.  It is incorrect
to deny a predicativist the right to make any assertion of this
type; for example, he surely can recognize that the standard
ordering on omega is a well-ordering.  Not only can he see that
no proper progressive subset may be found in whatever partial
universe is currently available, he can also recognize that it
is impossible for such a set to appear in any future universe.
But in the other direction, a predicativist should not be willing
to generally accept LEM for all statements of this type.  For more
complicated orderings of omega, the indefiniteness of the process
by which the universe of sets can be enlarged might entail
indefiniteness as to whether a proper progressive subset may
appear in some future universe.

If you do accept LEM, say with regard to the natural numbers,
I think this means that you grasp omega as a completed whole.
A true finitist would not accept the dichotomy "if we keep
searching through omega forever, either we will perpetually
continue finding new prime pairs or we will stop finding new
prime pairs at some point" because he would not accept that
"keep searching through omega forever" is a well-defined
concept.  If you are cetain it *is* a well-defined concept
then I think you do grasp omega as a completed whole.

I discuss the suitability of intuitionistic logic in the
predicativist setting at length in sections 2.2 and 2.3 of
my Gamma_0 paper.  (The "for all" versus "for any" distinction,
which goes back to Russell, is also discussed there.)  I am
optimistic that one day large numbers of people will read and
comprehend this paper!

Nik Weaver
Math Dept.
Washington University
St. Louis, MO 63130 USA
nweaver at math.wustl.edu