[FOM] Intuitionism and ill-defined domains

Nik Weaver nweaver at dax.wustl.edu
Sat Oct 22 01:52:55 EDT 2005

In reference to the question whether it is possible to imagine
checking the twin primes conjecture by mechanically running through
all natural numbers, Arnon Avron wrote:

> The upshot is that we cannot avoid thinking of *all* the integers.
> Neither do we need to be able to actually run through all the integers
> in order to know that for every integer n we come across it will be the
> case that n=n, or that either p(n) or its negation will be true,
> or that if we continue to check integers greater than n, then
> either we'll encounter a pair of twin primes or else we shall never
> encounter one. I cant imagine a third possiblity, and if you
> do than you have a much better imagination than me.
> In short: Classical logic is valid also for potential infinity!

That is well put, but I disagree on two grounds.  First, if you
accept that the term "all the integers" is meaningful, which I
understand you to say you do, then I don't think you're talking
about a potential infinity, you are talking about a completed
infinity which is grasped as a whole.  We must draw a distinction
between the quantifier "for any integer" (that may appear in some
indefinite future) and the quantifier "for all integers" (referring
to elements of a well-defined set, the set of all integers).  Only
the former can legitimately be used by someone who disbelieves in
a completed infinity.

It seems to me that you really are using the latter quantifier, not
the former.  This is shown by your comment about not seeing a third
possibility.  I think the existence of a third possibility in your
example seems absurd to you because you regard "continuing to check
integers greater than n" as a well-defined process, which I think
entails that you can conceive of the set of all integers as a
well-defined set.

(You might better have said "if you do then you have a much *worse*
imagination than I", because your dichotomy could only fail to be
recognized as meaningful by someone who lacks the ability to
imagine omega as a well-defined entity.)

My second point of disagreement is that I feel that if you are
willing to assert things like "if we continue to check integers
greater than n, then either we'll encounter a pair of twin primes
or else we shall never encounter one" then you are in fact able
to imagine running through all the integers.  (You seem to blur
the distinction between this being logically possible and it
being physically possible in our universe; the latter is not
at issue.)

Let's move this debate to an arena where it's easier to distinguish
between conceiving of something as a well-defined entity versus
conceiving of it as being necessarily incomplete and always capable
of extension.  Namely, consider the universe of sets.  I propose
the following test.  Let a "large cardinal axiom" be any statement
of the form "there exists a cardinal x such that P" where P is any
formula in the language of set theory.

Claim A: every large cardinal axiom has a definite truth-value.

Is it possible to accept this claim without being a platonist?  If
you disbelieve in the universe of sets as a well-defined complete
entity, have you any grounds for accepting claim A?  I say no.

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

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