[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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