1 Set theory with urelements

Set theory was traditionally often conceived as a theory of abstract collection taking place over a class of already-existing primitive objects, the urelements or atoms—we use the terms interchangeably—the irreducible mathematical objects, which are not sets but out of which the sets are to be formed. Perhaps we have numbers as these primitive urelements, or perhaps geometric points or some other kind of atomic, irreducible mathematical object. With these urelement atoms in hand, we proceed to form the sets of atoms, sets of sets of atoms, and so on. The cumulative set-theoretic universe thus grows out of the atoms.

So let us consider this urelement set theory. If A is the class of all urelement atoms, we denote the corresponding set-theoretic universe by $V(A)$ , reserving the symbol V to denote the class of pure sets, which have no urelements amongst their hereditary members; we do not intend to suggest with this notation that $V(A)$ was somehow constructed from V by adding urelements A, although we shall explain (page 3) senses in which $V(A)$ does admit a cumulative hierarchical structure. Rather, the central structure in focus is ${\left \langle V(A),\in ,{\mathbb A}\right \rangle }$ , a model of the theory we shall presently describe, where $V(A)$ denotes the class of all sets and atoms and ${\mathbb A}$ is a predicate picking out the atoms, so that ${\mathbb A}(a)$ holds exactly when a is an urelement atom. This predicate distinguishes the urelements from the empty set $\varnothing $ , which would otherwise appear like an urelement in having no elements, but amongst these $\in $ -minimal objects, only $\varnothing $ is a set. The predicate ${\mathbb A}$ is thus definable in ${\left \langle V(A),\in \right \rangle }$ from the parameter $\varnothing $ , and we could dispense with the predicate ${\mathbb A}$ by instead specifying $\varnothing $ as a constant.

1.3 The rank hierarchy

In ZFC set theory, the universe V of pure sets is stratified into the familiar cumulative $V_{\alpha }$ hierarchy of rank, which can be defined by iterating the power set operator, starting with $V_0=\varnothing $ , taking power set at successors $V_{\alpha +1}=P(V_{\alpha })$ , and unions at limits $V_{\lambda }=\bigcup _{\alpha <\lambda }V_{\alpha }$ . Alternatively, one can define the rank of a set directly, with the recursive definition $\rho (y)=\sup \{{\rho (x)+1\mid x\in y}\}$ . The rank $\rho (y)$ is the smallest ordinal for which $y\in V_{\rho (y)+1}$ , and so $V_{\alpha }$ is the collection of sets having rank less than $\alpha $ .

In the urelement context $V(A)$ , one can attempt to iterate the power set by defining $V_0(A)=A$ , taking the power set (power class) with $V_{\alpha +1}(A)=P(V_{\alpha }(A))$ and unions at limits $V_{\lambda }(A)=\bigcup _{\alpha <\lambda }V_{\alpha }(A)$ . When A is a set, this works completely fine. When A is a proper class, however, this recursion is not set-like, because every $V_{\alpha }(A)$ will be a proper class. Furthermore, in the proper class case this manner of definition is problematic, because one cannot generally undertake class recursions of this form without additional axiomatic power—the principle of elementary transfinite recursion ETR, asserting that all first-order class recursions along class well-ordered relations have solutions, is known to have consistency strength strictly stronger than GBC, even for class recursions of length $\omega $ (see [Reference Gitman, Hamkins, Caicedo, Cummings, Koellner and Larson6]).

Nevertheless, the rank recursion can be successfully undertaken in urelement set theory by defining the rank function directly on individual sets with the recursion $\rho (y)=\sup \{{\rho (x)+1\mid x\in y}\}$ , adding that for the atoms. Thus, we can define the class $V_{\alpha }(A)$ to be the atoms and sets with rank less than $\alpha $ . In this way, we achieve a rank hierarchy on $V(A)$ .

$$ \begin{align*}V(A)=\bigcup_{\alpha\in\mathord{\mathrm{Ord}}}V_{\alpha}(A).\end{align*} $$

An alternative way to realize this is to define $V_{\alpha }(A)$ to be the union of $V_{\alpha }(w)$ for any set of atoms $w\subseteq A$ , using the fact that the powerset-iteration construction works fine for $V_{\alpha }(w)$ , stratifying the class $V(w)$ , and every set in $V(A)$ is in some $V(w)$ , where w is the set of atoms supporting the given set.

$$ \begin{align*}V(A)=\bigcup_{\genfrac{}{}{0pt}{2}{\alpha\in\mathord{\mathrm{Ord}}}{w\subseteq A}}V_{\alpha}(w).\end{align*} $$

Thus, the universe $V(A)$ is realized as a two-dimensional set-like cumulative hierarchy, a potentialist system that allows both for increasing height as $\alpha $ increases and increasing width as one increases the set of urelements $w\subseteq A$ used for support.

2 Interpreting urelement set theory in ZFC

Let us describe next how to interpret urelement set theory inside the urelement-free set theories ZF and ZFC. The fact that this is possible, and furthermore the fact that it is possible in a way that achieves strong axioms for the interpreted urelement set theories, such as “there are a proper class of atoms” or even “the atoms can be well-ordered in type $\mathord {\mathrm {Ord}}$ ,” tends to support the view that the foundational roles sought for the urelement theories can also be undertaken ultimately without urelements by interpreting these urelement theories inside ZFC. On this view, one will not identify any new mathematical structure or relation in these urelement set theories that we cannot in principle find already in ZFC.

Assume V is a set-theoretic universe satisfying ZFC, and suppose A is any class of sets in V. (In the ZFC context, we assume A is a definable class, but we shall later move to the GBC context, where we drop the definability requirement.) Inside V we shall presently define a certain model of urelement set theory $V{\mathopen {\lbrack \!\lbrack }A\mathclose {\rbrack \!\rbrack }}$ , a model of $\mathrm {ZFCU}$ in which the urelements will be indexed by the class A, whose pure sets, furthermore, will be an exact copy of V.

We begin by making a copy of A with $\bar A=\{{0}\}\times A$ , representing every object $a\in A$ by its copy $\bar a={\left \langle 0,a\right \rangle }$ . These will be the urelements of $V{\mathopen {\lbrack \!\lbrack }A\mathclose {\rbrack \!\rbrack }}$ , and we accordingly place $\bar A\subseteq V{\mathopen {\lbrack \!\lbrack }A\mathclose {\rbrack \!\rbrack }}$ . Next, we simply close $V{\mathopen {\lbrack \!\lbrack }A\mathclose {\rbrack \!\rbrack }}$ under the operation: if $y\subseteq V{\mathopen {\lbrack \!\lbrack }A\mathclose {\rbrack \!\rbrack }}$ , then $\bar y={\left \langle 1,y\right \rangle }\in V{\mathopen {\lbrack \!\lbrack }A\mathclose {\rbrack \!\rbrack }}$ . That is, $V{\mathopen {\lbrack \!\lbrack }A\mathclose {\rbrack \!\rbrack }}$ is the smallest class of sets containing $\bar A$ and closed under that operation. One may undertake this as a set-like transfinite recursion in ZF by forming a hierarchy $V_{\alpha }{\mathopen {\lbrack \!\lbrack }A\cap V_{\alpha }\mathclose {\rbrack \!\rbrack }}$ , gradually adding atoms and closing under the operation as the recursion progresses. The central idea is that $\bar y={\left \langle 1,y\right \rangle }$ will be the set in $V{\mathopen {\lbrack \!\lbrack }A\mathclose {\rbrack \!\rbrack }}$ whose elements are the (actual) elements of y. Thus, we define the membership relation as $x\mathrel {\mkern 3mu\overline {\mkern -3mu\in \mkern -1.5mu}\mkern 1.5mu} \bar y$ if and only if $x\in y$ , where $\bar y={\left \langle 1,y\right \rangle }$ . The relation $\mathrel {\mkern 3mu\overline {\mkern -3mu\in \mkern -1.5mu}\mkern 1.5mu}$ is therefore set-like and well-founded in V, because $x\mathrel {\mkern 3mu\overline {\mkern -3mu\in \mkern -1.5mu}\mkern 1.5mu} \bar y$ requires that x has lower rank than $\bar y$ . In short, the definition is that every object in $V{\mathopen {\lbrack \!\lbrack }A\mathclose {\rbrack \!\rbrack }}$ will be an ordered pair, with the urelements having the form $\bar a={\left \langle 0,a\right \rangle }$ for $a\in A$ and fulfilling ${\mathbb A}(\bar a)$ and the sets having the form $\bar y={\left \langle 1,y\right \rangle }$ for some $y\subseteq V{\mathopen {\lbrack \!\lbrack }A\mathclose {\rbrack \!\rbrack }}$ , with the $\mathrel {\mkern 3mu\overline {\mkern -3mu\in \mkern -1.5mu}\mkern 1.5mu}$ -elements of $\bar y$ being simply the actual members of y.

The claim is that ${\left \langle V{\mathopen {\lbrack \!\lbrack }A\mathclose {\rbrack \!\rbrack }},\mathrel {\mkern 3mu\overline {\mkern -3mu\in \mkern -1.5mu}\mkern 1.5mu},{\mathbb A}\right \rangle }$ is a model of ZFCU. Extensionality holds because the urelements have no $\mathrel {\mkern 3mu\overline {\mkern -3mu\in \mkern -1.5mu}\mkern 1.5mu}$ -members and if $\bar y={\left \langle 1,y\right \rangle }$ and $\bar z={\left \langle 1,z\right \rangle }$ have the same $\mathrel {\mkern 3mu\overline {\mkern -3mu\in \mkern -1.5mu}\mkern 1.5mu}$ -elements, then necessarily $y=z$ and consequently $\bar y=\bar z$ . Foundation holds because $\mathrel {\mkern 3mu\overline {\mkern -3mu\in \mkern -1.5mu}\mkern 1.5mu}$ is well-founded—the $\mathrel {\mkern 3mu\overline {\mkern -3mu\in \mkern -1.5mu}\mkern 1.5mu}$ -elements of a set have lower rank in V. It is easy to show that $V{\mathopen {\lbrack \!\lbrack }A\mathclose {\rbrack \!\rbrack }}$ has the union, pairing, and power set axioms simply by constructing the appropriate representing set in each case. For example, if $u,v\in V{\mathopen {\lbrack \!\lbrack }A\mathclose {\rbrack \!\rbrack }}$ , then ${\left \langle 1,\{{u,v}\}\right \rangle }$ will represent the unordered pair $\{{u,v}\}$ in $V{\mathopen {\lbrack \!\lbrack }A\mathclose {\rbrack \!\rbrack }}$ . We get the separation, replacement, and collection axioms because every subset $y\subseteq V{\mathopen {\lbrack \!\lbrack }A\mathclose {\rbrack \!\rbrack }}$ that is realized in V is represented by the set $\bar y\in V{\mathopen {\lbrack \!\lbrack }A\mathclose {\rbrack \!\rbrack }}$ . Similarly, the axiom of choice holds in $V{\mathopen {\lbrack \!\lbrack }A\mathclose {\rbrack \!\rbrack }}$ , if it holds in V, because we can form a suitable set y of choices in V, and then consider the corresponding set $\bar y$ in $V{\mathopen {\lbrack \!\lbrack }A\mathclose {\rbrack \!\rbrack }}$ .

Every set u in V will be represented by a corresponding set $\check u={\left \langle 1,\{{\check v\mid v\in u}\}\right \rangle }$ in $V{\mathopen {\lbrack \!\lbrack }A\mathclose {\rbrack \!\rbrack }}$ , for it is easy to see that $v\in u\mathrel {\leftrightarrow } \check v\mathrel {\mkern 3mu\overline {\mkern -3mu\in \mkern -1.5mu}\mkern 1.5mu}\check u$ and consequently ${\left \langle V,\in \right \rangle }$ is isomorphic to ${\left \langle \smash {\hat V},\mathrel {\mkern 3mu\overline {\mkern -3mu\in \mkern -1.5mu}\mkern 1.5mu}\right \rangle }$ , where $\hat V=\{{\check u\mid u\in V}\}$ , which is a transitive class in $V{\mathopen {\lbrack \!\lbrack }A\mathclose {\rbrack \!\rbrack }}$ . Furthermore, using the fact that the rank of a set $\bar y$ in $V{\mathopen {\lbrack \!\lbrack }A\mathclose {\rbrack \!\rbrack }}$ is bounded by the rank of $\bar y$ in V, it follows that $\hat V$ consists precisely of the pure sets of $V{\mathopen {\lbrack \!\lbrack }A\mathclose {\rbrack \!\rbrack }}$ . In particular, the ordinals of $V{\mathopen {\lbrack \!\lbrack }A\mathclose {\rbrack \!\rbrack }}$ are precisely the $\check \alpha $ for the ordinals $\alpha $ of V.

Let us expand the urelement language somewhat by introducing the urelement enumeration predicate $\smash {\vec {{\mathbb A}}}$ , which holds exactly in the instances $\smash {\vec {{\mathbb A}}}(\check a,\bar a)$ when $a\in A$ . That is, $\smash {\vec {{\mathbb A}}}$ is the graph of the function $\check a\mapsto \bar a$ , for $a\in A$ . This predicate identifies the urelements—they are exactly the range of the map $\smash {\vec {{\mathbb A}}}$ —and so the urelement predicate ${\mathbb A}$ is definable from $\smash {\vec {{\mathbb A}}}$ . But $\smash {\vec {{\mathbb A}}}$ also provides the enumeration of the urelements by the elements of A, well, by the copy of A in the pure sets of $V{\mathopen {\lbrack \!\lbrack }A\mathclose {\rbrack \!\rbrack }}$ , that is, by the class $\hat A=\{{\check a\mid a\in A}\}$ . This enumeration $\smash {\vec {{\mathbb A}}}$ will not generally be definable from ${\mathbb A}$ alone, since permutations of the urelements will lead to automorphisms of $V{\mathopen {\lbrack \!\lbrack }A\mathclose {\rbrack \!\rbrack }}$ that preserve ${\mathbb A}$ but not $\smash {\vec {{\mathbb A}}}$ . We interpret the phrase, “there are A many atoms” as the assertion that $\smash {\vec {{\mathbb A}}}$ is a bijection between A and the class of urelements. The theories $\mathrm {ZF}\vec {\mathrm {U}}$ and $\mathrm {ZFC}\vec {\mathrm {U}}$ are the analogues of $\mathrm {ZFU}$ and $\mathrm {ZFCU}$ in the language with the atom-enumeration predicate $\smash {\vec {{\mathbb A}}}$ .

In light of these various bi-interpretations, we are justified in viewing these interpreted models of urelement set theory from two perspectives. On the one hand, $V{\mathopen {\lbrack \!\lbrack }A\mathclose {\rbrack \!\rbrack }}$ is a definable class in the ground model V, defined there as an interpreted class structure. On the other hand, since V is isomorphic to the pure sets of $V{\mathopen {\lbrack \!\lbrack }A\mathclose {\rbrack \!\rbrack }}$ , we can view the model instead as an extension $V({\mkern 3.5mu\overline {\mkern -3.5mu A\mkern -.5mu}\mkern .5mu})$ of the original universe V, obtained by adjoining a new class ${\mkern 3.5mu\overline {\mkern -3.5mu A\mkern -.5mu}\mkern .5mu}$ of urelements to V, enumerated by the class A. The bi-interpretation result explains exactly how these two perspectives are equivalent accounts of the same phenomenon.

Let us next record a further observation about the nature of the interpreted urelement structure.

The hypothesis that every set is equinumerous with a pure set is philosophically significant with regard to structuralism, because it follows that every mathematical structure is isomorphic to a structure in the pure sets. A structuralist in the philosophy of mathematics could therefore take this as evidence that urelements are not needed in such a situation, since one can already realize all possible mathematical structure in the universe of pure sets. Alternatively, since one mathematician’s modus ponens is another’s modus tollens, the observation is evidence that in order for urelements to be helpful to us in the foundations of mathematics, we should want or expect to have sets of urelements that are not equinumerous with any pure set. In particular, this would require a denial of the axiom of choice, since every well-order is isomorphic to an ordinal, which is a pure set. Do we expect that there are collections of irreducible atomic mathematical objects that cannot in principle be placed into bijection with any pure set? That seems strange and wonderful, but the puzzle is that such kinds of urelements would be very different from the traditional urelement proposals, such as numbers and geometric points, both of which find equinumerosity with pure sets. So the philosophical task for the urelement supporters is to explain why we would need a foundational theory to accommodate strange and bizarre set of urelements, if the intended sets of urelements are tame.

What we mean is that the assertion “there are A many atoms” is expressed in the language with the urelement enumeration predicate $\smash {\vec {{\mathbb A}}}$ by asserting that this predicate enumerates the urelements using the class A as it is defined in the class of pure sets of the $\mathrm {ZFC}\vec {\mathrm {U}}$ model. In particular, the following theories are bi-interpretable:

1. (1) ZFC,

2. (2) $\mathrm {ZFC}\vec {\mathrm {U}}+$ “there are $\omega $ many atoms,”

3. (3) $\mathrm {ZFC}\vec {\mathrm {U}}+$ “there are ${\mathbb R}$ many atoms,”

4. (4) $\mathrm {ZFC}\vec {\mathrm {U}}+$ “there are $\aleph _{\omega ^2+5}$ many atoms,”

5. (5) $\mathrm {ZFC}\vec {\mathrm {U}}+$ “there are $\mathord {\mathrm {Ord}}$ many atoms,”

6. (6) $\mathrm {ZFC}\vec {\mathrm {U}}+$ “there are V many atoms,”

where we express the theories in the language with the urelement enumeration predicate $\smash {\vec {{\mathbb A}}}$ . The corresponding theories with ZFCU in place of $\mathrm {ZFC}\vec {\mathrm {U}}$ also are bi-interpretable with each other and ZFC, and all the above theories, provided one allows a parameter—simply use a set parameter to enumerate the atoms, as explained in Theorem 2.5.

It follows that $\mathrm {ZFC}\vec {\mathrm {U}}$ is not a tight theory in the sense of [Reference Enayat, van Eijck, Iemhoff and Joosten4, Reference Freire and Hamkins5], since it admits these distinct bi-interpretable extensions. A theory is tight when any extensions of it are bi-interpretable if and only if they are identical.

If the class of urelements is a set, then we can dispense with the need for the urelement predicates ${\mathbb A}$ and $\smash {\vec {{\mathbb A}}}$ in the interpretation, since the enumeration will in effect be one of the sets available in $V{\mathopen {\lbrack \!\lbrack }A\mathclose {\rbrack \!\rbrack }}$ , which we can simply use as a parameter.

If A is a proper class, however, then it will turn out that ${\left \langle V,\in \right \rangle }$ is not bi-interpretable with ${\left \langle V{\mathopen {\lbrack \!\lbrack }A\mathclose {\rbrack \!\rbrack }},\mathrel {\mkern 3mu\overline {\mkern -3mu\in \mkern -1.5mu}\mkern 1.5mu},{\mathbb A}\right \rangle }$ , even with parameters, although these structures are mutually interpretable and semi-bi-interpretable, meaning that V can identify its copy inside $V{\mathopen {\lbrack \!\lbrack }A\mathclose {\rbrack \!\rbrack }}$ .

The proof relied on the fact that ZFC models are definably rigid, but models of ZFCU with a proper class of urelements have lots of parametrically definable automorphisms, fixing any desired finite list of parameters. In this situation, there can be no bi-interpretation of models. The basic lesson is that there is a fundamental difference for interpretative power between the theories $\mathrm {ZFCU}$ and $\mathrm {ZFC}\vec {\mathrm {U}}$ , between having the urelements merely as a class ${\mathbb A}$ versus having them as an explicit enumeration $\smash {\vec {{\mathbb A}}}$ by a class of pure sets. One can expect to achieve bi-interpretation of urelement set theory with urelement-free set theory only in the latter case, and otherwise it is the urelement-free theory with the stronger interpretative power, since V sees how it is copied to the pure sets of $V{\mathopen {\lbrack \!\lbrack }A\mathclose {\rbrack \!\rbrack }}$ , but ${\left \langle V{\mathopen {\lbrack \!\lbrack }A\mathclose {\rbrack \!\rbrack }},\in ,{\mathbb A}\right \rangle }$ cannot in general see how it arises from its pure sets.

3 Reflection

Let us recall the first-order reflection phenomenon in set theory. The Lévy–Montague reflection theorem of ZF set theory asserts that for any formula $\varphi (x)$ , there is an ordinal $\lambda $ such that $\varphi $ is absolute between $V_{\lambda }$ and V. A somewhat more refined version asserts of every meta-theoretically finite number n that there is an ordinal $\lambda $ such that all $\Sigma _n$ formulas are absolute between $V_{\lambda }$ and V. Indeed, there is a definable closed unbounded class C of such ordinals $\lambda $ , the club of $\Sigma _n$ -correct cardinals. This is a theorem scheme, with a separate statement for each number n. The reflection theorem is equivalent over the rest of ZFC to the collection axiom (hence to replacement over Zermelo set theory), since the existence of sufficiently large reflecting ordinals $\lambda $ provides $V_{\lambda }$ as a suitable collecting set for the formula asserting that indeed the witnesses exist.

In the urelement context, when there are a proper class of urelements, then we should not generally expect to reflect truths from $V(A)$ specifically to the class $V_{\lambda }(A)$ , since there will be sets of urelements $w\subseteq A$ in $V(A)$ of cardinality larger than $\lambda $ , which having rank $1$ will be in $V_{\lambda }(A)$ , but such w are not equinumerous with any ordinal in $V_{\lambda }(A)$ . This would be a violation of reflection between $V_{\lambda }(A)$ and $V(A)$ for the statement $\varphi =$ “every set of urelements is bijective with an ordinal.” What we should want to do with reflection, rather than reflecting specifically to some rank-initial-segment class $V_{\lambda }(A)$ , is to reflect truth to some transitive set.

If there is merely a set of urelements, then we can stratify the full universe $V(A)$ by the rank hierarchy $V_{\alpha }(A)$ , and each of these is a set. This provides a set-like continuous transfinite tower $V_{\alpha }(A)$ , and this is enough to run the usual ZFC proof of the reflection theorem, where we shall be able to reflect any $\varphi $ from $V(A)$ to some $v=V_{\alpha }(A)$ . Similarly, in a model of $\mathrm {ZFC}\vec {\mathrm {U}}+ \text {“}\mathord {\mathrm {Ord}}$ many atoms,” we can similarly stratify the universe in a set-like hierarchy, using $V_{\alpha +1}(\smash {\vec {{\mathbb A}}}\upharpoonright \alpha )$ , where $\smash {\vec {{\mathbb A}}}\upharpoonright \alpha $ restricts the urelement enumeration to the first $\alpha $ many elements, and taking unions at limit ordinals for continuity. This continuous cumulative hierarchy is again enough to prove the reflection theorem by the usual argument, reflecting any $\varphi $ from $V(A)$ to some $v=\bigcup _{\alpha <\lambda }V_{\alpha +1}(\smash {\vec {{\mathbb A}}}\upharpoonright \alpha )$ .

But in fact, we should like to prove that we get the first-order reflection theorem in ZFCU generally, making no assumption on the number or kinds of urelements. Let’s begin by establishing the following duplication and homogeneity lemmas, showing that over some fixed set of urelements, all additional sets of urelements of the same cardinality look exactly alike.

The urelement duplication property will not necessarily hold over any set u, since in the model Y discussed earlier, the uncountable set A of urelements cannot be duplicated to a disjoint set of urelements of the same size, since all the remaining sets of urelements are countable subsets of B, and so in that model Y, we do not achieve duplication over $\varnothing $ . In the model Y, however, we do achieve the duplication property over the set $u=A$ , since any set w disjoint from this u can be duplicated to another such set $w'$ disjoint from $u\cup w$ . Moreover, the second author [Reference Yao26] proves that it is consistent with ZFU + $\omega $ -DC that the duplication property holds over no set of urelements.

We cannot in general dispense with the fixed set u in the homogeneity claim of Lemma 3.3, for although as we have observed $V(B)$ is isomorphic to $V(C)$ whenever B and C are equinumerous classes of urelements, such an isomorphism is not necessarily realizable by an automorphism of $V(A)$ . Indeed, it is not true in general in ZFCU that whenever two sets of urelements are equinumerous, then they are automorphic, since perhaps there are exactly $\omega _1$ many urelements and w is the set of all urelements, but $w'$ has all but one of them. These sets are equinumerous by a bijection in $V(w)$ and $V(w)$ is isomorphic to $V(w')$ , but the sets are not automorphic by any automorphism of $V(w)$ .

4 Class theory

We should like next to undertake the entire analysis in the context of second-order set theory, developing the urelement analogues of Gödel–Bernays set theory and Kelley–Morse set theory. These are conveniently formalized as two-sorted theories, with a first-order domain of individuals, consisting of the atoms and the sets, and a second-order domain of classes, consisting of a family of classes of individuals.

Let us begin with a quick review of the second-order pure theories. A model of Gödel–Bernays set theory GBC has the form ${\left \langle M,\in ^M,\mathcal {M}\right \rangle }$ , where the first-order part ${\left \langle M,\in ^M\right \rangle }$ is a model of ZFC, and the second-order part $\mathcal {M}\subseteq P(M)$ is a collection of classes $X\subseteq M$ . The classes are allowed as atomic predicates into the first-order language, so that the model can assert $a\in X$ for the elements a of X in M, and we allow these class parameters into the ZFC collection and separation axiom schemes. The theory GBC also has the first-order class comprehension axiom, asserting that for any formula $\varphi $ involving only first-order quantifiers (over the sets) and class parameters $X_i$ , the collection $X=\{{a\mid \varphi (a,X_0,\ldots ,X_n)}\}$ forms a class in $\mathcal {M}$ . The theory GBC also includes the axiom of global choice, which can be equivalently expressed either as the assertion that there is a global choice function F for which $F(u)\in u$ for every nonempty set u, as the assertion that there is a global well-order relation $<$ on the universe of sets, a principle known also as the global well-order principle, or as the $\mathord {\mathrm {Ord}}$ enumeration principle, asserting specifically that there is such an order of type $\mathord {\mathrm {Ord}}$ . Although these formulations of global choice are equivalent for the pure-set purposes of GBC, we warn the reader that they are no longer equivalent in the urelement class theories.

The theory GBC is conservative over ZFC for assertions about sets, because we can equip any model $M\models \mathrm {ZFC}$ with all its first-order definable classes (with parameters), and this will satisfy all of GBC except possibly the global choice principle, since in some models there is no definable global well-order. But with the method of forcing we can add a generic global well-order to the universe, a forcing extension adding no sets, and then take the classes that are first-order definable using this new generic class parameter. The result is a model of GBC with the same sets as M, and so the conclusion is that any purely set-theoretic situation that can happen in ZFC can also happen in GBC, and this is an alternative equivalent way to express conservativity. It follows from the conservativity result that GBC is equiconsistent with ZFC. Despite what seems to be a very close semantic connection between GBC and ZFC, however, these theories (if consistent) are not mutually interpretable—the reason is that GBC is finitely axiomatizable and consequently cannot be interpreted in ZFC, for such an interpretation would make use of only finitely many axioms of ZFC, but ZFC proves the consistency of its finite fragments, and so such an interpretation of GBC in ZFC would violate the second incompleteness theorem.

Kelley–Morse set theory KM strengthens GBC with the second-order class comprehension axiom scheme, allowing one to define classes $\{{x\mid \varphi (x,Z)}\}$ by formulas $\varphi $ having second-order quantifiers ranging over the available classes of the model. This theory is not conservative over ZFC since it implies that there is a satisfaction class for first-order truth, and the existence of such a truth predicate implies $\mathop {\mathrm {Con}}(\mathrm {ZFC})$ and $\mathop {\mathrm {Con}}(\mathrm {ZFC}+\mathop {\mathrm {Con}}(\mathrm {ZFC}))$ and much more (see [Reference Hamkins10] an elementary account). Meanwhile, if $\kappa $ is an inaccessible cardinal, then ${\left \langle V_{\kappa },\in ,V_{\kappa +1}\right \rangle }$ is a model of Kelley–Morse set theory, and so the consistency strength of KM is strictly between ZFC and ZFC plus an inaccessible cardinal. So KM ultimately is just a small step up in consistency strength from ZFC or GBC.

Gitman and Hamkins [Reference Gitman and Hamkins7] (see also [Reference Gitman, Hamkins, Caicedo, Cummings, Koellner and Larson6, Reference Gitman, Hamkins, Holy, Schlicht and Williams9]) have observed that even Kelley–Morse set theory KM does not prove the class choice principle CC, which asserts that for every class I, if for every individual $i\in I$ there is a class X for which $\varphi (i,X,I)$ , then there is a class $X\subseteq I\times V$ such that for every $i\in I$ we have $\varphi (i,X_i,I)$ , where $X_i=\{{x\mid (i,x)\in X}\}$ is the ith section of the class X. For example, if for every n there is a class X with $\varphi (n,X)$ , then CC allows you to put such classes together into a single class $X\subseteq \omega \times V$ such that $\varphi (n,X_n)$ for all n. This is essentially a choice principle or collection principle for classes, and even the $\omega $ -indexed version $\omega $ -CC is not provable in KM.

Nevertheless, every model of KM admits a submodel with the same sets (but possibly fewer classes), which is a model of KM+CC (see [Reference Gitman and Hamkins7]). It follows that KM and KM+CC are mutually interpretable theories and consequently also equiconsistent. But they are not bi-interpretable, in light of [Reference Enayat, van Eijck, Iemhoff and Joosten4, Reference Freire and Hamkins5]. Meanwhile, the class choice principle supports important applications of Kelley–Morse set theory, and KM+CC should be seen as a natural, robust extension of KM.

5 Class theory with urelements

Let us now introduce the urelement analogues of GBC and KM. Namely, GBCU is the two-sorted theory for models of the form ${\left \langle M,\in ^M,{\mathbb A},\mathcal {M}\right \rangle }$ , where ${\left \langle M,\in ^M,{\mathbb A}\right \rangle }$ is a model of ZFCU, allowing class parameters in the collection and separation schemes, and $\mathcal {M}$ is a collection of classes in M that fulfills the first-order class comprehension axiom (in the language with $\in $ and ${\mathbb A}$ , allowing class parameters), plus the assertion that there is a global well-order $<$ of the universe. The global well-order $<$ is simply one of the classes in $\mathcal {M}$ , as is the class of all urelements. The theory KMU extends GBCU with the second-order class comprehension axiom. The theories $\mathrm {KMU}$ and $\mathrm {GBCU}$ are formulated analogously with the enumeration predicate $\smash {\vec {{\mathbb A}}}$ for the atoms rather than merely the predicate ${\mathbb A}$ . We warn the reader that the various formulations of global choice, although equivalent in pure set theory, come apart in the urelement context (see [Reference Howard, Rubin and Rubin13]). In particular, we won’t generally be able to convert every class well-order of the universe to a well-order of type $\mathord {\mathrm {Ord}}$ , since there may simply be too many urelements for this to be possible. We shall discuss this issue in further depth later.

In any model of Gödel–Bernays set theory ${\left \langle V,\in ,\mathcal {V}\right \rangle }$ , with any class $A\in \mathcal {V}$ , we can define the interpreted urelement model ${\left \langle V{\mathopen {\lbrack \!\lbrack }A\mathclose {\rbrack \!\rbrack }},\mathrel {\mkern 3mu\overline {\mkern -3mu\in \mkern -1.5mu}\mkern 1.5mu},\smash {\vec {{\mathbb A}}},\mathcal {V}{\mathopen {\lbrack \!\lbrack }A\mathclose {\rbrack \!\rbrack }}\right \rangle }$ in analogy with the first-order case, defining $V{\mathopen {\lbrack \!\lbrack }A\mathclose {\rbrack \!\rbrack }}$ , $\mathrel {\mkern 3mu\overline {\mkern -3mu\in \mkern -1.5mu}\mkern 1.5mu}$ and $\smash {\vec {{\mathbb A}}}$ just as we did for the ZFC context in Section 2, but now also interpreting the second-order part of the models by placing the classes $\mathcal {V}{\mathopen {\lbrack \!\lbrack }A\mathclose {\rbrack \!\rbrack }}=\{{B\in \mathcal {V}\mid B\subseteq V{\mathopen {\lbrack \!\lbrack }A\mathclose {\rbrack \!\rbrack }}}\}$ . That is, the new classes are simply the old classes that happen to be subclasses of $V{\mathopen {\lbrack \!\lbrack }A\mathclose {\rbrack \!\rbrack }}$ .

The following theories are consequently all bi-interpretable with parameters:

1. (1) GBC,

2. (2) $\mathrm {GBCU}+$ “there are $\omega $ many atoms,”

3. (3) $\mathrm {GBCU}+$ “there are $\mathord {\mathrm {Ord}}$ many atoms,”

4. (4) $\mathrm {GBCU}+$ “there are V many atoms,”

as well as the following theories:

1. (1) KM,

2. (2) $\mathrm {KMU}+$ “there are $\omega $ many atoms,”

3. (3) $\mathrm {KMU}+$ “there are $\mathord {\mathrm {Ord}}$ many atoms,”

4. (4) $\mathrm {KMU}+$ “there are V many atoms,”

and the following:

1. (1) KM + CC,

2. (2) $\mathrm {KMU}+\mathrm {CC}+$ “there are $\omega $ many atoms,”

3. (3) $\mathrm {KMU}+\mathrm {CC}+$ “there are $\mathord {\mathrm {Ord}}$ many atoms,”

4. (4) $\mathrm {KMU}+\mathrm {CC}+$ “there are V many atoms,”

5. (5) $\mathrm {ZFC}^-+$ “there is a largest cardinal $\kappa $ , which is inaccessible,”

6. (6) $\mathrm {ZFCU}^{-}+$ “there is a largest cardinal $\kappa $ , which is inaccessible, and $\kappa $ many atoms.”

Note that we can take a class parameter enumerating all the urelements, if desired. In the next section we shall explain how to achieve more than $\mathord {\mathrm {Ord}}$ many urelements in models of GBCU and KMU.

6 Abundant atoms

In urelement set theory, for any cardinal $\kappa $ we define the corresponding hereditary class $H_{\kappa }(A)$ to be $(H_{\kappa })^{V(A)}$ , that is, the class of atoms and sets in $V(A)$ of hereditary size less than $\kappa $ , the sets whose transitive closure has size less than $\kappa $ . If we are working in a second-order urelement theory in the ambient universe ${\left \langle V(A),\in ,\mathcal {V}\right \rangle }$ , then we may equip $H_{\kappa }(A)$ with the family of classes $\mathcal {H}=\mathcal {V}\upharpoonright H_{\kappa }(A)$ , consisting of all the classes $X\in \mathcal {V}$ for which $X\subseteq H_{\kappa }(A)$ . Since every atom $a\in A$ appears in $H_{\kappa }(A)$ , it follows that $A\subseteq H_{\kappa }(A)$ and consequently also every class of urelements $B\subseteq A$ is a class in the hereditary model $\mathcal {H}$ . Thus, although $H_{\kappa }(A)$ has many fewer sets than $V(A)$ , we nevertheless retain exactly the same classes of urelements in $\mathcal {H}$ as in $\mathcal {V}$ .

Let us look a little more closely at this model ${\left \langle H_{\kappa }(A),\in ,\mathcal {H}\right \rangle }$ , for there are some interesting things to notice. The ordinals of $H_{\kappa }(A)$ are exactly the ordinals below $\kappa $ , and so if A is larger than $\kappa $ in the original universe, then because it contains every urelement in A, the class $H_{\kappa }(A)$ will not be equinumerous with $\kappa $ . In other words, ${\left \langle H_{\kappa }(A),\in ,\mathcal {H}\right \rangle }$ will be a model of GBCU or KMU in which every class has a well-order, but the universe has no $\mathord {\mathrm {Ord}}$ -enumeration. In particular, the so-called limitation of size principle fails in this model, even though the global choice and global well-order principles hold.

Let us now assume specifically that A is equinumerous with $\mathord {\mathrm {Ord}}$ in the original model and look a little more closely at what kinds of classes there are in ${\left \langle H_{\kappa }(A),\in ,\mathcal {H}\right \rangle }$ . Since A has size $\mathord {\mathrm {Ord}}$ in the original universe ${\left \langle V(A),\in ,\mathcal {V}\right \rangle }$ , it follows that for every cardinal $\delta $ , including cardinals $\delta \geq \kappa $ , there will be a class $B\subseteq A$ of size $\delta $ in $V(A)$ . Such a class will have been a set in $V(A)$ , but is a proper class of urelements in ${\left \langle H_{\kappa }(A),\in ,\mathcal {H}\right \rangle }$ , although it is smaller than A in size, in the sense that these two proper classes are not equinumerous.

In this model ${\left \langle H_{\kappa }(A),\in ,\mathcal {H}\right \rangle }$ , let us say that a class is small, if it is strictly smaller in equinumerosity than the class A of all urelements. There are many small classes in this model, since all the cardinals between $\kappa $ and $\mathord {\mathrm {Ord}}$ in the original universe $V(A)$ are realized by small proper classes $B\subseteq A$ in $\mathcal {H}$ . Indeed, the small classes $B\subseteq A$ are exactly the classes $B\subseteq A$ that are sets in $V(A)$ . It follows from this that if $B\subseteq A$ is a small class of urelements, then it has a power set in $V(A)$ , and so we may find a class $D\subseteq I\times B$ , for some small class I, such that the sections $D_i=\{{b\in B\mid (i,b)\in D}\}$ are all different and every class $B'\subseteq B$ is realized as $D_i$ for some $i\in I$ . In other words, the model ${\left \langle H_{\kappa }(A),\in ,\mathcal {H}\right \rangle }$ can see that every small class of urelements has a small power class, indexed by a class D upon a small class I in that way. It also follows that for every small class I and every class $D\subseteq I\times A$ , such that $D_i$ is small for every $i\in I$ , then D itself is small. This is simply because the union of set many sets in $V(A)$ is a set, by the replacement axiom. This fact also is visible in ${\left \langle H_{\kappa }(A),\in ,\mathcal {H}\right \rangle }$ .

Let us give a name to these properties.

And so we established the following improvement on Theorem 6.1.

Let us deepen the analysis by revealing the bi-interpretation phenomenon at the heart of the situation.

Proof The relationships of the five bi-interpretable models are illustrated in Figure 1, but note that the model V at lower left is not itself used directly in the interpretations. Let us start by interpreting from the bottom right of the figure to the top left, interpreting from theory (1) to theory (4). Suppose ${\left \langle V(A),\in ,\mathcal {V}\right \rangle }$ satisfies $\mathrm {KMU}+\mathrm {CC}$ with the abundant atom axiom. We may follow the unrolling construction of Theorem 4.1, which actually works equally well in $\mathrm {KMU}+\mathrm {CC}$ as in $\mathrm {KM}+\mathrm {CC}$ . That is, we consider the class membership codes E, allowing class relations not just on $\mathord {\mathrm {Ord}}$ or on the pure sets V, but on the class of urelements, and define equivalence $E\sim F$ and membership $E\mathrel {\varepsilon } F$ . The arguments of Theorem 4.1 show that the resulting universe ${\left \langle W,\mathrel {\varepsilon }\right \rangle }$ is a model of $\mathrm {ZFC}^-$ . The ordinals of V become an inaccessible cardinal $\kappa $ in W. Because of the abundant atom axiom, however, there will be cardinals in W larger than $\kappa $ . The shortest well-ordering of the class of atoms will become an inaccessible cardinal $\lambda $ of W strictly above $\kappa $ . The cardinals below $\lambda $ in W are precisely those with class membership codes that are small in $\mathcal {V}$ . The abundant atom axiom ensures $\lambda $ is inaccessible in W, namely, a strong limit because any small class membership code will have a small class code for the power set and regular, because any small class code for a set of cardinals less than $\lambda $ will also be less than $\lambda $ . Thus, W is a model of $\mathrm {ZFC}^-$ with two $\kappa <\lambda $ , where $\lambda $ is the largest cardinal.

Figure 1 The bi-interpretable models.

The model ${\left \langle W,\mathrel {\varepsilon }\right \rangle }$ is in turn bi-interpretable with the extension ${\left \langle W(A),\mathrel {\varepsilon }\right \rangle }$ adjoining $\lambda $ many atoms. This bi-interpretation can be realized via the construction of $W{\mathopen {\lbrack \!\lbrack } \lambda \mathclose {\rbrack \!\rbrack }}$ described in Section 2 as with Theorem 2.1, but undertaken in $\mathrm {ZFC}^-$ , where the construction works fine, producing a model of $\mathrm {ZFCU}^{-}$ in which $\kappa $ and $\lambda $ remain inaccessible, with $\lambda $ remaining as the largest cardinal, and the set of urelements has size $\lambda $ . Thus, theories (4) and (5) are bi-interpretable.

The model ${\left \langle W,\mathrel {\varepsilon }\right \rangle }$ is bi-interpretable with ${\left \langle {\mkern 3.5mu\overline {\mkern -3.5muV\mkern -.5mu}\mkern .5mu}, \mathrel {\mkern 3mu\overline {\mkern -3mu\in \mkern -1.5mu}\mkern 1.5mu}, \overline {\mathcal {V}}\right \rangle }$ as in Theorem 4.1, where ${\mkern 3.5mu\overline {\mkern -3.5muV\mkern -.5mu}\mkern .5mu}=W_{\lambda }$ and $\overline {\mathcal {V}}$ consists of the subsets of ${\mkern 3.5mu\overline {\mkern -3.5muV\mkern -.5mu}\mkern .5mu}$ available in W. The model ${\left \langle {\mkern 3.5mu\overline {\mkern -3.5muV\mkern -.5mu}\mkern .5mu}, \mathrel {\mkern 3mu\overline {\mkern -3mu\in \mkern -1.5mu}\mkern 1.5mu}, \overline {\mathcal {V}}\right \rangle }$ has $\lambda $ as its class of ordinals, and $\kappa $ will still be inaccessible there. The model ${\left \langle {\mkern 3.5mu\overline {\mkern -3.5muV\mkern -.5mu}\mkern .5mu}, \mathrel {\mkern 3mu\overline {\mkern -3mu\in \mkern -1.5mu}\mkern 1.5mu},\overline {\mathcal {V}}\right \rangle }$ is in turn bi-interpretable with ${\left \langle {\mkern 3.5mu\overline {\mkern -3.5muV\mkern -.5mu}\mkern .5mu}(A),\mathrel {\mkern 3mu\overline {\mkern -3mu\in \mkern -1.5mu}\mkern 1.5mu},\overline {\mathcal {V}}(A)\right \rangle }$ , adjoining $\lambda $ many atoms, because of Theorem 5.1.

The original model ${\left \langle V(A),\in ,\mathcal {V}\right \rangle }$ is realized as the ${\left \langle H_{\kappa }(A),\mathrel {\varepsilon },\mathcal {H}\right \rangle }$ as defined in ${\left \langle W(A),\mathrel {\varepsilon }\right \rangle }$ , and it can see how it is realized in that way via the interpretation to W and then to $W{\mathopen {\lbrack \!\lbrack }A\mathclose {\rbrack \!\rbrack }}$ . So all five theories are bi-interpretable, as claimed.

Let us also describe how to achieve a direct bi-interpretation of ${\left \langle V(A),\in ,\mathcal {V}\right \rangle }$ with ${\left \langle W(A),\mathrel {\varepsilon }\right \rangle }$ and hence also with ${\left \langle {\mkern 3.5mu\overline {\mkern -3.5muV\mkern -.5mu}\mkern .5mu}(A), \mathrel {\mkern 3mu\overline {\mkern -3mu\in \mkern -1.5mu}\mkern 1.5mu},\overline {\mathcal {V}}(A)\right \rangle }$ . We can simply mount an urelement analogue of the unrolling construction, what might be called the urelement unrolling of the model. Specifically, define that an urelement membership code is a well-founded directed graph class relation E with a unique maximal node, which is extensional on all its non-minimal nodes, and where the E-minimal nodes are either elements of A or $\varnothing $ . For any set x in the urelement context, the canonical membership code ${\left \langle \mathop {\mathrm { TC}}(\{{x}\}),\in \right \rangle }$ is an example, since the $\in $ -minimal elements of this set will be atoms or $\varnothing $ . As before, we define an equivalence relation on the codes $E\sim F$ , which holds when they are isomorphic by a map fixing the atoms used as minimal elements. And we define $E\mathrel {\varepsilon } F$ if there is some $y\mathrel {F} x$ , where x is the maximal node of F, such that $E\sim F\downarrow y$ . In the theory KMU+CC, one can verify that these codes work as expected to code higher sets, just as in Theorem 4.1 with the pure membership codes. One shows that the isomorphisms witnessing equivalence are unique, that any two partial initial isomorphisms of codes agree on their common part, and that every two codes admit a maximal partial initial isomorphism. Using the urelement analogues of the arguments made earlier, one can thus verify that the structure ${\left \langle W(A),\mathrel {\varepsilon },\mathcal {W}\right \rangle }$ , where $W(A)$ consists of the equivalence classes $[E]_{\sim }$ of the urelement membership codes, and $\mathcal {W}$ consists of the classes $X\subseteq W(A)$ that are realized by a class of codes in $\mathcal {H}$ . One gets the power set axiom in ${\left \langle W(A),\mathrel {\varepsilon }\right \rangle }$ below $\lambda $ using that the small classes are closed under power class by the abundant atom axiom. To verify collection, one uses both CC as before and also the part of the abundant atom axiom asserting that the small classes are closed under small unions. The cardinal $\kappa $ arising from $\mathord {\mathrm {Ord}}^V$ is inaccessible in $W(A)$ , as is the cardinal $\lambda $ that comes from the class of atoms.

If one considers only the urelement membership codes arising from small classes, the result is the model ${\left \langle {\mkern 3.5mu\overline {\mkern -3.5muV\mkern -.5mu}\mkern .5mu}(A), \mathrel {\mkern 3mu\overline {\mkern -3mu\in \mkern -1.5mu}\mkern 1.5mu},\overline {\mathcal {V}}(A)\right \rangle }$ , and if one considers only the small class membership codes (with no urelements), then one realizes ${\left \langle {\mkern 3.5mu\overline {\mkern -3.5muV\mkern -.5mu}\mkern .5mu}, \mathrel {\mkern 3mu\overline {\mkern -3mu\in \mkern -1.5mu}\mkern 1.5mu},\overline {\mathcal {V}}\right \rangle }$ . In this way, we can directly interpret in a natural way between any two of these five models.

Finally, let us observe that we can take only the pure sets and classes of ${\left \langle V(A),\in ,\mathcal {V}\right \rangle }$ to get the model ${\left \langle V,\in ,\mathcal {V}_{\tiny pure}\right \rangle }$ of $\mathrm {KM}+\mathrm {CC}$ indicated in the lower left of Figure 1, but what we should like specifically to observe is that this model is not bi-interpretable with the other five models mentioned in the proof of Theorem 6.4. This model has no classes of size larger than $\kappa $ and has no way to represent the cardinals of ${\mkern 3.5mu\overline {\mkern -3.5muV\mkern -.5mu}\mkern .5mu}$ and W above $\kappa $ . If we were to perform the unrolling construction of this model, we would get a model of $\mathrm {ZFC}^-$ in which $\kappa $ was the largest cardinal. In fact, it would be $H_{\kappa ^+}^W=H_{\kappa ^+}^{{\mkern 3.5mu\overline {\mkern -3.5muV\mkern -.5mu}\mkern .5mu}}$ , which is considerably smaller than the other models we are considering here.

7 Second-order reflection in pure set theory

Let us now consider the topic of reflection in models ${\left \langle V,\in ,\mathcal {V}\right \rangle }$ of second-order set theory. Of course, we immediately get first-order reflection as with the Lévy–Montague reflection theorem—every first-order statement $\varphi (X)$ that is true in ${\left \langle V,\in ,\mathcal {V}\right \rangle }$ even with a class parameter $X\in \mathcal {V}$ is also true at some rank in the cumulative hierarchy $V_{\lambda }\models \varphi (X\upharpoonright V_{\lambda })$ .

Generalizing this, the second-order reflection principle (evidently first considered in [Reference Bernays1]) is the scheme of assertions that every second order assertion $\varphi (X)$ true in ${\left \langle V,\in ,\mathcal {V}\right \rangle }$ is already true in some rank-initial segment of the universe $V_{\lambda }\models \varphi (X\cap V_{\lambda })$ . It would be equivalent to say that every $\varphi (X)$ true in ${\left \langle V,\in ,\mathcal {V}\right \rangle }$ is true in some transitive set v, that is, the statement $\varphi (X\cap v)$ holds in the structure ${\left \langle v,\in ,P(v)\right \rangle }$ , equipped with all its subsets. The reason is that by simply including GBC as part of what $\varphi $ asserts, we can thereby assume ${\left \langle v,\in ,P(v)\right \rangle }$ is a model of GBC, and since the model has all subsets of v this will be a model of second-order $\mathrm {ZFC}_2$ , but by Zermelo’s quasi-categoricity theorem, this implies that v must be $V_{\kappa }$ for some inaccessible cardinal $\kappa $ . So in fact, the second-order reflection principle is equivalently stated as: every second-order assertion $\varphi (X)$ true in the full universe ${\left \langle V,\in ,\mathcal {V}\right \rangle }$ reflects to some inaccessible cardinal ${\left \langle V_{\kappa },\in ,V_{\kappa +1}\right \rangle }\models \varphi (X\cap V_{\kappa })$ . Note that we can equivalently state reflection as the principle that every formula $\varphi (x,X)$ with a free variable x and class parameter X is absolute to some inaccessible $V_{\kappa }$ , since we can incorporate as a parameter the class of satisfying instances x, and reflecting that class and the fact that it is defined by $\varphi $ amounts to absoluteness of $\varphi (x,X)$ between $V_{\kappa }$ and V.

By pushing on this inaccessibility argument a little, we get a stationary proper class of inaccessible cardinals. Namely, for any class club $C\subseteq \mathord {\mathrm {Ord}}$ , we can reflect the assertion that C is a class club down to some inaccessible $V_{\kappa }$ , and from this it follows that $\kappa $ is a limit point of C and hence in C. In other words, second-order reflection implies that every closed unbounded class of cardinals contains an inaccessible cardinal, and this is what it means to say that $\mathord {\mathrm {Ord}}$ is Mahlo. Pushing still harder achieves even better lower bounds, as follows.

For a quick upper bound on the strength of second-order reflection, let us observe the following (see also [Reference Solovay, Reinhardt and Kanamori23]).

Kanamori [Reference Kanamori14, Exercise 9.18] shows that an $\omega $ -Erdős cardinal suffices for this, and this is interesting because these cardinals are consistent with $V=L$ and consequently the large cardinal upper bound on the strength of second-order reflection is comparatively low in the large cardinal hierarchy, amongst the large cardinals that can exist in L.

Let us also observe a curiosity, namely, that the second-order reflection principle erases the difference between GBC and KM. Indeed, one even gets global choice, as a consequence of AC for sets only, as well as the principle of class choice CC, all for free as a consequence of second-order reflection. Let GBc be the theory $\mathrm {GB}+\mathrm {AC}$ , that is, where we omit global choice and have just AC for sets.

8 Second-order reflective cardinals and supercompactness

We introduce a notion of second-order reflection from higher structures to smaller substructures, which will turn out to characterize the supercompact cardinals.

One can equivalently drop the requirement that m is elementary in M, requiring only that m is a submodel, by Skolemizing the language to make these notions agree. It is also equivalent to consider only finite languages, rather than languages of size less than $\kappa $ , since one can index the language elements by ordinals below some $\gamma <\kappa $ , with $\gamma $ named as a constant, and then $m\cap \kappa \in \kappa $ will ensure that all desired indices are in m. One can accommodate individuals $a\in M$ and class parameters $X\subseteq M$ into $\varphi $ simply by including them in the signature ${\left \langle M,a,X,\dots \right \rangle }$ . And it is equivalent to ask for the absoluteness of a formula $\varphi (x)$ between m and M, rather than merely a sentence, simply by including the class of satisfying instances into the language.

The $\lambda $ -reflective cardinals have a natural affinity with Magidor’s [Reference Magidor16] characterization of supercompactness, namely, that $\kappa $ is supercompact if and only if for every ordinal $\lambda \geq \kappa $ there is $\alpha <\kappa $ and an elementary embedding $j:V_{\alpha }\to V_{\lambda }$ sending its critical point to $\kappa $ , and more to the point, an affinity with Magidor’s proof that the smallest supercompact cardinal is the least cardinal $\kappa $ such that every $\Pi ^1_1$ sentence true in any structure M (in a finite language) is true in a substructure of size less than $\kappa $ . Despite the similarity of this hypothesis with our notion of $\lambda $ -reflectivity above—it lacks only the requirement that $m\cap \kappa \in \kappa $ —our notion exactly does not appear in [Reference Magidor16]. Nevertheless, as we prove in Theorem 8.2 and Corollary 8.3, the concept of $\lambda $ -reflectivity successfully characterizes supercompactness generally, not just for the least supercompact cardinal. Furthermore, Theorem 8.4 provides a level-by-level characterization of the $\lambda $ -reflective cardinals as exactly the nearly $\lambda $ -supercompact cardinals. Our results can therefore be taken as a refinement of Magidor’s characterizations.

The second-order reflectivity hierarchy is therefore simply a different, finer manner of stratifying the supercompactness hierarchy. Indeed, let us mount a somewhat more delicate analysis, which will enable us to provide an exact level-by-level equivalence using the notion of near supercompactness. Specifically, Schanker [Reference Schanker18–Reference Schanker20], in dissertation work undertaken with the first author of this paper, defines that a cardinal $\kappa $ is nearly $\lambda $ -supercompact, if for every $A\subseteq \lambda $ there is a transitive model $M\models \mathrm {ZFC}^-$ , closed under -sequences, with $\lambda ,A\in M$ , and there is an elementary embedding $j:M\to N$ with critical point $\kappa $ for which $\lambda <j(\kappa )$ and . He proves an abundance of equivalent characterizations in [Reference Schanker19, Theorem 2.1.3], one of which is the normal filter property: $\kappa $ is nearly $\lambda $ -supercompact if and only if for every collection M having at most $\lambda $ many subsets of $P_{\kappa }\lambda $ and at most $\lambda $ many functions $f:P_{\kappa }\lambda \to \lambda $ , there is a $\kappa $ -complete fine filter on $P_{\kappa }\lambda $ measuring every subset in M that is also M-normal, meaning that every regressive function f in M is constant on a measure one set.

Every nearly $\lambda $ -supercompact cardinal $\kappa $ is also nearly $\lambda ^{<\kappa }$ -supercompact, by [Reference Schanker19, Lemma 2.1.5], and so we could have alternatively stated this theorem as the claim: a cardinal $\kappa $ is $\lambda ^{<\kappa }$ -reflective for $\Pi ^1_1$ if and only if it is nearly $\lambda $ -supercompact.

Theorem 8.4 thus provides a more refined analysis than Theorem 8.2 in the characterization of the cardinals that are $\lambda $ -reflective for $\Pi ^1_1$ assertions, which are exactly the nearly $\lambda $ -supercompact cardinals. This theorem is part of a constellation of closely related, similar results in the research literature, including [Reference Carr2, Theorems 3.5 and 4.7], [Reference Cody3, Theorem 1.4], and [Reference Hayut and Magidor12, Lemma 2.8].

In the KM context, we say that $\kappa $ is second-order X-reflective, for any class $X\supseteq \kappa $ , if any such assertion $\varphi $ true in a class structure M on domain X is also true in a set structure $m\prec M$ of size less than $\kappa $ with $m\cap \kappa \in \kappa $ . And similarly with X-reflective for $\Pi ^1_1$ assertions, restricting the complexity of $\varphi $ . If $\kappa $ is $\mathord {\mathrm { Ord}}$ -reflective for $\Pi ^1_1$ , then it is $\lambda $ -reflective for $\Pi ^1_1$ for all $\lambda \geq \kappa $ , and so by Theorem 8.2, it follows that $\kappa $ is supercompact. In this sense, $\mathord {\mathrm {Ord}}$ -reflectivity is a strengthening of supercompactness.

9 Second-order reflection with urelements

Let us now analyze the situation of second-order reflection in the urelement context. A model ${\left \langle V(A),\in ,\mathcal {V}\right \rangle }$ of second-order urelement set theory GBCU satisfies the second-order reflection principle if every second-order statement $\varphi (X)$ true in that model of some class parameter $X\in \mathcal {V}$ reflects to some transitive set $v\in V(A)$ , meaning that ${\left \langle v,\in ,P(v)\right \rangle }\models \varphi (X\cap v)$ . By including $\mathrm {GBCU}$ as part of $\varphi $ , we may assume ${\left \langle v,\in ,P(v)\right \rangle }\models \mathrm {GBCU}$ , and from this it follows that v must have the form $H_{\kappa }(w)$ for some inaccessible cardinal $\kappa $ and set of urelements w, because v will be closed under power sets and will have to contain all the size-less-than- $\kappa $ subsets of itself, where $\kappa =v\cap \mathord {\mathrm {Ord}}$ , which will consequently be inaccessible. (Note that when w has size $\kappa $ or larger, this is not the same as $V_{\kappa }(w)$ , since w itself has rank $1$ and would appear as a set, but in $H_{\kappa }(w)$ it would be a proper class.) The second author proved in [Reference Yao25] that KMU with at most $\mathord {\mathrm {Ord}}$ many urelements and second-order reflection is bi-interpretable with KM plus second-order reflection. The main contribution of this article, in contrast, will be to observe the dramatic increase in large-cardinal strength of second-order reflection when there are more than $\mathord {\mathrm {Ord}}$ many urelements.

Let us begin with the observation that from supercompactness, we can produce models of urelement set theory with more than $\mathord {\mathrm {Ord}}$ many urelements, yet with second-order reflection. This observation is one of the core ideas and main results proved by the second author in [Reference Yao25, Reference Yao26], showing that if $\kappa $ is $\kappa ^+$ -supercompact, then there is a model of KMU with strictly more than $\mathord {\mathrm {Ord}}$ many urelements and the second-order reflection principle. Here, we generalize the observation to higher levels of supercompactness.

Next, we improve on the hypothesis by observing that the proof can be undertaken with the weaker assumption that $\kappa $ is merely nearly $\lambda $ -supercompact, instead of $\lambda $ -supercompact. We simply place $H_{\kappa }{\mathopen {\lbrack \!\lbrack } \lambda \mathclose {\rbrack \!\rbrack }}$ into a transitive $M\prec H_{\lambda ^+}$ of size $\lambda $ , closed under -sequences, and then apply Theorem 8.4. This provides:

A consequence of Theorem 9.2 is that to produce a model of $\mathrm {KMU}+\mathrm {CC}$ with strictly more than $\mathord {\mathrm {Ord}}$ many atoms and second-order reflection, it to have a nearly $\kappa ^+$ -supercompact cardinal, which is strictly weaker than $\kappa ^+$ -supercompact, although this hypothesis remains very strong in large cardinal terms—for example, by [Reference Schanker19] it implies $\mathrm {AD}^{L({\mathbb R})}$ .

10 Second-order reflection with abundant atoms implies supercompact

We come now finally to the main result of this article, where we aim to prove conversely that supercompactness is required for second-order reflection in the context of urelement set theory when there are abundant atoms, even just for $\Pi ^1_1$ reflection.