Ord is not definably weakly compact

[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:

  1. The definable tree property fails; there is a definable Ord-tree with no definable cofinal branch.
  2. The definable partition property fails; there is a definable 2-coloring of a definable proper class, with no homogeneous definable proper subclass.
  3. The definable compactness property fails for $\mathcal{L}_{\mathrm{Ord,\omega}}$; there is a definable theory in this logic, all of whose set-sized subtheories are satisfiable, but the whole theory has no definable class model.

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.

Diamond on the ordinals

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

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.

Theorem. The following are equivalent over $\ZFC$.

  1. There is a definable well-ordering of the universe, using some set parameter $p$.
  2. $V=\HOD_{\{p\}}$, for some set $p$.
  3. $\Diamond_{\Ord}$ holds for definable classes. That is, there is a set parameter $p$ and a definable sequence $\vec A=\langle A_\alpha\mid\alpha<\Ord\rangle$, such that for any definable class $A\of\Ord$ and definable class club $C\of\Ord$, there is some $\alpha\in C$ with $A\cap\alpha=A_\alpha$.

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.

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!

Incomparable $\omega_1$-like models of set theory

[bibtex key=FuchsGitmanHamkins2017:IncomparableOmega1-likeModelsOfSetTheory]

This is joint work with Gunter Fuchs and Victoria Gitman.

Abstract. We show that the analogues of the Hamkins embedding theorems, proved for the countable models of set theory, do not hold when extended to the uncountable realm of $\omega_1$-like models of set theory. Specifically, under the $\diamondsuit$ hypothesis and suitable consistency assumptions, we show that there is a family of $2^{\omega_1}$ many $\omega_1$-like models of $\text{ZFC}$, all with the same ordinals, that are pairwise incomparable under embeddability; there can be a transitive $\omega_1$-like model of ZFC that does not embed into its own constructible universe; and there can be an $\omega_1$-like model of PA whose structure of hereditarily finite sets is not universal for the $\omega_1$-like models of set theory.

In this article, we consider the question of whether the embedding theorems of my article, Every countable model of set theory embeds into its own constructible universe, which concern the countable models of set theory, might extend to the realm of uncountable models. Specifically, in that paper I had proved that (1) any two countable models of set theory are comparable by embeddability; indeed, (2) one countable model of set theory embeds into another just in case the ordinals of the first order-embed into the ordinals of the second; consequently, (3) every countable model of set theory embeds into its own constructible universe; and furthermore, (4) every countable model of set theory embeds into the hereditarily finite sets $\langle\text{HF},{\in}\rangle^M$ of any nonstandard model of arithmetic $M\models\text{PA}$. The question we consider here is, do the analogous results hold for uncountable models? Our answer is that they do not. Indeed, we shall prove that the corresponding statements do not hold even in the special case of $\omega_1$-like models of set theory, which otherwise among uncountable models often exhibit a special affinity with the countable models. Specifically, we shall construct large families of pairwise incomparable $\omega_1$-like models of set theory, even though they all have the same ordinals; we shall construct $\omega_1$-like models of set theory that do not embed into their own $L$; and we shall construct $\omega_1$-like models of \PA\ that are not universal for all $\omega_1$-like models of set theory.

The embedding theorems are expressed collectively in the theorem below. An embedding of one model $\langle M,{\in^M}\rangle$ of set theory into another $\langle N,{\in^N}\rangle$ is simply a function $j:M\to N$ for which $x\in^My\longleftrightarrow j(x)\in^Nj(y)$, for all $x,y\in M$, and in this case we say that $\langle M,{\in^M}\rangle$ embeds into $\langle N,{\in^N}\rangle$; note by extensionality that every embedding is injective. Thus, an embedding is simply an isomorphism of $\langle M,{\in^M}\rangle$ with its range, which is a submodel of $\langle N,{\in^N}\rangle$. Although this is the usual model-theoretic embedding concept for relational structures, the reader should note that it is a considerably weaker embedding concept than commonly encountered in set theory, because this kind of embedding need not be elementary nor even $\Delta_0$-elementary, although clearly every embedding as just defined is elementary at least for quantifier-free assertions. So we caution the reader not to assume a greater degree of elementarity beyond quantifier-free elementarity for the embeddings appearing in this paper.


1. For any two countable models of set theory $\langle M,\in^M\rangle$ and $\langle N,\in^N\rangle$, one of them embeds into the other.

2. Indeed, such an $\langle M,{\in^M}\rangle$ embeds into $\langle N,{\in^N}\rangle$ if and only if the ordinals of $M$ order-embed into the ordinals of $N$.

3. Consequently, every countable model $\langle M,\in^M\rangle$ of set theory embeds into its own constructible universe $\langle L^M,\in^M\rangle$.

4. Furthermore, every countable model of set theory embeds into the hereditary finite sets $\langle \text{HF},{\in}\rangle^M$ of any nonstandard model of arithmetic $M\models\text{PA}$. Indeed, $\text{HF}^M$ is universal for all countable acyclic binary relations.

One can begin to get an appreciation for the difference in embedding concepts by observing that ZFC proves that there is a nontrivial embedding $j:V\to V$, namely, the embedding recursively defined as follows $$j(y)=\bigl\{\ j(x)\ \mid\ x\in y\ \bigr\}\cup\bigl\{\{\emptyset,y\}\bigr\}.$$

We leave it as a fun exercise to verify that $x\in y\longleftrightarrow j(x)\in j(y)$ for the embedding $j$ defined by this recursion. (See my paper Every countable model of set theory embeds into its own constructible universe; but to give a hint here for the impatient, note that every $j(y)$ is nonempty and also $\emptyset\notin j(y)$; it follows that inside $j(y)$ we may identify the pair $\{\emptyset,y\}\in j(y)$; it follows that $j$ is injective and furthermore, the only way to have $j(x)\in j(y)$ is from $x\in y$.} Contrast this situation with the well-known Kunen inconsistency, which asserts that there can be no nontrivial $\Sigma_1$-elementary embedding $j:V\to V$. Similarly, the same recursive definition applied in $L$ leads to nontrivial embeddings $j:L\to L$, regardless of whether $0^\sharp$ exists. But again, the point is that embeddings are not necessarily even $\Delta_0$-elementary, and the familiar equivalence of the existence of $0^\sharp$ with a nontrivial “embedding” $j:L\to L$ actually requires a $\Delta_0$-elementary embedding.)

We find it interesting to note in contrast to the theorem above that there is no such embedding phenomenon in the the context of the countable models of Peano arithmetic (where an embedding of models of arithmetic is a function preserving all atomic formulas in the language of arithmetic). Perhaps the main reason for this is that embeddings between models of PA are automatically $\Delta_0$-elementary, as a consequence of the MRDP theorem, whereas this is not true for models of set theory, as the example above of the recursively defined embedding $j:V\to V$ shows, since this is an embedding, but it is not $\Delta_0$-elementary, in light of $j(\emptyset)\neq\emptyset$. For countable models of arithmetic $M,N\models\text{PA}$, one can show that there is an embedding $j:M\to N$ if and only if $N$ satisfies the $\Sigma_1$-theory of $M$ and the standard system of $M$ is contained in the standard system of $N$. It follows that there are many instances of incomparability. Meanwhile, it is a consequence of statement (4) that the embedding phenomenon recurs with the countable models of finite set theory $\text{ZFC}^{\neg\infty}$, that is, with $\langle\text{HF},{\in}\rangle^M$ for $M\models\text{PA}$, since all nonstandard such models are universal for all countable acyclic binary relations, and so in the context of countable models of $\text{ZFC}^{\neg\infty}$ there are precisely two bi-embeddability classes, namely, the standard model, which is initial, and the nonstandard countable models, which are universal.

Our main theorems are as follows.


1. If $\diamondsuit$ holds and ZFC is consistent, then there is a family $\mathcal C$ of $2^{\omega_1}$ many pairwise incomparable $\omega_1$-like models of ZFC, meaning that there is no embedding between any two distinct models in $\mathcal C$.

2. The models in statement (1) can be constructed so that their ordinals order-embed into each other and indeed, so that the ordinals of each model is a universal $\omega_1$-like linear order. If ZFC has an $\omega$-model, then the models of statement (1) can be constructed so as to have precisely the same ordinals.

3. If $\diamondsuit$ holds and ZFC is consistent, then there is an $\omega_1$-like model $M\models\text{ZFC}$ and an $\omega_1$-like model $N\models\text{PA}$ such that $M$ does not embed into $\langle\text{HF},{\in}\rangle^N$.

4. If there is a Mahlo cardinal, then in a forcing extension of $L$, there is a transitive $\omega_1$-like model $M\models\text{ZFC}$ that does not embed into its own constructible universe $L^M$.

Note that the size of the family $\mathcal C$ in statement (1) is as large as it could possibly be, given that any two elements in a pairwise incomparable family of structures must be non-isomorphic and there are at most $2^{\omega_1}$ many isomorphism types of $\omega_1$-like models of set theory or indeed of structures of size $\omega_1$ in any first-order finite language. Statement (2) shows that the models of the family $\mathcal C$ serve as $\omega_1$-like counterexamples to the assertion that one model of set theory embeds into another whenever the ordinals of the first order-embed into the ordinals of the second.

Degrees of rigidity for Souslin trees

[bibtex key=FuchsHamkins2009:DegreesOfRigidity]

We investigate various strong notions of rigidity for Souslin trees, separating them under Diamond into a hierarchy. Applying our methods to the automorphism tower problem in group theory, we show under Diamond that there is a group whose automorphism tower is highly malleable by forcing.

Changing the heights of automorphism towers by forcing with Souslin trees over $L$

[bibtex key=FuchsHamkins2008:ChangingHeightsOverL]

We prove that there are groups in the constructible universe whose automorphism towers are highly malleable by forcing. This is a consequence of the fact that, under a suitable diamond hypothesis, there are sufficiently many highly rigid non-isomorphic Souslin trees whose isomorphism relation can be precisely controlled by forcing.

In an earlier paper with Simon Thomas, “Changing the heights of automorphism towers,” we had added such malleable groups by forcing, and the current paper addresses the question as to whether there are such groups already in L.

Diamond (on the regulars) can fail at any strongly unfoldable cardinal

[bibtex key=DzamonjaHamkins2006:DiamondCanFail]

If $\kappa$ is any strongly unfoldable cardinal, then this is preserved in a forcing extension in which $\Diamond_\kappa(\text{REG})$ fails. This result continues the progression of the corresponding results for weakly compact cardinals, due to Woodin, and for indescribable cardinals, due to Hauser.

A class of strong diamond principles

[bibtex key=Hamkins:LaverDiamond]

In the context of large cardinals, the classical diamond principle $\Diamond_\kappa$ is easily strengthened in natural ways. When $\kappa$ is a measurable cardinal, for example, one might ask that a $\Diamond_\kappa$ sequence anticipate every subset of $\kappa$ not merely on a stationary set, but on a set of normal measure one. This is equivalent to the existence of a function $\ell:\kappa\to V_\kappa$ such that for any $A\in H(\kappa^+)$ there is an embedding $j:V\to M$ having critical point $\kappa$ with $j(\ell)(\kappa)=A$. This and similar principles formulated for many other large cardinal notions, including weakly compact, indescribable, unfoldable, Ramsey, strongly unfoldable and strongly compact cardinals, are best conceived as an expression of the Laver function concept from supercompact cardinals for these weaker large cardinal notions. The resulting Laver diamond principles can hold or fail in a variety of interesting ways.