Lacuna in "Is Frege's Definition of the Ancestral Adequate?"

Ran Lanzet has pointed out a significant lacuna in the proof of the main result in my paper "Is Frege's Definition of the Ancestral Adequate?" This has been repaired in the 'pre-publication' version of the paper, which can be downloaded here. See p.21 of that document.

I had certainly thought of the missing case, and seem to recall that at some point I'd introduced a 'simplifying assumption' that allowed me to ignore it. But that assumption is not mentioned in the published version of the paper, and it isn't nearly as easy as I'd supposed to see that it's permissible (which is perhaps why I removed it, but without fixing the affected part of the proof).

The Squeezing Argument in "Is Frege's Definition of the Ancestral Correct?"

I mentioned in an earlier post my paper "Is Frege's Definition of the Ancestral Correct?" It has now been refereed at Philosophia Mathematica and officially accepted for publication. One of the two reports was unusually helpful and led to some significant improvements in the final version, which is now available online.

The most helpful comment caught a nasty thinko in one of the proofs. But the most interesting observation this referee made was that perhaps the central argument of the paper is a so-called "squeezing argument". (See Robbie Williams's discussion of such arguments here, and Peter Smith's here, a version of which was eventually published in Analysis.)

The rough structure of such arguments is as follows. Suppose there is some intuitive notion I and you want to show that some rigorous notion R is co-extensive with I. Then one way to do so is as follows. Suppose that it is uncontroversial that R gives a necessary condition for I. And suppose further that we can find a different rigorous notion Q that uncontroversially gives a sufficient condition for I. So, to put it set-theoretically, we have:

Q ⊆ I ⊆ R

Then if we can show rigorously that R is sufficient for Q, i.e., that R ⊆ Q, then it will follow that both Q and R are co-extensive with I. As it's put, I has been "squeezed" between  Q and R.

The way this works in the paper is that I is the intutive notion of the ancestral; R is Frege's definition; and Q is an alternative definition that I give and claim, in fact, is intensionally correct. In response to an objection to the intensional correctness of that definition, however, I fall back on this squeezing argument.

This makes at least three instances of this sort of argument: The original, in Kreisel, which is meant to show that the model-theoretic account of validity is extensionally correct; Smith's, which is supposed to show that Turing's analysis of computability is extensionally correct; and now this one. Are there others? I'm guessing maybe there are?

New Paper: Is Frege's Definition of the Ancestral Correct?

I've posted a new paper to my website, titled "Is Frege's Definition of the Ancestral Correct?" (PDF here) The paper is scheduled to appear in a special issue of Philosophia Mathematica edited by Roy Cook and Erick Rech.
Abstract:

Why should one think that Frege's definition of the ancestral is correct? It can be proven to be extensionally correct, but the argument uses arithmetical induction, and that fact might seem to undermine Frege's claim to have justified induction in purely logical terms—a worry that goes back to Bruno Kerry and Henri Poincaré. In this paper, I discuss such circularity objections and then offer a new definition of the ancestral, one that is intended to be intensionally correct; its extensional correctness then follows without proof. It can then be proven to be equivalent to Frege's definition, without any use of arithmetical induction. This constitutes a proof that Frege's definition is extensionally correct that does not make any use of arithmetical induction, thus answering the circularity objections.

In the general case, the new definition is fairly complicated. But in the special case of the concept of natural number, it reduces to:

n is a natural number iff there exists a Dedekind finite concept (or set) F such that F0, Fn, and ∀x∀y[Fx & Pxy & x≠n → Fy]

The last condition says that F is closed under successors, except that it need not be true of the successor of n. The point, which has also been noted (independently) by Aldo Antonelli and Albert Visser, is that Dedekind finitude can be used here, so the definition is non-circular. The intuitive idea is just that, if n is not finite, then the other three conditions force every natural number to be F, in which case of course F is Dedekind infinite. If n is finite, by contrast, then F is just [x: 0 ≤ x ≤ n].
One can then go onto prove induction from this definition, using many of the results Frege proves for his own definition of the ancestral.