[FOM] The boundary of objective mathematics

[Paul Budnik]

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).
>   
As a practical matter I agree that arithmetical conclusions reached in 
the first way and probably the second are correct and thus the 
statements are meaningful in having useful consequences. However I do 
not think you can conclude a statement is objective from its 
implications.  It is possible to construct all sorts of crazy theories 
that have the same arithmetical consequences. The objectivity of a 
statement depends on the assumptions that lead to it and not the 
conclusions it leads to.

I think we need a different formulation for the foundations of 
mathematics that explicitly and internally recognizes the unavoidable 
limitations of any formal system. In the light of the Lowenheim Skolem 
theorem, I think Cantor's proof that the reals are not countable should 
be thought of as the first major incompleteness theorem. Reals are human 
constructions and Cantor proved one can always construct more of them 
and not that there exist more reals than integers.


> 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.)
>
> My own view is that any statement about sets of bounded rank is 
> meaningful, and that statements like GCH which involve universal 
> quantification for sets of arbitrary rank are vague. In between these 
> two classes of statements are existential statements with no bound upon 
> the rank -- statements like "a measurable cardinal exists". (In other 
> words, the statement "no measurable cardinal exists" is vague while "a 
> measurable cardinal exists" is less so; this asymmetry is not 
> unreasonable and is analogous the asymmetry between the statements "the 
> Riemann Hypothesis is not provable" and "the Riemann Hypothesis is 
> provable" which have different epistemological statuses).
>
> -- JS
>   
I suspect all of ZFC and many of its proposed extensions have an 
interpretation in terms of properties of recursive processes running in 
a potentially infinite universe. I think all of the implications of 
systems that have such an interpretation are meaningful  and they are 
objective in that interpretation, but not objective as formulated in 
ZFC.  Figuring out this interpretation is extremely difficult beyond a 
certain point. It almost certainly requires computer aided proofs and a 
somewhat different approach to developing mathematics. However every 
finite formal system is a computer program for enumerating theorems and 
one can in theory come to a deep understanding of the combinatorial 
implications of a relatively short computer program like the axioms of ZFC.

Paul Budnik
www.mtnmath.com