[FOM] The boundary of objective mathematics

[Monroe Eskew]

Analysts seem to take many infinitary statements as meaningful,
without regard to their potential consequences for number theory.  And
set theory has found many statements in real analysis to be
independent of ZFC.

Further, it is rare to run across a modern mathematician in any area
who (1) does not do logic, and (2) has doubts about the meaningfulness
of well-formed statements.  But that may just be a skewed impression
from my limited experience.

On Thu, Mar 12, 2009 at 8:22 AM,  <joeshipman at aol.com> wrote:

> The practical attitude many mathematicians seem to take is that
> statements in the first-order language of arithmetic OR statements of
> higher type which have arithmetical consequences are meaningful. For
> example, Schanuel's conjecture cannot be formulated in arithmetic but
> using it one can prove that (e^(e^n)) is never an integer when n is an
> integer, a statement which can be given an arithmetical formulation in
> terms of convergence of computations. The existence of a (countably
> additive) real-valued measure on all subsets of the continuum is a
> statement of even higher type which has useful arithmetical
> consequences such as Con(ZFC).
>
> Many mathematicians would also declare as meaningful statements those
> which are set-theoretically absolute (have the same truth value in all
> transitive models of ZFC). For example, the Invariant Subspace
> Conjecture (all bounded linear operators on Hilbert space have
> nontrivial invariant subspaces) does not have any arithmetical
> consequences that I know of, but is considered to be one of the major
> open problems in mathematics. (Of course one can't prove that the
> Invariant Subspace conjecture has no arithmetical consequences without
> proving it consistent, which might be no easier than proving it
> outright.)
>