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

[Timothy Y. Chow]

On Mon, 9 Mar 2015, Arnon Avron wrote:
> (And there is a really no difference here between formulas with 5 
> alternations of quantifiers and formula with 6 alternations of 
> quantifiers... If you think otherwise, can you point out what is the 
> minimal number of alternations that makes formulas doubtful, and what is 
> special about that number?).

Again to play devil's advocate, suppose that one accepts the induction 
schema outright only for quantifier-free formulas?  Specific formulas with 
quantifiers could also be accepted on a case-by-case basis, but only after 
one has gone through the mental process of thinking about the formula and 
convincing oneself that induction for that property is "obvious."

> I do not see the connection between the two parts of this paragraph 
> (before and after the "Hence..."). Yes, I do know "many ways of 
> weakening the induction axiom to produce a system whose consistency 
> doesn't imply the consistency of PA". The simplest of them is to throw 
> induction away altogether (like in Q). Still, I do not have any picture 
> (let alone "fairly clear" one) what it could mean for the *natural 
> numbers* to exist, yet for PA to be inconsistent.

"Fairly clear" might be an overstatement, but as an analogy, let's 
consider the power set axiom P of ZF.  Someone might feel that they can 
grasp the set-theoretic universe (including of course the set of natural 
numbers) clearly enough to be convinced that ZF - P is consistent, yet not 
grasp the concept of an "arbitrary subset" with enough clarity to rule out 
the possibility that ZF is inconsistent.

Do you agree with that?  If so, then returning to PA, someone might feel 
that they understand what is meant by "1, 2, 3, 4, and so on indefinitely" 
yet not understand what an "arbitrary first-order property" of the natural 
numbers is with enough clarity to rule out the possibility that PA is 
inconsistent.  The induction that you describe for first-order properties 
is conceptually more complicated than the induction needed to grasp the 
successor operation.  And of course, we have technically precise ways of 
demonstrating that the greater conceptual complexity is real and not 
illusory.  Given this, I don't find it convincing to argue that induction 
for first-order formulas is part of the *fundamental meaning* of the 
natural numbers, any more than I find it convincing to argue that the 
power set axiom is part of the *fundamental meaning* of the set-theoretic 
universe.

Tim