[FOM] What is second order ZFC?

[Harvey Friedman]

Thanks to Richard Heck for replying to my question designed to get more
specific about just what 2nd order systems are in general and in the case
of ZFC - and what jobs they do or not do.

>
> Second-order ZFC is the theory, formulated in a second-order language,
> whose axioms are the second-order versions of the axioms of first-order
> ZFC. Its theorems are the second-order consequences of those axioms. The
> problem is now to explain "second-order consequence". But, as always,
> consequence is truth-preservation, so what we need to know is how to define
> truth for a second-order language. Since quantification will be handled
> largely as Tarski taught, all we really need to know is what "instances" a
> quantified formula has.
>

This is one of the more common and reasonable ways of talking about second
order ZFC as a system.

But first note that there is no notion of derivation present in the above
version. Thus 2nd order ZFC is not even remotely doing the work for f.o.m.
that first order ZFC does.

Also note that we have no way of even determining whether or not, say, the
continuum hypothesis CH, is or is not a second order consequence. In fact,
we easily see that

1) CH is a second order consequence if and only if CH is true.
2) not CH is a second order consequence if and only if CH is false.

This is one of the worst possible states of affairs for a "foundation" for
mathematics, which makes the above formulation completely unusable for
various purposes.

But how to fix these fatal flaws?

Easy. Just move to the first order associate, which is something like
Morse-Kelly class theory, or more meekly, NBG. These are minor variants of
ordinary first order ZFC.

Boolos's solution was to appeal to plural quantification, the locus
> classicus for that discussion being his paper "To Be Is To Be a Value of a
> Variable". Boolos insists, plausibly enough, that there are some sets none
> of which is an element of itself, but denies, sensibly enough, that there
> is any set of all the sets that are not elements of themselves. The
> locution "there are some sets" should not, therefore, be regarded as
> quantifying over sets.
>
> Boolos seems to have regarded plural quantification (i) as perfectly
> intelligible in its own right and (ii) as a reasonable way of interpreting
> second-order quantification. ...
>

I am interested in this idea in principle. But I have never seen a workable
version of what second order ZFC is based on this idea, that gets around
what I have said above. E.g., is there now a version of pluralistic ZFC
where we avoid things like 1),2),, and also where pluralistic ZFC is really
in some interesting sense more powerful than ordinary ZFC? E.g., something
really between first and second order ZFC?


> Another option is Frege's: regard second-order quantifiers as quantifying
> over what he called "concepts", which are essentially a kind of function,
> not regarded as any kind of set. Frege's explanation of what these are
> appeals to an abstract notion of predication (unsaturatedness), which many
> have regarded as difficult to understand but which, in my own view, is
> actually pretty well understood nowadays. On this kind of view, the
> semantics of a second-order language is itself simply to be given in a
> higher-order language, as in work by Agustín Rayo and Gabriel Uzquiano.
>

Once again, same question. Of course, there are obvious continuations of
Morse-Kelly involving classes of classes of sets, etcetera.

>
> There is, of course, an obvious question how it is to be guaranteed that
> the "second-order domain", if we want to speak that way (Boolos did not),
> is as "big" as it is supposed to be. There is literature on this question,
> as well. One option, pursued by Vann McGee (and, in a somewhat similar way,
> by me) is to appeal to a notion of "open-endedness" that is also found in
> some of Sol Feferman's work, though I'm sure Sol would not approve of the
> use we make of it. Here there are more questions than answers, no doubt,
> but the matter is hardly closed or hopeless.
>

Roughly in this arena of thought might be Concept Calculus, especially see
The Eight Supernatural Consistency Proofs for Mathematics, and also the
Divine Consistency Proof for Mathematics. In these, we never use the
cumulative hierarchy of sets, and instead focus on objects and classes of
objects only.

>
> As Harvey has said, there are features of second-order logic, perhaps
> illustrated by (4), that make second-order theories useless for
> foundational purposes *of certain kinds*. There is very nice work by Peter
> Koellner exploring exactly why and how, but without simply assuming, as
> Harvey seems to do, that completeness is non-negotiable. I don't myself
> think completeness is non-negotiable: Partial axiomatization is perfectly
> possible, in many cases, and will do much of what we want.
>
> Actually, I am not at all sure that I am regarding completeness as
nonnegotiable. I am actually quite interested in whether or not I am doing
that!

What I regard as nonnegotiable for a foundation of mathematics is that
there be a "formalism" and that mathematical proofs are modeled as finite
combinatorial objects - i.e., as certificates.

This amounts, in practice, to spelling out up front just what "substantive
assumptions" one is making. Yes, there is an issue of where logic stops and
substantive issues start. But I am still adhering to the idea of
"certificates".

So if you make a case that "logic" whatever that is, is "open ended", I
will say that if you act on this open endedness to periodically argue for
new logical principles, then that is unsatisfactory as a foundation for
mathematics.

So the very fact that there is some completeness theorem lying around is at
least very refreshing, since it says that you are not going to be acting on
any open endedness.

But I can at least imagine that we all agree that we won't act on any
possible open endedness, and not have a completeness theorem, but still
claim empirically and otherwise that we have put a large umbrella over
mathematics.

CAUTION: I have another life where I try to destroy current f.o.m. through
45 years of intense revolutionary activity (smile).

Harvey Friedman
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