- J. D. Hamkins, “A multiverse perspective on the axiom of constructibility,” in Infinity and Truth, World Sci. Publ., Hackensack, NJ, 2014, vol. 25, pp. 25-45.
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This article expands on an argument that I made during my talk at the Asian Initiative for Infinity: Workshop on Infinity and Truth, held July 25–29, 2011 at the Institute for Mathematical Sciences, National University of Singapore, and will be included in a proceedings volume that is being prepared for that conference.

**Abstract.** I argue that the commonly held $V\neq L$ via maximize position, which rejects the axiom of constructibility $V=L$ on the basis that it is restrictive, implicitly takes a stand in the pluralist debate in the philosophy of set theory by presuming an absolute background concept of ordinal. The argument appears to lose its force, in contrast, on an upwardly extensible concept of set, in light of the various facts showing that models of set theory generally have extensions to models of $V=L$ inside larger set-theoretic universes.

In section two, I provide a few new criticisms of Maddy’s proposed concept of `restrictive’ theories, pointing out that her concept of *fairly interpreted in *is not a transitive relation: there is a first theory that is fairly interpreted in a second, which is fairly interpreted in a third, but the first is not fairly interpreted in the third. The same example (and one can easily construct many similar natural examples) shows that neither the *maximizes over* relation,* nor *the *properly maximizes over *relation, nor the *strongly maximizes over* relation is transitive. In addition, the theory ZFC + `there are unboundedly many inaccessible cardinals’ comes out as formally restrictive, since it is strongly maximized by the theory ZF + `there is a measurable cardinal, with no worldly cardinals above it’.

To support the main philosophical thesis of the article, I survey a series of mathemtical results, which reveal various senses in which the axiom of constructibility $V=L$ is compatible with strength in set theory, particularly if one has in mind the possibility of moving from one universe of set theory to a much larger one. Among them are the following, which I prove or sketch in the article:

**Observation.** The constructible universe $L$ and $V$ agree on the consistency of any constructible theory. They have models of the same constructible theories.

**Theorem. **The constructible universe $L$ and $V$ have transitive models of exactly the same constructible theories in the language of set theory.

**Corollary.** (Levy-Shoenfield absoluteness theorem) In particular, $L$ and $V$ satisfy the same $\Sigma_1$ sentences, with parameters hereditarily countable in $L$. Indeed, $L_{\omega_1^L}$ and $V$ satisfy the same such sentences.

**Theorem.** Every countable transitive set is a countable transitive set in the well-founded part of an $\omega$-model of V=L.

**Theorem.** If there are arbitrarily large $\lambda<\omega_1^L$ with $L_\lambda\models\text{ZFC}$, then every countable transitive set $M$ is a countable transitive set inside a structure $M^+$ that is a pointwise-definable model of ZFC + V=L, and $M^+$ is well founded as high in the countable ordinals as desired.

**Theorem.** (Barwise) Every countable model of ZF has an end-extension to a model of ZFC + V=L.

**Theorem.** (Hamkins, see here) Every countable model of set theory $\langle M,{\in^M}\rangle$, including every transitive model, is isomorphic to a submodel of its own constructible universe $\langle L^M,{\in^M}\rangle$. In other words, there is an embedding $j:M\to L^M$, which is elementary for quantifier-free assertions.

Another way to say this is that every countable model of set theory is a submodel of a model isomorphic to $L^M$. If we lived inside $M$, then by adding new sets and elements, our universe could be transformed into a copy of the constructible universe $L^M$.

(Plus, the article contains some nice diagrams.)

Related Singapore links:

A very, very interesting article! Thanks. A couple of questions pop into mind:

i) Considering the view you espouse in the paper, why should V=L not be considered in some sense a ‘background theory’ which defines the notion of what it means to be a set?

ii) (regarding the relation between ‘toy models of V’ and V) Paul Cohen, in his paper “The Discovery of Forcing” (pp 1090-1091), makes the following observation: “…one cannot prove the existence of any uncountable standard model in which AC holds, and CH is false (this does not mean that in the universe CH is true, merely that one cannot prove the existence of such a model even granting the existence of standard models, or even any of the higher axioms of infinity).” He then goes on to prove this observation as follows: “If M is an uncountable standard model in which AC holds, it is easy to see that M contains all countable ordinals. If the axiom of constructibility is assumed, this means that all the real numbers are in M and are constructible in M. Hence CH holds. I only saw this after I was asked at a lecture why I only worked with countable models, whereupon the above proof occurred to me…The above result refers to standard models only.” If one assumes the existence of a countable standard model of ZFC in which CH does not hold, then how can one infer there exists an uncountable model of ZFC where CH does not hold, given Cohen’s ‘proof’. Is there a flaw in his proof, and if so, then where?

Thanks for your comments as usual! Regarding Cohen’s proof, he assumes V=L, but this contradicts all of the stronger “higher axioms of infinity”. For example, if one assumes that there is a measurable cardinal, then there certainly are uncountable transitive models of ZFC+ not CH. My paper on Inner models with large cardinal features usually obtained by forcing contains some further examples, where we can prove the existence of even proper class inner models of certain theories usually obtained by forcing.

Thanks. When I read Cohen’s proof of his observation I questioned his assumption of V=L. Should I have? Regarding the existence of uncountable models of ZFC+ not-CH given the existence of a measurable cardinal assume the existence of a measurable cardinal. Then there is a uncountable model of ZFC+not-CH. Taking the training wheels off (so to speak) one adds to this model sufficient new sets and elements to transform ZFC+not-CH into a copy of the constructible universe.

In this new universe, is the measurable cardinal one assumed to exist in order to prove the existence of the uncountable model of ZFC+not-CH still measurable (since the actual model would seem to be a model of ZFC+not-CH+”There exists a measurable cardinal”?

Writing “ORD” for the ordinal (and cardinal) of V, consider the following maximality principle: “ORD is large with respect to all the cardinals in V.” Call this principle “MP” (MP is a version of the third maximality principle in Garvin Melles’ “Natural internal forcing schemata extending ZFC”). Now, as long as V itself is taken to be the universe, any large-cardinal properties of ORD are not actually “realized” or “manifested,” even if we accept MP; their manifestation requires some larger universe V-plus to which V belongs. Suppose that V is a model of “ZFC + there exists a proper class of supercompact cardinals;” then MP implies that ORD is (at least) supercompact in V-plus. In that case, (i) V=L does not hold in V-plus, and (ii) V cannot be regarded as a countable transitive set in V-plus. The claim that V=L holds in V-plus requires restricting the largeness of V (and of ORD) by rejecting MP. Hence, it appears that V=L is restrictive after all, and does not accord with the rule of thumb “maximize” that is expressed in MP.

Thanks for your comments. I’d rather not call that principle MP, in light of my work on A Simple Maximality Principle, which has another maximality principle denoted by MP. But leaving that aside, I think that more care is needed in your principle. What exactly does your MP assert? It cannot be first-order expressible in the language of set theory, since because of what you say it won’t be preserved to elementary submodels, and no actually countable model of set theory can seem to satisfy it. So I’m not clear on what your principle is asserting. Meanwhile, since it seems that your MP is false in every toy universe, why should we think it is true in our current universe, which for all we know is a toy universe in a much larger universe?

Thanks for the reply; I’ll try to provide some clarification. First, I’ll change the name of the principle from MP to POM (“Principle of Maximization”). Now, if we take V to belong to a suitably larger universe V+, then POM can be expressed as a first-order schema: namely, for all cardinals k such that k is a member of ORD, and for phi a first-order formula (in particular, for phi a property of cardinals), if k has the property phi, AND if k’s having phi implies that k is inaccessible [which is a way of saying that phi is a large-cardinal property], then ORD has the property phi.

If we consider a universe V by itself, without taking V to belong to a larger universe V+, then POM is simply inexpressible in V, since ORD isn’t a member of V. The motivation for taking POM to hold in V+ is that it captures part of the rule of thumb “maximize”; specifically, it makes ORD large in relation to ORD’s members. And taking POM to hold in V+ implies that V=L doesn’t hold in V+, assuming that V is a model of ZFC + a proper class of supercompact cardinals; for, it implies that ORD is supercompact in V. Of course, if V is a toy universe in V+, then POM fails in V+. But that simply shows that taking V to be a toy universe in V+ is a restrictive assumption that goes against the spirit of “maximize”. That doesn’t show such an assumption to be wrong, of course; but it does point to a tension between V=L and “maximization”.

Correction: “ORD is supercompact in V” (in the second paragraph) should read: “ORD is supercompact in V+”. Sorry for the error.

This is the kind of toy model formalization that I menttion in my paper, which is used also to formalize the Inner Model Hypothesis of Friedman. But another point: I still don’t really get your principle, since what you say seems to make it simply inconsistent with large cardinals: let phi(kappa) be `kappa is an inaccessible limit of inaccessible cardinals’. And let psi(kappa) be `kappa is inacessible, but not a limit of inaccessible cardinals’. If there are a proper class of inaccessible cardinals, then we have a proper class of kappa with phi(kappa) and a proper class of kappa with psi(kappa). On my understanding of what you describe with POM, this would lead us to say both phi(Ord) and psi(Ord) in V+. But those properties contradict each other.

Here’s an improved attempt to formalize POM: for phi a formula with (proper) subformulas psi-sub-1,…, psi-sub-n, IF [(there exists kappa in ORD such that [phi(kappa) & (phi(kappa) implies kappa is inaccessible)]) & NOT ([either there exists kappa such that (psi-sub-1(kappa) & kappa is not inaccessible)] OR … [there exists kappa such that (psi-sub-n(kappa) & kappa is not inaccessible)])], THEN phi(ORD).

The idea is that in order for phi(ORD) to hold, none of phi’s subformulas may ascribe to kappa any properties other than those that imply kappa’s inaccessibility itself (e.g., phi should not say anything about kappa’s being a limit of inaccessibles).

Regarding Friedman’s Inner Model Hypothesis (IMH), his “outer models” are themselves countable (see, e.g., p. 1 of Friedman et al., “On the consistency strength of the IMH”); but this does not seem to be the case for your larger universe V+ that looks upon V as a mere toy. (If V+ itself is countable, then I apologize for misunderstanding you.)

Now, if V is a model of ZFC + supercompacts, then POM implies that ORD is supercompact; and in this context, “ORD is supercompact” would seem to mean “ORD is supercompact in V+” (at least, I don’t see how else to interpret it). Hence, ORD is supercompact in a larger universe that is not a countable model, which implies that (i) V is not in fact a toy model after all (assuming POM, that is), and (ii) V=L fails in V+ (again, assuming POM). I take this to indicate that viewing V as a toy model is a “restrictive” view of V – at least in a context in which V+ itself is not a toy model – and that the maximizing principle POM does away with this restrictive view, and with V=L.

I think your new ersion of POM is equivalent to the old version of POM, and thus inconsistent with large cardinals. To see that they are equivalent, if phi(kappa) is any property, to which we want to assert the old POM. Consider the new property, to be asserted with the new POM: exists y such that kappa in y and phi*(kappa), where phi* is the formula obtained by adding “and y=y” in all atomic clauses. This new formula has y free in all nontrivial subformulas, and so there are no phi_0(kappa) etc., since no subformulas with x free have only x free. So the new POM applied to the new formula is equivalent to the old POM applied with phi.

As far as I can tell, for each subformula psi*_n in which y is free, there is a logically equivalent formula in which y is not free (because “y” does not occur at all), and only kappa is free. Hence, it seems that we can take POM to apply to formulas phi that have subformulas psi_n each of which is logically equivalent to a formula psi*_n in which only kappa is free, and we can reformulate POM by replacing occurrences of “psi_n” by occurrences of “psi*_n”.

(One other correction: in the formulation of POM that I gave above, the conjunct “phi(kappa) implies kappa is inaccessible” needs to be preceded by “for all kappa,” i.e. kappa here needs to be universally quantified.)

Leaving aside the question of how to formulate POM, the fact remains that if ORD contains supercompacts, it is not unreasonable to think that ORD itself might be supercompact. And if we say that ORD is supercompact, then the argument I gave above goes through, regardless of our ability to encapsulate the reasonableness of “ORD is supercompact” in a general principle such as POM Furthermore, I don’t see any motivation for denying that ORD here is supercompact.

I lack your confidence that this kind of subformula approach to maximality will ultimately be satisfying. The more you introduce such syntactic qualifications into the principle, the more it looks like ad hoc avoidance of the particular counterexamples presented, and the less philosophical support the principle finds on maximality-type grounds.

Nevertheless, as I explain in my paper, I do agree with the main thrust of your position, that the Ord of V may often exhibit large cardinal features in a larger universe V+. But I do not agree with you that maximality considerations should lead us to think that it must exhibit large cardinal features in every such V+. Indeed, we can easily imagine a forcing extension V+[G] in which it becomes a countable ordinal, simply by forcing to collapse it. So it seems that if Ord is supercompact in some V+, then it is a countable ordinal and definitely not supercompact in some other V++. The position that I argue in the paper is that there seems plenty of support on maximality-type grounds that V itself becomes a countable set in the L of a larger universe V+.

Finally, I object to your idea that just from having a proper class of supercompact cardinals, we should think that Ord is supercompact in some V+. For example, this seems unreasonable unless there are also many measurable limits of supercompact cardinals, and kappa+-supercompact kappa that are limits of supercompact cardinals, and so on, since if Ord were supercommpact then it would be a supercompact limit of supercompact cardinals and this would reflect down in such a way. For example, it is easy to see by such reasoning that if $\kappa$ is the least inaccessible limit of supercompact cardinals, then $V_\kappa$ has ZFC plus a proper class of supercompact cardinals, but $\kappa$ cannot be supercompact or even weakly compact in any top-extension of $V_\kappa$, since if it were, then there would have to be inaccessible limits of supercompact cardinals below, which there aren’t.

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Joel,

In your theorem about a modek embedding in its L: can we arrange that the embedding preserves Sigma_0 formulas?

Unfortunately, no, that would be too much. (And every cofinal $\Sigma_0$-elementary embedding is fully elementary.) The embedding I construct is not $\Sigma_0$-elementary, and doesn’t even preserve ordinals or even the natural numbers (although I believe one could arrange at least that—we’re working on it). So this is definitely not the usual kind of “embedding” we consider in set theory, although it is an “embedding” in the usual model-theoretic sense of the term for binary relational structures.

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