[FOM] "Proof" of the consistency of PA published by Oxford UP

Gyorgy Sereny sereny at math.bme.hu
Mon Mar 2 17:19:42 EST 2015

Dear Fomers,

I would like to inform you about a strange publication.
I have just come across a book newly published by
Oxford University Press:

    The Consistency of Arithmetic: And Other Essays Hardcover
    24 Jul 2014 by Storrs McCall (Author)


The last sentence of the review found in this page of amazon
about the book is intriguing:

    The eponymous first essay contains the proof of
    a fact that in 1931 Kurt Gödel had claimed to be
    unprovable, namely that the set of arithmetic
    truths forms a consistent system.

And indeed, the first one in the collection of essays by the
philosopher Storrs McCall has the title: "The Consistency
of Arithmetic". I do not have the book at hand, so I was able
to look into it only as a Google book. The first lines of
the essay on p.8. are:

   Is Peano arithmetic (PA) consistent? This paper contains a
   proof that it is: a proof moreover, that does not lie in
   deducing its consistency as a theorem in a system with axioms
   and rules of inference. [...]  If there is to be a genuine
   proof of PA's consistency, it cannot be a proof
   relative to the consistency of some other, stronger system,
   but an absolute proof, such as the proof of consistency of
   two-valued propositional logic using truth-tables. Axiomatic
   proofs we may categorize as "syntactic", meaning that they
   concern only symbols and the derivation of one string of
   symbols from another, according to set rules. "Semantic"
   proofs, on the other hand, differ from syntactic proofs
   in being based not only on symbols but on a non-symbolic,
   non-linguistic component, a domain of objects. If the sole
   paradigm of "proof" in mathematics is "axiomatic proof",
   in which to prove a formula means to deduce it from axioms
   using specified rules of inference, then Gödel indeed appears
   to have had the last word on the question of PA
   consistency. But in addition to axiomatic proofs there is
   another kind of proof. In this paper I give a proof of
   PA's consistency based on a formal semantics for PA. To my
   knowledge, no semantic consistency proof of Peano
   arithmetic has yet been constructed.

Later, at the bottom of the p.9, one can read the following
remark concerning the method of the proof:

   The proof constructed in this paper [...] is based on a
   non-linguistic component, [...] a _physical domain of
   three-dimensional cube-shaped blocks_.
   (emphasis mine)

Once more, this text was published by Oxford University Press.
It's bizarre, isn't it?

Gyorgy Sereny

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