Maximal Realism and Continuum Hypotehsis

Question

I have read in multiple places, per example here, that there is a unique model of second order logic that axiomatizes the real numbers. If this is true, then we should be able to decide everything about the real numbers in second order logic, namely the continuuum hypothesis.

In this question I read

«First of all, the claim "second-order logic decides CH" requires us to already believe that the full powerset of an arbitrary structure exists; this level of set-theoretic realism already commits us to the claim that the continuum hypothesis has a definite truth value.».

So my questions are:

1. If the real numbers only have one model in second order logic, we can say the continuuum hypothesis (CH) is true or false in second order logic, right? This settles for me the issue of CH. I assume we can't prove which one it is in second order logic because of incompletness.

2. About the citation I put above, if I believe in some kind of maximal model, where there are as many sets as there can possible be, I am assuming «full powerset of an arbitrary structure exists». So that should imply the continuuum hypothesis, is that correct? I say this because if my model is as large as it can be, the axiom of choice should be true, and GCH imples Choice. Although from that I cannot conclude GCH is true (and so CH), it gives me intuition that when the Universe of Sets gets larger, it goes in favor of CH.

Thank you

Answer 1

It doesn't provide an argument one way or another.

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Second-order logic is a bit of a side issue; the real point is the ontological commitment(s) mentioned in the OP. Those commitments immediately imply that CH has a definite truth value, since (granting said commitment) either there is an "anomalous" set of reals or there isn't one. That's all it gets you, unfortunately.

Even augmenting it with additional commitments, like "maximality," doesn't help much. Specifically, forcing indicates that there's not going to be a simple way to use considerations about maximality to decide CH one way or the other, since any model of ZFC (countable or not, well-founded or not) can in a precise sense imagine an even larger model in which CH is decided differently. (This holds for AC as well, if all we take for granted is ZF, by combining forcing with a bit of inner model theory.)

Digression: This is not to say that maximality concerns are irrelevant to the topic, merely that their use will have to be rather subtle. There are a couple different ideas along these lines. The Inner Model Hypothesis, a personal favorite of mine, implies that the continuum is extremely large. Forcing axioms - specifically, PFA - start from the assumption that the universe should have as many "pretty generic" objects as possible, and wind up implying that $2^{\aleph_0}=\aleph_2$ while also implying a kind of "effective CH" (basically: no nontrivial cardinal characteristics of the continuum). And in the pro-CH direction we have Foreman's generic large cardinals program and Woodin's $\Sigma^2_1$ absoluteness theorem, which in my opinion are substantially more technical but still grow out of a "maximality" intuition. This Mathoverflow thread is a good source of discussion on the topic. The point I'm making above is just that there aren't going to be any simple arguments one way or the other, and in my opinion this should be the main philosophical takeaway of forcing.

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OK, now what about second-order logic?

The same set-theoretic commitment mentioned above also means that second-order logic (with the standard semantics) is a well-defined thing. In particular, it means that "second-order validity" is a well-defined notion. We already know that (under the above commitment) CH has a definite truth value, but we can additionally show that CH is "reducible" to second-order logic: there are sentences $\varphi,\psi$ such that $\varphi$ is a second-order validity iff CH is true and $\psi$ is a second-order validity iff CH is false.

But this really says more about second-order logic than about CH - the takeaway should be that second-order logic really is seriously entangled with set-theoretic reality (in contrast to first-order logic or even $\mathcal{L}_{\omega_1,\omega}$).

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Basically, there are two separate things going on. First, we have the fact - which doesn't involve second-order logic at all - that enough set-theoretic ontologyentails that CH has a definite truth value. Second, we then have the fact that that same commitment means that second-order logic with the standard semantics is actually well-defined, which in turn lets us equate CH with a question about validity in second-order logic.

Note that neither piece sheds any light on whether CH is true or false. It's possible to view this pessimistically, but in my opinion that's not necessary since we can rephrase the situation as follows: the set theoretic commitment is enough to tell you something but not everything about CH. That's not so surprising - it says that one set of hypotheses isn't complete yet.