[FOM] Re: Absoluteness of Clay prize problems
Dmytro Taranovsky
dmytro at mit.edu
Mon Aug 23 15:11:20 EDT 2004
Harvey Friedman wrote:
>In fact, in my opinion, the essence of mathematics is mostly Pi01 and
>predominantly at most Pi02.
I do not agree with this. I think that mathematics can be roughly separated
into three large areas: the study of finite structures or integers, the study
of real numbers and certain well-behaved functions on real numbers, and general
or set theoretical mathematics.
Natural mathematical theorems tend to avoid many alterating unbounded
quantifiers, so in number theory and related areas, the vast majority of
theorems can naturally be expressed as Pi-0-2 statements. Commonly used
functions whose domain is a subset of R tend to be continuous or otherwise
Delta-0-2. Statements about analysis tend to be expressible (after some
coding) as Pi-1-2 statements.
Many theorems in analysis--such as the theorem stating compactness of the unit
interval--are Pi-1-2 statements and are not reducible over RCA_0 (a weak base
theory) to arithmetical or Pi-1-1 statements. Of course, they are reducible
to arithmetical statements over ZFC, but the easiest to achieve the reduction
may be to prove them outright. Graph minor theorem is a Pi-1-1 statement that
is (as far as I know) not implied by any consistent arithmetical statement
over Pi-1-1-CA_0 (a theory that tends to suffice for ordinary as opposed to
"set theoretical" mathematics).
Pi-1-2 statements (provably) have the same truth value in all inner models and
generic extensions of V, so "ordinary" mathematics largely escapes the
independence results.
It should be noted that much of general mathematics is general primarily for
convenience and that the use of uncountable sets can be partially avoided. For
example, one can study countable fields instead of arbitrary fields with only a
partial loss of content.
Also, regarding the Continuum Hypothesis, the problem is an important basic
mathematical (or, some would say, metamathematical) problem in its own right,
so study related to CH is important even if it has no known (non Sigma-0-1)
Pi-0-1 or commercial applications. We still know relatively little and
there are many places where a compelling answer to the CH hypothesis could
come from. The theory of reasonably definable using subsets of omega_1 as
parameters sets of subsets of omega_1 is largery unexplored, and it could be
that the only plausible theory of such sets implies the CH. Based on my
beliefs about unity of knowledge, human potential, and importance of
basic study, I would conjecture that the eventual solution to the Continuum
Hypothesis will have important practical applications.
Dmytro Taranovsky
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