FOM: revised assessment of mathematical undecidability
Stephen G Simpson
simpson at math.psu.edu
Wed Mar 3 20:07:30 EST 1999
To: John Kadvany
John,
In your paper `Reflections on the Legacy of Kurt G"odel: Mathematics,
Skepticism, Postmodernism', you surveyed the `postmodernist condition'
in f.o.m., as follows:
G"odel's First Theorem showed that undecidability was endemic to
mathematical reasoning and a major portion of foundational research
since the 1930s has been involved in exploring *mathematically*
important conjectures that are undecidable with respect to various
``foundational'' systems-- the ironic quotation marks now being
mathematically necessary. G"odel's original undecidable sentence
is of no known mathematical interest apart from its undecidability
in elementary arithmetic and its primary role in logical research,
and for many years it was an open problem to find a
non-metamathematical sentence similarly undecidable in Peano
arithmetic; the first such sentence was found in 1977 by Jeff Paris
and Leo Harrington. Another line of research following from the
model set by the First Theorem turns the undecidability issue
around. Why not investigate alternative mathematical systems that
*do* decide some of the mathematically interesting undecidable
propositions one way or the other? Within any one of these
systems, one can investigate those new theorems which are provable
or refutable, but undecidable with respect to relatively lean
systems such as Zermelo-Fraenkel (ZF) set theory without the axiom
of choice. Cantor's Continuum Hypothesis, regarding the order of
infinity of the real number line, is the most famous undecidable
statement investigated in this way, but there are today several
such propositions of mathematical, and not only metamathematical,
interest. One of the dominant strategies therefore in
post-G"odelian foundational studies is reflected in these attempts
to prove to mathematicians that metamathematics has a direct
bearing on concrete mathematical problems, but this progress has
been made mostly *across* several foundational theories, primarily
ones with specialized consequences for real analysis and properties
of the real number line.
My question for you is, how would you revise this assessment now? It
appears that a major revision is needed in light of spectacular
results obtained subsequent to 1977. Consider independent
combinatorial statements such as, most recently, propositions 2 and 6
in Friedman's `Free Sets/Large Cardinals' abstract in his FOM posting
of 27 Feb 1999 01:43:18. Here Friedman is producing absolute, finite
combinatorial, mathematically natural statements that require large
cardinals to prove. He is no longer talking about the real line but
rather finite combinatorics, yet the independence is from ZF and
stronger systems, not from PA. Also, the large cardinal hierarchy is
linearly ordered by consistency strength, so this perhaps brings some
amount of order to your `postmodern chaos'.
-- Steve
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