[FOM] SOL vs. ZFC

José Ferreirós Domínguez josef at us.es
Mon Jun 24 03:37:22 EDT 2013

```Dear Joe Shipman,

We agree to an important extent: our disagreement comes down to whether the logical notion of "property" and the combinatorial notion of "subset" are fundamentally at odds, and this can already be sharply clarified in the case of properties of natural numbers and subsets of the natural numbers. -- Well, more exactly my claim is not that "property" and "set" are fundamentally at odds, but that their relation is fundamentally asymmetric. Properties can always be modelled by sets, but the inverse would be a metaphysical claim going beyond logic.

You write:

> I'm not going to let you get away without choosing one of the following alternatives or giving another one which they do not subsume:
>
> 1) My combinatorial notion of "property" is not only an improper use of that English word, but also incoherent and unsuitable for reasoning
>
> 2) My combinatorial notion of "property" is a coherent notion, however Hilbert's deductive calculus is an inappropriate formalization of that notion
>
> 3) My combinatorial notion of "property" is a coherent notion, and Hilbert's deductive calculus is an appropriate formalization of that notion, however that notion is not a "logical" one and Hilbert's calculus is really mathematics rather than logic

Ok, the question is fair. As you probably suspected, my views fall within 2), but let me comment a bit.

1) The combinatorial notion of “property” is of course an improper use of that English word, indeed of that word in any language -- even if we take into account its usage by logicians of the past all the way up to Russell (included), e.g. the way Frege used ‘Begriff’ and ‘Funktion’. It is only in mathematese of the last 100 years that such a usage of "property" can be found, so this is certainly far from being "classical".
The notion may be inevitably vague as well, i.e. characterizable only partially and mainly in a negative way (adopting arbitrary subsets means to reject any requirement of definability). But I would not claim that it is incoherent and unsuitable for reasoning. We do reason about it and can recognize e.g. that it grounds principles such as the Axiom of Choice.

2) Hilbert’s deductive calculus does not appropriately formalize that notion. I do not mean this just in the well-known sense that full SOL is incomplete. I mean to say that Hilbert's calculus falls very much short of formalizing the notion -- it just provides a partial characterization in the sense I mean above. Try to picture the complexity of the realm of arbitrary subsets of N, you may think of the binary tree, etc. It seems clear that we should go far beyond AC if we want to capture that structure. And this intuitive idea is of course supported by metatheoretic results.
In fact, no known principle formalizes the notion of arbitrary set of natural numbers adequately, at least not one that is uncontroversial. Woodin has insisted that the structure of Second-Order Arithmetic < P(N), N, +, ·, ∈ > is correctly characterized when we include a form of Projective Determinacy. If Woodin is right, according to the “standard model-theoretic" view of SOL, which identifies its validities purely semantically, Projective Determinacy is a purely logical consequence of the known principles of arithmetic. As you may well imagine, this seems to me close to a refutation of the "standard" view. (Notice also that the issue of the evidence supporting PD is complex and disputed; there is relevant work by Martin and by Steel on the conceptual or philosophical side of the problem.)

3) I would also argue that the combinatorial notion of property is not a logical one, and I have indeed suggested that full SOL is applied mathematics. Actually it is applied naïve set theory, and for that reason it is quite close to the ideas of the practising mathematician -- most mathematicians are naïve about set theory. Here it suffices if by ‘naïve’ we mean the opinion that the powerset operation is a perfectly clear and well-defined one. See again the remarks of Shelah I was quoting in a previous post. Also, I need not remind the members of FOM that relevant people (from Baire and Brouwer to Martin-Löf, and beyond) have insisted in the past that the extensional counterpart of your notion of property –the combinatorial notion of infinite set – is so far from being “logical” that it is not even a good candidate for a “mathematical” notion. (The imagined "full" semantics seems to know much more than any human being presently knows, even more than we can know.)

Best wishes,

Jose
Prof. José Ferreirós
Departamento de Filosofia y Logica