[bibtex key=HamkinsLinetskyReitz2013:PointwiseDefinableModelsOfSetTheory]
One occasionally hears the argument—let us call it the math-tea argument, for perhaps it is heard at a good math tea—that there must be real numbers that we cannot describe or define, because there are are only countably many definitions, but uncountably many reals. Does it withstand scrutiny?
This article provides an answer. The article has a dual nature, with the first part aimed at a more general audience, and the second part providing a proof of the main theorem: every countable model of set theory has an extension in which every set and class is definable without parameters. The existence of these models therefore exhibit the difficulties in formalizing the math tea argument, and show that robust violations of the math tea argument can occur in virtually any set-theoretic context.
A pointwise definable model is one in which every object is definable without parameters. In a model of set theory, this property strengthens V=HOD, but is not first-order expressible. Nevertheless, if ZFC is consistent, then there are continuum many pointwise definable models of ZFC. If there is a transitive model of ZFC, then there are continuum many pointwise definable transitive models of ZFC. What is more, every countable model of ZFC has a class forcing extension that is pointwise definable. Indeed, for the main contribution of this article, every countable model of Godel-Bernays set theory has a pointwise definable extension, in which every set and class is first-order definable without parameters.
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Regarding Theorem 11, i.e. “Every Countable model of ZFC has a pointwise definable class forcing extension”, can one, using Boolean-valued models show that every model of ZFC has a pointwise definable class forcing extension? Also, in the proof of Theorem 11 ( since HOD is being forced) does the Leibniz-Myceielski axiom also hold in the class forcing extension and can the model constructed in Theorem 11 satisfy not-CH (since in the proof the construction forces failures of the GCH)?
Thomas, no, it would be impossible to extend the theorem to uncountable models, since pointwise definable models must be countable, as there are only countably many definitions. This is in a sense the residue of the math tea argument, but applied in the right way—one may enumerate the definable elements in a realm where one has the satisfaction relation, external to the model. Meanwhile, every pointwise definable model must have V=HOD, since in fact you don’t need the ordinal parameters, and it is easy to arrange not CH or CH. One can have GCH or lots of failures of GCH simply by using a different coding method to force V=HOD.
The mathematics is undoubtedly correct, but I’m not sold on the more philosophical conclusion that the math tea-argument does not withstand scrutiny. I will give a ZFC-centric rebuttal, but we can replace ZFC with pretty much any other sufficiently well-motivated set theory and the argument will still go through.
Here’s my thinking. Just as a “complete lattice” is not just “a model of the first-order theory of complete lattices” but must necessarily be “a model of the second-order theory of complete lattices,” similarly I would argue that a “model of set theory” is not just “a model of ZFC” but is necessarily “a model of ZFC-2” (i.e. second-order ZFC with its standard semantics). After all, the ultimate motivation for ZFC is of course ZFC-2; the latter is what we actually “believe,” while the former is merely a recursively-ennumerable approximation to (the consequences of) those beliefs. We put up with first-order set theory because we’re interested in founding mathematics on set theory; the moment we want to know what’s actually *true* in the universe of sets according to a different foundations (e.g. homotopy theory theory), we immediately move to second-order set theory.
It should be clear where I’m going with this. Since the language of ZFC-2 is countable, but every model of ZFC-2 is uncountable, thus any model of set theory must have elements that cannot be defined without parameters. Along a similar vein, since every model of set theory contains an isomorphic copy of the real line, which is therefore uncountable, the tea party argument works perfectly; without parameters, set theory cannot “see” every real number.
Thanks for the comment. You are on Zermelo’s side, because Zermelo’s original formulation of the axioms of set theory used essentially a second-order formulation. This was later changed to the first-order formulation we know so well today, in large part because many mathematicians find $\text{ZFC}_2$ to be essentially incoherent. How are we to make sense of a second-order model of a theory, except with respect to some background concept of set? But it is the background concept of set we are trying to describe with our theory! So I think that your argument is sensible only to those who believe already that there is a unique absolute background concept of set. I have argued at length in my various papers on the set-theoretic multiverse that what we have is actually a plurality of concepts of set. To speak of a second-order logic is to speak of set theory itself. Daniel Isaacson has argued that CH and other set-theoretic assertions are settled on the basis of $\text{ZFC}_2$, since the initial segments of the cumulative universe are unique in models of full second-order logic, and either CH holds there or it doesn’t. But meanwhile, we can’t say which way CH is determined. I don’t find this sense of “being determined in second-order logic” to be very meaningful.
“Daniel Isaacson has argued that CH and other set-theoretic assertions are settled on the basis of ZFC2.”
Look the way I see it, $\mathrm{ZFC_2}$ is just a finite list of strings. It doesn’t even come with a deductive system to deduce further strings (unless we’re interested in Henkin semantics, which we’re not.) So from this viewpoint, nothing is settled on the basis of $\mathrm{ZFC_2}$, not even whether or not $\aleph_0$ comes before $2^{\aleph_0}$! So I’m going to have to strongly disagree with Daniel Isaacson position; if $\mathrm{ZFC_2}$ is defined as above, it settles nothing.
And you know what, THAT’S OKAY. After all, $\mathrm{ZFC}_2$ isn’t meant to settle anything. Notwithstanding Zermelo’s original intentions, we know today that its not a foundations on which mathematics can be built. Does it have any use, then? It sure does. In particular, $\mathrm{ZFC}_2$ it tells us how to define the phrase “model of set theory” correctly, in much the same way that the second-order axioms for the real numbers tell us how to define the phrase “Dedekind-complete ordered field” correctly, in much the same way that the second-order Peano axioms tell us how to define the phrase “model for arithmetic” correctly.
By the way, I think this is perfectly compatible with the multiverse perspective. However, my thoughts on the matter are very unclear, so maybe I’ll just stop writing.
I’m not sure that your position is self-consistent. On the one hand, you want to regard the models of set theory as models of $\text{ZFC}_2$, which requires them to have *all* the subsets of natural numbers and all the sets of set of natural numbers, but on the other hand, you seem to want there to be different models of set theory with different outcomes for CH. These two expectations seem contradictory, since if every model of set theory has all the reals and all the sets of reals, then they will all agree on CH, which is determined by those sets.
Oh its perfectly consistent; the multiverse is populated by all models of $\mathrm{ZFC}$, but the phrase “model of set theory” ought be reserved for models of $\mathrm{ZFC_2}$. That is really all I was trying to say; everything else I’ve said is just motivation, it was just meant to explain why this convention is optimal.
Now you may say: “Well if you’re merely advocating a particular convention, then its kind of no big deal right? They’re just definitions.” However I think the stakes are very high. Here’s why.
If my younger brother comes to me tomorrow and asks: “What is a real number system?” and I reply, “its a real closed field,” then I think you and I both agree that this was the wrong answer. The right answer was, of course “its a complete ordered field.” Similarly if he asks: “What is a model of set theory?” and I reply “Its a model of $\mathrm{ZFC}$,” then by analogy with the previous case I think I have given the wrong answer. What I should have said was: “Its a model of $\mathrm{ZFC_2}$.”
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