FOM: revised assessment of mathematical undecidability

Stephen G Simpson simpson at
Wed Mar 3 20:07:30 EST 1999

To: John Kadvany


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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