This will be a talk for the Set Theory in the UK, STUK 10, held in Oxford 14 June 2023, organized by my students Clara List, Emma Palmer, and Wojciech Wołoszyn.
Abstract. I shall speak on the surprising strength of the second-order reflection principle in the context of set theory with abundant urelements. The theory GBcU with the abundant urelement axiom and second-order reflection is bi-interpretable with a strengthening of KM with a supercompact cardinal. This is joint work with Bokai Yao.
[bibtex key=”HamkinsYao:Reflection-in-second-order-set-theory-with-abundant-urelements”]
Download pdf at arXiv:2204.09766
Abstract. After reviewing various natural bi-interpretations in urelement set theory, including second-order set theories with urelements, we explore the strength of second-order reflection in these contexts. Ultimately, we prove, second-order reflection with the abundant atom axiom is bi-interpretable and hence also equiconsistent with the existence of a supercompact cardinal. The proof relies on a reflection characterization of supercompactness, namely, a cardinal $\kappa$ is supercompact if and only if every $\Pi^1_1$ sentence true in a structure $M$ (of any size) containing $\kappa$ in a language of size less than $\kappa$ is also true in a substructure $m\prec M$ of size less than $\kappa$ with $m\cap\kappa\in\kappa$.
See also my talk at the CUNY Set Theory Seminar: The surprising strength of reflection in second-order set theory with abundant urelements
This was an online talk 15 April 12:15 for the CUNY Set Theory Seminar. Held on Zoom at 876 9680 2366.
Abstract. I shall give a general introduction to urelement set theory and the role of the second-order reflection principle in second-order urelement set theory GBCU and KMU. With the abundant atom axiom, asserting that the class of urelements greatly exceeds the class of pure sets, the second-order reflection principle implies the existence of a supercompact cardinal in an interpreted model of ZFC. The proof uses a reflection characterization of supercompactness: a cardinal $\kappa$ is supercompact if and only if for every second-order sentence $\psi$ true in some structure $M$ (of any size) in a language of size less than $\kappa$ is also true in a first-order elementary substructure $m\prec M$ of size less than $\kappa$. This is joint work with Bokai Yao.
See the article at: Reflection in second-order set theory with abundant urelements bi-interprets a supercompact cardinal
This will be a talk for the Seminaire de Logique Lyon-Paris on 14 April 2021 4pm Paris time (3pm UK). The talk will be held on Zoom at
875 1148 7359.
Abstract. The principle of open determinacy for class games — two-player games of perfect information with plays of length ω, where the moves are chosen from a possibly proper class, such as games on the ordinals — is not provable in Zermelo-Fraenkel set theory ZFC or Gödel-Bernays set theory GBC, if these theories are consistent, because provably in ZFC there is a definable open proper class game with no definable winning strategy. In fact, the principle of open determinacy and even merely clopen determinacy for class games implies Con(ZFC) and iterated instances Con(Con(ZFC)) and more, because it implies that there is a satisfaction class for first-order truth, and indeed a transfinite tower of truth predicates $\text{Tr}_\alpha$ for iterated truth-about-truth, relative to any class parameter. This is perhaps explained, in light of the Tarskian recursive definition of truth, by the more general fact that the principle of clopen determinacy is exactly equivalent over GBC to the principle of elementary transfinite recursion ETR over well-founded class relations. Meanwhile, the principle of open determinacy for class games is strictly stronger, although it is provable in the stronger theory GBC+$\Pi^1_1$-comprehension, a proper fragment of Kelley-Morse set theory KM.
Abstract.
We show that Kelley-Morse (KM) set theory does not prove the class Fodor principle, the assertion that every regressive class function $F:S\to\newcommand\Ord{\text{Ord}}\Ord$ defined on a stationary class $S$ is constant on a stationary subclass. Indeed, it is relatively consistent with KM for any infinite $\lambda$ with $\omega\leq\lambda\leq\Ord$ that there is a class function $F:\Ord\to\lambda$ that is not constant on any stationary class. Strikingly, it is consistent with KM that there is a class $A\subseteq\omega\times\Ord$, such that each section $A_n=\{\alpha\mid (n,\alpha)\in A\}$ contains a class club, but $\bigcap_n A_n$ is empty. Consequently, it is relatively consistent with KM that the class club filter is not $\sigma$-closed.
The class Fodor principle is the assertion that every regressive class function $F:S\to\Ord$ defined on a stationary class $S$ is constant on a stationary subclass of $S$. This statement can be expressed in the usual second-order language of set theory, and the principle can therefore be sensibly considered in the context of any of the various second-order set-theoretic systems, such as Gödel-Bernays (GBC) set theory or Kelley-Morse (KM) set theory. Just as with the classical Fodor’s lemma in first-order set theory, the class Fodor principle is equivalent, over a weak base theory, to the assertion that the class club filter is normal. We shall investigate the strength of the class Fodor principle and try to find its place within the natural hierarchy of second-order set theories. We shall also define and study weaker versions of the class Fodor principle.
If one tries to prove the class Fodor principle by adapting one of the classical proofs of the first-order Fodor’s lemma, then one inevitably finds oneself needing to appeal to a certain second-order class-choice principle, which goes beyond the axiom of choice and the global choice principle, but which is not available in Kelley-Morse set theory. For example, in one standard proof, we would want for a given $\Ord$-indexed sequence of non-stationary classes to be able to choose for each member of it a class club that it misses. This would be an instance of class-choice, since we seek to choose classes here, rather than sets. The class choice principle $\text{CC}(\Pi^0_1)$, it turns out, is sufficient for us to make these choices, for this principle states that if every ordinal $\alpha$ admits a class $A$ witnessing a $\Pi^0_1$-assertion $\varphi(\alpha,A)$, allowing class parameters, then there is a single class $B\subseteq \Ord\times V$, whose slices $B_\alpha$ witness $\varphi(\alpha,B_\alpha)$; and the property of being a class club avoiding a given class is $\Pi^0_1$ expressible.
Thus, the class Fodor principle, and consequently also the normality of the class club filter, is provable in the relatively weak second-order set theory $\text{GBC}+\text{CC}(\Pi^0_1)$. This theory is known to be weaker in consistency strength than the theory $\text{GBC}+\Pi^1_1$-comprehension, which is itself strictly weaker in consistency strength than KM.
But meanwhile, although the class choice principle is weak in consistency strength, it is not actually provable in KM; indeed, even the weak fragment $\text{CC}(\Pi^0_1)$ is not provable in KM. Those results were proved several years ago by the first two authors, but they can now be seen as consequences of the main result of this article (see corollary 15. In light of that result, however, one should perhaps not have expected to be able to prove the class Fodor principle in KM.
Indeed, it follows similarly from arguments of the third author in his dissertation that if $\kappa$ is an inaccessible cardinal, then there is a forcing extension $V[G]$ with a symmetric submodel $M$ such that $V_\kappa^M=V_\kappa$, which implies that $\mathcal M=(V_\kappa,\in, V^M_{\kappa+1})$ is a model of Kelley-Morse, and in $\mathcal M$, the class Fodor principle fails in a very strong sense.
In this article, adapting the ideas of Karagila to the second-order set-theoretic context and using similar methods as in Gitman and Hamkins’s previous work on KM, we shall prove that every model of KM has an extension in which the class Fodor principle fails in that strong sense: there can be a class function $F:\Ord\to\omega$, which is not constant on any stationary class. In particular, in these models, the class club filter is not $\sigma$-closed: there is a class $B\subseteq\omega\times\Ord$, each of whose vertical slices $B_n$ contains a class club, but $\bigcap B_n$ is empty.
Main Theorem. Kelley-Morse set theory KM, if consistent, does not prove the class Fodor principle. Indeed, if there is a model of KM, then there is a model of KM with a class function $F:\Ord\to \omega$, which is not constant on any stationary class; in this model, therefore, the class club filter is not $\sigma$-closed.
We shall also investigate various weak versions of the class Fodor principle.
Definition.
We shall separate these principles as follows.
Theorem. Suppose KM is consistent.
Finally, we show that the class Fodor principle can neither be created nor destroyed by set forcing.
Theorem. The class Fodor principle is invariant by set forcing over models of $\text{GBC}^-$. That is, it holds in an extension if and only if it holds in the ground model.
Let us conclude this brief introduction by mentioning the following easy negative instance of the class Fodor principle for certain GBC models. This argument seems to be a part of set-theoretic folklore. Namely, consider an $\omega$-standard model of GBC set theory $M$ having no $V_\kappa^M$ that is a model of ZFC. A minimal transitive model of ZFC, for example, has this property. Inside $M$, let $F(\kappa)$ be the least $n$ such that $V_\kappa^M$ fails to satisfy $\Sigma_n$-collection. This is a definable class function $F:\Ord^M\to\omega$ in $M$, but it cannot be constant on any stationary class in $M$, because by the reflection theorem there is a class club of cardinals $\kappa$ such that $V_\kappa^M$ satisfies $\Sigma_n$-collection.
Read more by going to the full article: [bibtex key=”GitmanHamkinsKaragila:KM-set-theory-does-not-prove-the-class-Fodor-theorem”]
[bibtex key=”HamkinsWoodin2018:Open-class-determinacy-is-preserved-by-forcing”]
Abstract. The principle of open class determinacy is preserved by pre-tame class forcing, and after such forcing, every new class well-order is isomorphic to a ground-model class well-order. Similarly, the principle of elementary transfinite recursion ETR${}_\Gamma$ for a fixed class well-order $\Gamma$ is preserved by pre-tame class forcing. The full principle ETR itself is preserved by countably strategically closed pre-tame class forcing, and after such forcing, every new class well-order is isomorphic to a ground-model class well-order. Meanwhile, it remains open whether ETR is preserved by all forcing, including the forcing merely to add a Cohen real.
The principle of elementary transfinite recursion ETR — according to which every first-order class recursion along any well-founded class relation has a solution — has emerged as a central organizing concept in the hierarchy of second-order set theories from Gödel-Bernays set theory GBC up to Kelley-Morse set theory KM and beyond. Many of the principles in the hierarchy can be seen as asserting that certain class recursions have solutions.
In addition, many of these principles, including ETR and its variants, are equivalently characterized as determinacy principles for certain kinds of class games. Thus, the hierarchy is also fruitfully unified and organized by determinacy ideas.
This hierarchy of theories is the main focus of study in the reverse mathematics of second-order set theory, an emerging subject aiming to discover the precise axiomatic principles required for various arguments and results in second-order set theory. The principle ETR itself, for example, is equivalent over GBC to the principle of clopen determinacy for class games and also to the existence of iterated elementary truth predicates (see Open determinacy for class games); since every clopen game is also an open game, the principle ETR is naturally strengthened by the principle of open determinacy for class games, and this is a strictly stronger principle (see Hachtman and Sato); the weaker principle ETR${}_\text{Ord}$, meanwhile, asserting solutions to class recursions of length Ord, is equivalent to the class forcing theorem, which asserts that every class forcing notion admits a forcing relation, to the existence of set-complete Boolean completions of any class partial order, to the existence of Ord-iterated elementary truth predicates, to the determinacy of clopen games of rank at most Ord+1, and to other natural set-theoretic principles (see The exact strength of the class forcing theorem).
Since one naturally seeks in any subject to understand how one’s fundamental principles and tools interact, we should like in this article to consider how these second-order set-theoretic principles are affected by forcing. These questions originated in previous work of Victoria Gitman and myself, and question 1 also appears in the dissertation of Kameryn Williams, which was focused on the structure of models of second-order set theories.
It is well-known, of course, that ZFC, GBC, and KM are preserved by set forcing and by tame class forcing, and this is true for other theories in the hierarchy, such as GBC$+\Pi^1_n$-comprehension and higher levels of the second-order comprehension axiom. The corresponding forcing preservation theorem for ETR and for open class determinacy, however, has not been known.
Question 1. Is ETR preserved by forcing?
Question 2. Is open class determinacy preserved by forcing?
We intend to ask in each case about class forcing as well as set forcing. Question 1 is closely connected with the question of whether forcing can create new class well-order order types, longer than any class well-order in the ground model. Specifically, Victoria Gitman and I had observed earlier that ETR${}_\Gamma$ for a specific class well-order $\Gamma$ is preserved by pre-tame class forcing, and this would imply that the full principle ETR would also be preserved, if no fundamentally new class well-orders are created by the forcing. In light of the fact that forcing over models of ZFC adds no new ordinals, that would seem reasonable, but proof is elusive, and the question remains open. Can forcing add new class well-orders, perhaps very tall ones that are not isomorphic to any ground model class well-order? Perhaps there are some very strange models of GBC that gain new class well-order order types in a forcing extension.
Question 3. Assume GBC. After forcing, must every new class well-order be isomorphic to a ground-model class well-order? Does ETR imply this?
The main theorem of this article provides a full affirmative answer to question 2, and partial affirmative answers to questions 2 and 3.
Main Theorem.
We should like specifically to highlight the fact that questions 1 and 3 remain open even in the case of the forcing to add a Cohen real. Is ETR preserved by the forcing to add a Cohen real? After adding a Cohen real, is every new class well-order isomorphic to a ground-model class well-order? One naturally expects affirmative answers, especially in a model of ETR.
For more, click through the arxiv for a pdf of the full article.
[bibtex key=”HamkinsWoodin2018:Open-class-determinacy-is-preserved-by-forcing”]
Kameryn J. Williams successfully defended his dissertation under my supervision at the CUNY Graduate Center on April 6th, 2018, earning his Ph.D. degree in May 2018. He has accepted a position in mathematics at the University of Hawaii, to begin Fall 2018.
What a pleasure it was to work with Kameryn, an extremely talented mathematician with wide interests and huge promise.
Kameryn J Williams | MathOverflow | ar$\chi$iv
Kameryn J. Williams, The Structure of Models of Second-order Set Theories, Ph.D. dissertation for The Graduate Center of the City University of New York, May, 2018. arXiv:1804.09526.
Abstract. This dissertation is a contribution to the project of second-order set theory, which has seen a revival in recent years. The approach is to understand second-order set theory by studying the structure of models of second-order set theories. The main results are the following, organized by chapter. First, I investigate the poset of T-realizations of a fixed countable model of ZFC, where T is a reasonable second-order set theory such as GBC or KM, showing that it has a rich structure. In particular, every countable partial order embeds into this structure. Moreover, we can arrange so that these embedding preserve the existence/nonexistence of upper bounds, at least for finite partial orders. Second I generalize some constructions of Marek and Mostowski from KM to weaker theories. They showed that every model of KM plus the Class Collection schema “unrolls” to a model of ZFC− with a largest cardinal. I calculate the theories of the unrolling for a variety of second-order set theories, going as weak as GBC + ETR. I also show that being T-realizable goes down to submodels for a broad selection of second-order set theories T. Third, I show that there is a hierarchy of transfinite recursion principles ranging in strength from GBC to KM. This hierarchy is ordered first by the complexity of the properties allowed in the recursions and second by the allowed heights of the recurions. Fourth, I investigate the question of which second-order set theories have least models. I show that strong theories—such as KM or $\Pi^1_1$-CA—do not have least transitive models, while weaker theories—from GBC to GBC + ETR${}_{\text{Ord}}$—do have least transitive models.
In addition to his dissertation work and the research currently arising out of it, Kameryn has undertaken a number of collaborations with various international research efforts, including the following:
This will be a talk for the Prague set theory seminar, January 24, 11:00 am to about 2pm (!).
Abstract. The class forcing theorem is the assertion that every class forcing notion admits corresponding forcing relations. This assertion is not provable in Zermelo-Fraenkel ZFC set theory or Gödel-Bernays GBC set theory, if these theories are consistent, but it is provable in stronger second-order set theories, such as Kelley-Morse KM set theory. In this talk, I shall discuss the exact strength of this theorem, which turns out to be equivalent to the principle of elementary transfinite recursion ETRord for class recursions on the ordinals. The principle of clopen determinacy for class games, in contrast, is strictly stronger, equivalent over GBC to the full principle of ETR for class recursions over arbitrary class well-founded relations. These results and others mark the beginnings of the emerging subject I call the reverse mathematics of second-order set theory.
The exact strength of the class forcing theorem | Open determinacy for class games
A few years ago, I had observed after hearing a talk by Benjamin Rin that the principle of first-order transfinite recursion for set well-orders is equivalent to the replacement axiom over Zermelo set theory, and thus we may take transfinite recursion as a fundamental set-theoretic principle, one which yields full ZFC when added to Zermelo’s weaker theory (plus foundation).
In later work, Victoria Gitman and I happened to prove that the principle of elementary transfinite recursion ETR, which allows for first-order class recursions along proper class well-orders (not necessarily set-like) is equivalent to the principle of determinacy for clopen class games [1]. Thus, once again, a strong recursion principle exhibited robustness as a fundamental set-theoretic principle.
The theme continued in recent joint work on the class forcing theorem, in which Victoria Gitman, myself, Peter Holy, Philipp Schlicht and Kameryn Williams [2] proved that the principle $\text{ETR}_{\text{Ord}}$, which allows for first-order class recursions of length $\text{Ord}$, is equivalent to twelve natural set-theoretic principles, including the existence of forcing relations for class forcing notions, the existence of Boolean completions for class partial orders, the existence of various kinds of truth predicates for infinitary logics, the existence of $\text{Ord}$-iterated truth predicates, and to the principle of determinacy for clopen class games of rank at most $\text{Ord}+1$.
A few days ago, a MathOverflow question of Alec Rhea’s — Is there a stronger form of recursion? — led me to notice that one naturally gains additional strength by pushing the recursion principles further into second-order set theory.
So let me introduce the second-order recursion principle STR and make the comparatively simple observation that over Gödel-Bernays GBC set theory this is equivalent to Kelley-Morse set theory KM. Thus, we may take this kind of recursion as a fundamental set-theoretic principle.
Definition. In the context of second-order set theory, the principle of second-order transfinite recursion, denoted STR, asserts of any formula $\varphi$ in the second-order language of set theory, that if $\Gamma=\langle I,\leq_\Gamma\rangle$ is any class well-order and $Z$ is any class parameter, then there is a class $S\subset I\times V$ that is a solution of the recursion, in the sense that
$$S_i=\{\ x\ \mid\ \varphi(x,S\upharpoonright i,Z)\ \}$$
for every $i\in I$, where where $S_i=\{\ x\ \mid\ (i,x)\in S\ \}$ is the section on coordinate $i$ and where $S\upharpoonright i=\{\ (j,x)\in S\ \mid\ j<_\Gamma i\ \}$ is the part of the solution at stages below $i$ with respect to $\Gamma$.
Theorem. The principle of second-order transfinite recursion STR is equivalent over GBC to the second-order comprehension principle. In other words, GBC+STR is equivalent to KM.
Proof. Kelley-Morse set theory proves that every second-order recursion has a solution in the same way that ZFC proves that every set-length well-ordered recursion has a solution. Namely, we simply consider the classes which are partial solutions to the recursion, in that they obey the recursive requirement, but possibly only on an initial segment of the well-order $\Gamma$. We may easily show by induction that any two such partial solutions agree on their common domain (this uses second-order comprehension in order to find the least point of disagreement, if any), and we can show that any given partial solution, if not already a full solution, can be extended to a partial solution on a strictly longer initial segment. Finally, we show that the common values of all partial solutions is therefore a solution of the recursion. This final step uses second-order comprehension in order to define what the common values are for the partial solutions to the recursion.
Conversely, the principle of second-order transfinite recursion clearly implies the second-order comprehension axiom, by considering recursions of length one. For any second-order assertion $\varphi$ and class parameter $Z$, we may deduce that $\{x\mid \varphi(x,Z)\}$ is a class, and so the second-order class comprehension principle holds. $\Box$
It is natural to consider various fragments of STR, such as $\Sigma^1_n\text{-}\text{TR}_\Gamma$, which is the assertion that every $\Sigma^1_n$-formula $\varphi$ admits a solution for recursions of length $\Gamma$. Such principles are provable in proper fragments of KM, since for a given level of complexity, we only need a corresponding fragment of comprehension to undertake the proof that the recursion has a solution. The full STR asserts $\Sigma^1_\omega\text{-}\text{TR}$, allowing any length. The theorem shows that STR is equivalent to recursions of length $1$, since once you get the second-order comprehension principle, then you get solutions for recursions of any length. Thus, with second-order transfinite recursion, a little goes a long way. Perhaps it is more natural to think of transfinite recursion in this context not as axiomatizing KM, since it clearly implies second-order comprehension straight away, but rather as an apparent strengthening of KM that is actually provable in KM. This contrasts with the first-order situation of ETR with respect to GBC, where GBC+ETR does make a proper strengthening of GBC.
This will be a talk for the Mid-Atlantic Mathematical Logic Seminar at Virginia Commonwealth University, a conference to be held April 1-2, 2017.
Abstract. The principle of open determinacy for class games — two-player games of perfect information with plays of length $\omega$, where the moves are chosen from a possibly proper class, such as games on the ordinals — is not provable in Zermelo-Fraenkel set theory ZFC or Gödel-Bernays set theory GBC, if these theories are consistent, because provably in ZFC there is a definable open proper class game with no definable winning strategy. In fact, the principle of open determinacy and even merely clopen determinacy for class games implies Con(ZFC) and iterated instances Con(Con(ZFC)) and more, because it implies that there is a satisfaction class for first-order truth, and indeed a transfinite tower of truth predicates $\text{Tr}_\alpha$ for iterated truth-about-truth, relative to any class parameter. This is perhaps explained, in light of the Tarskian recursive definition of truth, by the more general fact that the principle of clopen determinacy is exactly equivalent over GBC to the principle of elementary transfinite recursion ETR over well-founded class relations. Meanwhile, the principle of open determinacy for class games is provable in the stronger theory GBC+$\Pi^1_1$-comprehension, a proper fragment of Kelley-Morse set theory KM. New work by Hachtman and Sato, respectively has clarified the separation of clopen and open determinacy for class games.
This is joint work with Victoria Gitman. See our article, Open determinacy for class games.
Regula Krapf successfully defended her PhD dissertation January 12, 2017 at the University of Bonn, with a dissertation entitled, “Class forcing and second-order arithmetic.” I was a member of the dissertation examining committee. Peter Koepke was the dissertation supervisor.
Regula Krapf, Class forcing and second-order arithmetic, dissertation 2017, University of Bonn. (Slides)
Abstract. We provide a framework in a generalization of Gödel-Bernays set theory for performing class forcing. The forcing theorem states that the forcing relation is a (definable) class in the ground model (definability lemma) and that every statement that holds in a class-generic extension is forced by a condition in the generic filter (truth lemma). We prove both positive and negative results concerning the forcing theorem. On the one hand, we show that the definability lemma for one atomic formula implies the forcing theorem for all formulae in the language of set theory to hold. Furthermore, we introduce several properties which entail the forcing theorem. On the other hand, we give both counterexamples to the definability lemma and the truth lemma. In set forcing, the forcing theorem can be proved for all forcing notions by constructing a unique Boolean completion. We show that in class forcing the existence of a Boolean completion is essentially equivalent to the forcing theorem and, moreover, Boolean completions need not be unique.
The notion of pretameness was introduced to characterize those forcing notions which preserve the axiom scheme of replacement. We present several new characterizations of pretameness in terms of the forcing theorem, the preservation of separation, the existence of nice names for sets of ordinals and several other properties. Moreover, for each of the aforementioned properties we provide a corresponding characterization of the Ord-chain condition.
Finally, we prove two equiconsistency results which compare models of ZFC (with large cardinal properties) and models of second-order arithmetic with topological regularity properties (and determinacy hypotheses). We apply our previous results on class forcing to show that many important arboreal forcing notions preserve the $\Pi^1_1$-perfect set property over models of second-order arithmetic and also give an example of a forcing notion which implies the $\Pi^1_1$-perfect set property to fail in the generic extension.
Regula has now taken up a faculty position at the University of Koblenz.
[bibtex key=EnayatHamkins2018:Ord-is-not-definably-weakly-compact]
In ZFC the class of all ordinals is very like a large cardinal. Being closed under exponentiation, for example, Ord is a strong limit. Indeed, it is a beth fixed point. And Ord is regular with respect to definable classes by the replacement axiom. In this sense, ZFC therefore proves that Ord is definably inaccessible. Which other large cardinal properties are exhibited by Ord? Perhaps you wouldn’t find it unreasonable for Ord to exhibit, at least consistently with ZFC, the definable proper class analogues of other much stronger large cardinal properties?
Meanwhile, the main results of this paper, joint between myself and Ali Enayat, show that such an expectation would be misplaced, even for comparatively small large cardinal properties. Specifically, in a result that surprised me, it turns out that the class of ordinals NEVER exhibits the definable proper class analogue of weak compactness in any model of ZFC.
Theorem. The class of ordinals is not definably weakly compact. In every model of ZFC:
The proof uses methods from the model theory of set theory, including especially the fact that no model of ZFC has a conservative $\Sigma_3$-elementary end-extension.
Theorem. The definable $\Diamond _{\mathrm{Ord}}$ principle holds in a model of ZFC if and only if the model has a definable well-ordering.
We close the paper by proving that the theory of the spartan models of Gödel-Bernays set theory GB — those equipped with only their definable classes — is $\Pi^1_1$-complete.
Theorem. The set of sentences true in all spartan models of GB is $\Pi_{1}^{1}$-complete.
[bibtex key=Hamkins:The-Vopenka-principle-is-inequivalent-to-but-conservative-over-the-Vopenka-scheme]
Abstract. The Vopěnka principle, which asserts that every proper class of first-order structures in a common language admits an elementary embedding between two of its members, is not equivalent over GBC to the first-order Vopěnka scheme, which makes the Vopěnka assertion only for the first-order definable classes of structures. Nevertheless, the two Vopěnka axioms are equiconsistent and they have exactly the same first-order consequences in the language of set theory. Specifically, GBC plus the Vopěnka principle is conservative over ZFC plus the Vopěnka scheme for first-order assertions in the language of set theory.
The Vopěnka principle is the assertion that for every proper class $\mathcal{M}$ of first-order $\mathcal{L}$-structures, for a set-sized language $\mathcal{L}$, there are distinct members of the class $M,N\in\mathcal{M}$ with an elementary embedding $j:M\to N$ between them. In quantifying over classes, this principle is a single assertion in the language of second-order set theory, and it makes sense to consider the Vopěnka principle in the context of a second-order set theory, such as Godel-Bernays set theory GBC, whose language allows one to quantify over classes. In this article, GBC includes the global axiom of choice.
In contrast, the first-order Vopěnka scheme makes the Vopěnka assertion only for the first-order definable classes $\mathcal{M}$ (allowing parameters). This theory can be expressed as a scheme of first-order statements, one for each possible definition of a class, and it makes sense to consider the Vopěnka scheme in Zermelo-Frankael ZFC set theory with the axiom of choice.
Because the Vopěnka principle is a second-order assertion, it does not make sense to refer to it in the context of ZFC set theory, whose first-order language does not allow quantification over classes; one typically retreats to the Vopěnka scheme in that context. The theme of this article is to investigate the precise meta-mathematical interactions between these two treatments of Vopěnka’s idea.
Main Theorems.
It follows that the Vopěnka principle VP and the Vopěnka scheme VS are not equivalent, but they are equiconsistent and indeed, they have the same first-order consequences.
Corollaries.
These results grew out of my my answer to a MathOverflow question of Mike Shulman, Can Vopěnka’s principle be violated definably?, inquiring whether there would always be a definable counterexample to the Vopěnka principle, whenever it should happen to fail. I interpret the question as asking whether the Vopěnka scheme is necessarily equivalent to the Vopěnka principle, and the answer is negative.
The proof of the main theorem involves the concept of a stretchable set $g\subset\kappa$ for an $A$-extendible cardinal, which has the property that for every cardinal $\lambda>\kappa$ and every extension $h\subset\lambda$ with $h\cap\kappa=g$, there is an elementary embedding $j:\langle V_\lambda,\in,A\cap V_\lambda\rangle\to\langle V_\theta,\in,A\cap V_\theta\rangle$ such that $j(g)\cap\lambda=h$. Thus, the set $g$ can be stretched by an $A$-extendibility embedding so as to agree with any given $h$.
I was recently surprised to discover that if there is a definable well-ordering of the universe, then the diamond principle on the ordinals holds for definable classes, automatically. In fact, the diamond principle for definable classes is simply equivalent in ZFC to the existence of a definable well-ordering of the universe. It follows as a consequence that the diamond principle for definable classes, although seeming to be fundamentally scheme-theoretic, is actually expressible in the first-order language of set theory.
In set theory, the diamond principle asserts the existence of a sequence of objects $A_\alpha$, of growing size, such that any large object at the end is very often anticipated by these approximations. In the case of diamond on the ordinals, what we will have is a definable sequence of $A_\alpha\subseteq\alpha$, such that for any definable class of ordinals $A$ and any definable class club set $C$, there are ordinals $\theta\in C$ with $A\cap\theta=A_\theta$. This kind of principle typically allows one to undertake long constructions that will diagonalize against all the large objects, by considering and reacting to their approximations $A_\alpha$. Since every large object $A$ is often correctly approximated that way, this enables many such constructions to succeed.
Let me dive right in to the main part of the argument.$\newcommand\restrict\upharpoonright
\newcommand\of\subseteq
\newcommand\Ord{\text{Ord}}
\newcommand\HOD{\text{HOD}}\newcommand\ZFC{\text{ZFC}}$
Theorem. In $\ZFC$, if there is a definable well-ordering of the universe, then $\Diamond_{\Ord}$ holds for definable classes. That is, there is a $p$-definable sequence $\langle A_\alpha\mid\alpha<\Ord\rangle$, such that for any definable class $A\of\Ord$ and any definable closed unbounded class of ordinals $C\of\Ord$ (allowing parameters), there is some $\theta\in C$ with $A\cap\theta=A_\theta$.
Proof. The theorem is proved as a theorem scheme; namely, I shall provide a specific definition for the sequence $\vec A=\langle A_\alpha\mid\alpha<\Ord\rangle$, using the same parameter $p$ as the definition of the global well-order and with a definition of closely related syntactic complexity, and then prove as a scheme, a separate statement for each definable class $A\of\Ord$ and class club $C\of\Ord$, that there is some $\alpha\in C$ with $A\cap\alpha=A_\alpha$. The definitions of the classes $A$ and $C$ may involve parameters and have arbitrary complexity.
Let $\lhd$ be the definable well-ordering of the universe, definable by a specific formula using some parameter $p$. I define the $\Diamond_{\Ord}$-sequence $\vec A=\langle A_\alpha\mid\alpha<\Ord\rangle$ by transfinite recursion. Suppose that $\vec A\restrict\theta$ has been defined. I shall let $A_\theta=\emptyset$ unless $\theta$ is a $\beth$-fixed point above the rank of $p$ and there is a set $A\of\theta$ and a closed unbounded set $C\of\theta$, with both $A$ and $C$ definable in the structure $\langle V_\theta,\in\rangle$ (allowing parameters), such that $A\cap\alpha\neq A_\alpha$ for every $\alpha\in C$. In this case, I choose the least such pair $(A,C)$, minimizing first on the maximum of the logical complexities of the definitions of $A$ and of $C$, and then minimizing on the total length of the defining formulas of $A$ and $C$, and then minimizing on the Gödel codes of those formulas, and finally on the parameters used in the definitions, using the well-order $\lhd\restrict V_\theta$. For this minimal pair, let $A_\theta=A$. This completes the definition of the sequence $\vec A=\langle A_\alpha\mid\alpha\in\Ord\rangle$.
Let me remark on a subtle point, since the meta-mathematical issues loom large here. The definition of $\vec A$ is internal to the model, and at stage $\theta$ we ask about subsets of $\theta$ definable in $\langle V_\theta,\in\rangle$, using the truth predicate for this structure. If we were to run this definition inside an $\omega$-nonstandard model, it could happen that the minimal formula we get is nonstandard, and in this case, the set $A$ would not actually be definable by a standard formula. Also, even when $A$ is definable by a standard formula, it might be paired (with some constants), with a club set $C$ that is defined only by a nonstandard formula (and this is why we minimize on the maximum of the complexities of the definitions of $A$ and $C$ together). So one must give care in the main argument keeping straight the distinction between the meta-theoretic natural numbers and the internal natural numbers of the object theory $\ZFC$.
Let me now prove that the sequence $\vec A$ is indeed a $\Diamond_{\Ord}$-sequence for definable classes. The argument follows in spirit the classical proof of $\Diamond$ in $L$, subject to the mathematical issues I mentioned. If the sequence $\vec A$ is not a diamond sequence, then there is some definable class $A\of\Ord$, defined in $\langle V,\in\rangle$ by a specific formula $\varphi$ and parameter $z$, and definable club $C\of\Ord$, defined by some $\psi$ and parameter $y$, with $A\cap\alpha\neq A_\alpha$ for every $\alpha\in C$. We may assume without loss that these formulas are chosen so as to be minimal in the sense of the construction, so that the maximum of the complexities of $\varphi$ and $\psi$ are as small as possible, and the lengths of the formulas, and the Gödel codes and finally the parameters $z,y$ are $\lhd$-minimal, respectively, successively. Let $m$ be a sufficiently large natural number, larger than the complexity of the definitions of $\lhd$, $A$, $C$, and large enough so that the minimality condition we just discussed is expressible by a $\Sigma_m$ formula. Let $\theta$ be any $\Sigma_m$-correct ordinal above the ranks of the parameters used in the definitions. It follows that the restrictions $\lhd\restrict V_\theta$ and also $A\cap\theta$ and $C\cap\theta$ and $\vec A\restrict\theta$ are definable in $\langle V_\theta,\in\rangle$ by the same definitions and parameters as their counterparts in $V$, that $C\cap\theta$ is club in $\theta$, and that $A\cap\theta$ and $C\cap\theta$ form a minimal pair using those definitions with $A\cap\alpha\neq A_\alpha$ for any $\alpha\in C\cap\theta$. Thus, by the definition of $\vec A$, it follows that $A_\theta=A\cap\theta$. Since $C\cap\theta$ is unbounded in $\theta$ and $C$ is closed, it follows that $\theta\in C$, and so $A_\theta=A\cap\theta$ contradicts our assumption about $A$ and $C$. So there are no such counterexample classes, and thus $\vec A$ is a $\Diamond_{\Ord}$-sequence with respect to definable classes, as claimed.
QED
Theorem. The following are equivalent over $\ZFC$.
Proof. Let me first give the argument, and then afterward discuss some issues about the formalization, which involves some subtle issues.
($1\to 2$) $\newcommand\rank{\text{rank}}$Suppose that $\lhd$ is a $p$-definable well-ordering of $V$, which means that every set has a $\lhd$-minimal element. Let us refine this order by defining $x\lhd’ y$, just in case $\rank(x)<\rank(y)$ or $\rank(x)=\rank(y)$ and $x\lhd y$. The new order is also a well-order, which now respects rank. In particular, the order $\lhd’$ is set-like, and so every object $x$ is the $\alpha^{th}$ element with respect to the $\lhd’$-order, for some ordinal $\alpha$. Thus, every object is definable from $p$ and an ordinal, and so $V=\HOD_{\{p\}}$, as desired.
($2\to 1$) If $V=\HOD_{\{p\}}$, then we have the canonical well-order of $\HOD$ using parameter $p$, similar to how one shows that the axiom of choice holds in $\HOD$. Namely, define $x\lhd y$ if and only if $\rank(x)<\rank(y)$, or the ranks are the same, but $x$ is definable from $p$ and ordinal parameters in some $V_\theta$ with a smaller $\theta$ than $y$ is, or the ranks are the same and the $\theta$ is the same, but $x$ is definable in that $V_\theta$ by a formula with a smaller Gödel code, or with the same formula but smaller ordinal parameters. It is easy to see that this is a $p$-definable well-ordering of the universe.
($1\to 3$) This is the content of the theorem above.
($3\to 1$) If $\vec A$ is a $p$-definable $\Diamond_{\Ord}$-sequence for definable classes, then it is easy to see that if $A$ is a set of ordinals, then $A$ must arise as $A_\alpha$ for unboundedly many $\alpha$. In $\ZFC$, using the axiom of choice, it is a standard fact that every set is coded by a set of ordinals. So let us define that $x\lhd y$, just in case $x$ is coded by a set of ordinals that appears earlier on $\vec A$ than any set of ordinals coding $y$. This is clearly a well-ordering, since the map sending $x$ to the ordinal $\alpha$ for which $A_\alpha$ codes $x$ is an $\Ord$-ranking of $\lhd$. So there is a $p$-definable well-ordering of the universe.
QED
An observant reader will notice some meta-mathematical issues concerning the previous theorem. The issue is that statements 1 and 2 are known to be expressible by statements in the first-order language of set theory, as single statements, but for statement 3 we have previously expressed it only as a scheme of first-order statements. So how can they be equivalent? The answer is that the full scheme-theoretic content of statement 3 follows already from instances in which the complexity of the definitions of $A$ and $C$ are bounded. Basically, once one gets the global well-order, then one can construct a $\Diamond_{\Ord}$-sequence that works for all definable classes. In this sense, we may regard the diamond principle $\Diamond_{\Ord}$ for definable classes as not really a scheme of statements, but rather equivalent to a single first-order assertion.
Lastly, let me consider the content of the theorems in Gödel-Bernays set theory or Kelley-Morse set theory. Of course, we know that there can be models of these theories that do not have $\Diamond_{\Ord}$ in the full second-order sense. For example, it is relatively consistent with ZFC that an inaccessible cardinal $\kappa$ does not have $\Diamond_\kappa$, and in this case, the structure $\langle V_\kappa,\in,V_{\kappa+1}\rangle$ will satisfy GBC and even KM, but it won’t have $\Diamond_{\Ord}$ with respect to all classes, even though it has a definable well-ordering of the universe (since there is such a well-ordering in $V_{\kappa+1}$). But meanwhile, there will be a $\Diamond_{\Ord}$-sequence that works with respect to classes that are definable from that well-ordering and parameters, simply by following the construction given in the theorem.
This leads to several extremely interesting questions, about which I am currently thinking, concerning instances where we can have $\Diamond_\kappa$ for definable classes in $V_\kappa$, even when the full $\Diamond_\kappa$ fails. Stay tuned!
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