[FOM] Remedial mathematics?
Timothy Y. Chow
tchow at alum.mit.edu
Mon May 23 11:57:53 EDT 2011
I would normally consider the comments below to be too elementary to post
to FOM, but in light of some of the recent discussions, I think it may be
helpful to state the obvious explicitly. The point I wish to make is that
the consistency of PA is trivial to prove by "ordinary mathematical
In a first-year graduate course on commutative ring theory, one typically
begins by stating the axioms for a commutative ring, and then giving some
examples. The first example everyone gives is Z, the ring of integers.
The verification that Z satisfies the axioms for a commutative ring is
usually considered too trivial to discuss in detail.
Perhaps because of its seeming triviality, some people may pass over this
event without noticing what is going on. In not so many words, the
lecturer or textbook writer for such a course is *proving the consistency
of the axioms for a commutative ring*, and doing so by saying that the
concrete example of Z satisfies all the axioms. Notably, the *existence
of Z is taken for granted*. There is no caveat in the books saying, "If
your philosophical prejudices permit you to believe in the existence of
the ring of integers as a completed infinite totality, then the ring of
integers is a model of the axioms for a commutative ring." The student is
not being asked to assume anything other than standard mathematical facts.
I repeat: Standard mathematical practice takes for granted the existence
of Z and its basic properties without comment or reservation.
Now let us consider PA, the axioms of first-order arithmetic. Since it
is standard mathematical practice to assume the existence of Z, and a
fortiori the existence of N, the only question is whether N satisfies the
axioms of PA. The only axiom that could possibly create an issue is the
induction axiom. But it is clear that a first-order formula defines a
precise property of N, on which we can of course perform induction. So N
is indeed an example of something that satisfies the axioms of PA. (I use
the word "example" here to underline the analogy with the graduate class
in commutative ring theory; of course, what is being proved is the
consistency of PA, but some people seem to have a knee-jerk reaction that
as soon as the word "consistency" is introduced then we're no longer doing
mathematics but doing philosophy.)
I've belabored this point because it strikes me that some of the skeptics
of the consistency of PA aren't fully cognizant of this "trivial" proof of
the consistency of PA. Indeed, f.o.m. experts may unwittingly encourage
non-experts to ignore this trivial proof, by focusing on more interesting
proofs such as Gentzen's. By citing Gentzen's proof, they may create the
impression that the consistency of PA is, by usual mathematical standards,
a highly non-trivial fact whose proof demands some sophisticated argument.
In my view, skeptics of the consistency of PA should first of all explain
why they are dissatisfied with the trivial proof, when under normal
circumstances they would not only accept it without objection but might
even regard it as being too simple to bother verifying in detail to a
class of graduate students. I think this would be more productive then
jumping ahead to objections to Gentzen's proof prior to addressing the
elephant in the room.
Some skeptics, like Nelson, of course have an answer---they don't believe
in N. But this is a pretty radical position, and I think skeptics should
be forced to acknowledge how radical a position they are taking and how
much standard mathematical practice they are throwing out with the
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