# Re: Paper and slides on indefiniteness of CH

On Mon, Oct 20, 2014 at 1:29 PM, Penelope Maddy wrote:

Dear Claudio and Sy, … I’m at a loss.

I’m also at a loss, for similar reasons, and have been for a while. I’d have thought that a true “multiverser” would want to replace all talk of “V” –understood as the universe of [absolutely] all ordinals (and sets, etc.)–with some more benign term, such as some very large, (perhaps maximally) fat, transitive model of (here a ref to ZFC + some very large card axioms).

But, for most mathematical purposes outside the higher reaches of set theory itself, I thought that it wouldn’t matter. Several messages back, ref was made to how a group theorist, for instance, might choose. But couldn’t either view accommodate any new axiom that might possibly matter to a group theorist? That seems to be the case with my modal version of multiverse, in which the possible structures are, up to isomorphism, linearly ordered by “end-extension”. The V theorist can almost certainly model the possibilities as set-models inside V. (One of the few things s/he can’t do is respect the unrestricted Extendability Principle, applied in context of second-order logic (or logic or plurals), since, as Kreisel “complained” decades ago, responding to Putnam’s “Mathematics without Foundations”, V is a counterexample!)

Geoffrey

# Re: Paper and slides on indefiniteness of CH

Pen,

You wrote to Sy,

“OK!  So we have an answer to the question Peter has been asking:  are you an actualist or a potentialist?  Answer:  a potentialist.  So you aren’t really out to settle CH in the ordinary way people think of that project; you aren’t out to discover new things about V (because there is no V).”

But here, ‘V’ can be replaced by “any standard initial segment of a (not “the”!) cumulative hierarchy of sets with full power sets up to and including rank $\omega+2$, for CH is determinate (semantically) there. And to accommodate the proof theory of “the ordinary way people think of that project”, one can respect that by replacing “new things about V” with “new things about any standard (well-founded with full power sets) model of T, where T is the relevant extension of ZFC for the result in question. This seems to be at least one way in which a potentialist can respect mathematical practice of higher set theory.

# Re: Paper and slides on indefiniteness of CH

That’s reassuring.

Geoffrey

# Re: Paper and slides on indefiniteness of CH

Ok, I’ll look at your [Penelope Maddy's] RSL paper. Does it articulate a view of the continuum? Are you now a predicativist? A Bishop constructivist?

Geoffrey

# Re: Paper and slides on indefiniteness of CH

Thanks for sharing this bit of personal history, Pen. My only question/comment here is this: while I’m as convinced now as you are now that set theory as a whole doesn’t track an objective reality–in part because I think (with Zermelo [1930], Mac Lane and others) that any universe of discourse can in principle be properly extended, ruling out a maximal “reality of sets”, etc.–this doesn’t speak to the question whether limited domains, like the integers or the reals, have genuine power sets, which is all it takes to “determine CH”, as already elaborated in earlier messages. For me, it’s very easy and natural to be skeptical of “absolutely all ordinals”, and equally easy to accept “absolutely all sets of integers, sets of reals, etc.” as mathematically definite. But again, I readily concede that anything like a conclusive proof of CH or its negation may well elude us, so in this sense, our best mathematical minds may be incapable of “tracking” the facts of third-order arithmetic (or even second-order arithmetic, or even first-order interpreted over standard models, i.e. arithmetical truth).

Yes.

# Re: Paper and slides on indefiniteness of CH

Well, I’m not enough of a mathematician or set theorist to judge fruitfulness of research programs in mathematics, even on a foundation question such as CH. But I entirely see why it can easily be that, although CH does make a definite statement, relative to widely accepted mathematical ideas–viz. “all subsets of the integers” and “all subsets of the reals”–nevertheless research on trying to “prove” or “refute” CH is not promising. Since the relevant standard of truth-determinateness I take to be the semantic one appealing to these notions, not the vague “decidable in some acceptable mathematical framework”, there’s plenty of room for doubt about the value of working on “deciding” CH. For the same reasons, I agree with Sol and others that offering a prize for “a proof or refutation of CH” wold be quite inappropriate.

# Re: Paper and slides on indefiniteness of CH

For now, I’ll just confine myself to two remarks:

1. I’d say your statement about Mozart and Beethoven is a good candidate for “indeterminate”, as “greater musician” leaves it open just how that is to be assessed. A great many ordinary statements involving vague terms are likewise indeterminate. “Geoffrey Hellman is bald” is a simpler example of the phenomenon. Of course, such statements can be “precisified”: “GH is totally bald” is definitely false; “GH is partially bald” is definitely true; and so forth.
2. Consider the sentence, “The sentence CH is determinate”, where that means has the same truth value in all full models of classical third-order number theory. I claim this statement is itself clearly true, demonstrably so, in virtue of the quantifiers occurring in CH (as I framed it a few messages back). So by the natural mathematical analogue of an empiricist verificationist standard, viz. with logico-mathematical proof corresponding to confirmation by empirical observation or test, this sentence is both meaningful and true. But what about CH itself (or its negation)? But by the criterion of “capable of proof or refutation from currently accepted axiom systems”, CH comes out indeterminate, even though it is determinate by the model-theoretic standard of invariance over the class of intended models. So we have two distinct standards of “determinate” in play. Whatever physicists may say, the situation seems clear enough if we keep track of the meanings of the terms involved.

# Re: Paper and slides on indefiniteness of CH

Ok, I agree that we can look at scientists’ actual practice, apart from what they say about meaning and truth, etc., and then draw conclusions about their actual standards on crucial matters such as the meaningfulness (or lack thereof) of statements that are unconfirmable/unfalsifiable in principle, and so forth. If one does this, however, then the picture that emerges is quite at odds with the sort of positivist views suggested in your previous message. Some examples are described in an old paper of mine, “Realist Principles,” Philosophy of Science, 50 (1983): 227-249.

For example, all kinds of statements about gravitational or other field values at various precise locations in the very early universe, say, prior to the decoupling of radiation and matter, are regarded as meaningful despite their being completely inaccessible empirically as a matter of physical principle. In practice, physicists (and even more clearly biologists) adopt realist positions on meaningfulness of statements whose actual truth value may be impossible to determine. (For an insightful critique of logical empiricists’ (positivists’) attempts to frame criteria of cognitive significance, a “must-read” is Hempel’s landmark paper, “Empiricist Criteria of Cognitive Significance: Problems and Changes”, in Aspects of Scientific Explanation (New York: MacMillan, 1965), pp 101-119. Hempel appeals to scientific practice, rather than pronouncements, much as we’re suggesting here.)

The upshot of these studies and reflections is that, contrary to what was suggested in your earlier message, scientific practice, if anything, tends to favor the view that statements like CH, undecidable in current set theory, should be regarded as truth-determinate even if we may never be able to decide them. (I say, “if anything”, because moving from the empirical sciences to pure mathematics is a risky business, although in the future that of course may change.)

PS Nothing in the above diminishes the importance of empirical testing and observation in the process of evaluating hypotheses and theories in the sciences (as true, or approximately probably true, etc.). This–not a narrow standard of meaningfulness–is what has been around in the sciences for a long time. The analogue here in mathematics would be the role of proofs: no one, regardless of whether they favor classicist or constructivist approaches, seriously asserts mathematical claims without proof (available in the community, not necessarily to the person making the assertion); otherwise they are properly put as conjectures.