2 Preliminaries on urelement set theory

3 Basic Law V in urelement set theory

A map $X \mapsto \epsilon X$ from classes to first-order objects fulfills Basic Law V if for any classes X and Y, $\epsilon X = \epsilon Y \leftrightarrow \forall x (x \in X \leftrightarrow x \in Y)$ . Hamkins [Reference Hamkins10] shows that in every model of ZF equipped with definable classes, there is a second-order definable map that fulfills Basic Law V. We shall first present Hamkins’ theorem in $\text {ZFU}_{\text {R}}$ . The proof uses Scott’s trick, namely, every class can be represented by the set of its elements with minimal rank, which is available as long as the urelements form a set.

The definability of the map $X \mapsto \epsilon X$ , as Hamkins notes, is a major difference between his result and the earlier consistency proofs of Basic Law V given by Parsons [Reference Parsons15], Bell [Reference Bell1] and Burgess [Reference Burgess5]. However, in urelement set theory, such definable maps do not always exist.

Proof. Suppose for reductio that in U some second-order formula $\varphi (X, \epsilon X, P)$ with a parameter P defines an extension-assignment map such that for every definable classes $X, Y$ of U,

$$ \begin{align*}\epsilon X = \epsilon Y \leftrightarrow \forall x (x \in X \leftrightarrow x \in Y).\end{align*} $$

P is a definable class of U so $P = \{ x \in U : U \models \psi (x, z)\}$ for some first-order formula $\psi $ and $z \in U$ . In U, let $A \subseteq \mathscr {A}$ be an infinite set of urelements disjoint from $ker(z)$ and define $X = \{B \subseteq \mathscr {A} : B - A \text { is finite} \}$ .

Claim 2.1. $A \cap ker(\epsilon X)$ is not empty.

Proof of the Claim

Suppose otherwise. Let $A' \subseteq \mathscr {A}$ be such that $A' \sim \omega $ and $A' \cap (A \cup ker(\epsilon X) \cup ker(z)) = \emptyset $ , which exists by (*). Let $A_1$ be a countably infinite subset of A and $\pi $ be an automorphism that swaps $A_1$ and $A'$ while point-wise fixing everything else (in particular, $\epsilon X$ and P). Since $\varphi (X, \epsilon X, P)$ , $\varphi (\pi X, \pi \epsilon X, \pi P)$ and so $\varphi (\pi X, \epsilon X, P)$ . Thus, $\pi X$ and X must be co-extensional. But $\pi X = \{ B \subseteq \mathscr {A}: B - \pi A \text { is finite}\}$ , and $A_1 \in X$ but $A_1 \notin \pi X$ —contradiction.

Fix some urelements $a \in ker(\epsilon X) \cap A$ and $b \notin ker(\epsilon X)\cup ker(z)$ . Let $\sigma $ be the automorphism that only swaps a and b. Consequently, $\sigma \epsilon X \neq \epsilon X$ . As X and $\sigma X$ are co-extensional, it follows that $\epsilon \sigma X = \epsilon X$ . Moreover, $\varphi (\sigma X, \sigma \epsilon X, P)$ so $\sigma \epsilon X = \epsilon \sigma X = \epsilon X$ —contradiction.

The Axiom of Countable Choice (AC $_\omega $ ), which states that every countable family of non-empty sets admits a choice function, implies that every infinite set has a countably infinite subset.

The equivalence above holds without Collection if we assume a stronger choice principle. The Axiom of Dependent Choice (DC), which is stronger than AC $_\omega $ , states that for every relation on a set without terminal nodes, there is an infinite sequence threading the relation. The Dependence Choice Scheme is a class version of the DC.

• (DC-scheme) If for every $x,$ there is some y such that $\varphi (x, y, u)$ , then for every $p,$ there is an infinite sequence s such that $s(0) = p$ and $\varphi (s(n), s(n+1), u)$ for every $n<\omega $ .

$\text {ZFCU}_{\text {R}}$ + DC-scheme does not prove Collection [Reference Yao22, theorem 2.17].

Note that the argument above also shows that $\text {ZFCU}_{\text {R}}$ does not prove the DC-scheme, because (*) fails in the finite-kernel model.

Plenitude is the axiom that for every well-ordered cardinal $\kappa $ , there is a set of urelements of size $\kappa $ . Next, we show that Plenitude is inconsistent with Basic Law V without using any choice principle. We also note that over $\text {ZFU}_{\text {R}}$ , Plenitude does not imply either Collection or the DC-scheme [Reference Yao24, theorems 17 and 36] although it implies both of them assuming AC [Reference Yao22, theorem 2.5].

While Hamkins’ Theorem 1 suggests that Basic Law V is a natural principle in the context of ZF, the results in this section indicate otherwise: when we quantify over non-sets, Basic Law V entails a specific ontological commitment. For example, Basic Law V implies that there cannot be unboundedly many urelements and that the urelements must form a set when, say, both Collection and AC $_\omega $ hold.

It is important to note that restricting to first-order definable classes does not weaken our argument against Basic Law V. This is because Russell’s Paradox shows that Basic Law V is inconsistent if we consider all classes. Specifically, the class $R = \{x \mid \exists X (x = \epsilon X \land x \notin X)\}$ cannot have an extension. A related issue is that Basic Law V seems to hold if there exists a global well-ordering $<$ , since we can then assign every class $\{x \mid \varphi (x)\}$ to the $<$ -least ordered-pair $\left \langle \ulcorner \psi _n \urcorner , u\right \rangle $ as in the proof of Theorem 1. The status of the Axiom of Choice and its second-order variants in urelement set theory will be discussed later. However, the problem with this argument is that there is no first-order definable global well-ordering if the urelements form a proper class. For, if $\varphi (x ,y ,p)$ defines a global well-ordering $<$ and the urelements do not form a set, then swapping two urelements a and b outside $ker(p)$ such that $a < b$ would yield $b < a$ , contradicting the assumption of a well-ordering.

4 Hume’s Principle in urelement set theory

A map $x \mapsto \# x$ from sets to first-order objects fulfills Hume’s Principle if for any sets x and y, x is equinumerous to y ( $x \sim y$ ) just in case $\# x = \# y$ . We say that Hume’s Principle holds for sets when such a map exists. In this section, we investigate the independence of Hume’s Principle for sets in ZF with urelements. The philosophical implications of this independence will be discussed at the end.

To begin with, $\text {ZFU}_{\text {R}}$ is consistent with Hume’s Principle for sets. When the urelements form a set, we can again use Scott’s trick to map each set x to $\# x = \{y : y \sim x \land \forall z (z \sim x \rightarrow \rho (z) \leq \rho (y))\}$ . And no matter if the urelements form a set, AC implies that Hume’s Principle holds for sets since we can map each set x to the least ordinal to which x is equinumerous.

However, we note that neither AC nor urelements’ forming a set is necessary for Hume’s Principle to hold for sets. A set A of urelements is said to be universal if every set x is equinumerous with some set y in $V(A)$ .

Proof. We simply use Scott’s trick within $V(A)$ . For every set x, let

$$ \begin{align*} \# x = \{y \in V(A) : y \sim x \land \forall z \in V(A) (z \sim x \rightarrow \rho (z) \leq \rho(y))\}. \end{align*} $$

$\# x$ is a well-defined set because A is universal. It is then routine to check that the map $x \mapsto \# x$ fulfills Hume’s Principle.

There are models of $\text {ZFU}_{\text {R}}$ + Collection + $\neg $ AC in which the urelements form a proper class but a universal set of urelements exists (see, e.g., the model W in [Reference Yao24, theorem 36]). Although it is not known whether Hume’s Principle for sets implies the existence of a universal set of urelements, we note that the following weaker implication holds, the proof of which will be useful later.

We now move on to the independence of Hume’s Principle for sets in $\text {ZFU}_{\text {R}}$ . A known result is the following.

As we noted in §2.1, since $\text {ZFU}_{\text {R}}$ is a rather weak theory, the independence of Hume’s Principle will be more robust if the counter-models of Hume’s Principle also satisfy stronger axioms such as RP. In other words, given the theorem by Gauntt and Lévy, it is natural to ask whether Hume’s Principle depends upon the axiomatization of urelement set theory (in the choiceless context, of course). To address this question, we shall construct a model where Hume’s Principle fails for sets but RP holds. Thus, Hume’s Principle for sets is independent of ZF with urelements regardless of one’s axiomatization of the theory.

Our proof that RP holds in the model will again appeal to two aforementioned axioms, Collection and the DC-scheme, due to the following observation made in [Reference Gitman, Hamkins and Johnstone9, p. 397] (though in a different context).

In fact, our result will also clarify the relationship between Hume’s Principle and AC by showing that no fragment of AC suffices for Hume’s Principle. By “fragment of AC”, we mean the DC $_\kappa $ -hierarchy. We shall review some facts about the DC $_\kappa $ -hierarchy that will be useful. To begin with, for every infinite well-ordered cardinal $\kappa $ , DC $_\kappa $ is the following axiom generalizing DC.

• (DC $_\kappa $ ) For every set x and relation $r\subseteq x^{<\kappa } \times x$ , if for every $s \in x^{<\kappa }$ , there is some $w \in x$ such that $\left \langle s, w \right \rangle \in r$ , then there is an $f: \kappa \rightarrow x$ such that $\left \langle f\mathord {\upharpoonright }\alpha , f(\alpha )\right \rangle \in r$ for all $\alpha < \kappa $ .

As in ZF, $\text {ZFU}_{\text {R}}$ proves that AC is equivalent to $\forall \kappa $ DC $_\kappa $ , so each DC $_\kappa $ is a fragment AC. Moreover, for each $\kappa $ , we can formulate the DC $_\kappa $ -scheme as the class version of DC $_\kappa $ .

• (DC $_\kappa $ -scheme) If for every $x,$ there is some y such that $\varphi (x, y, u)$ , then there is some function $f : \kappa \rightarrow U$ such that $\varphi (f\mathord {\upharpoonright } \alpha , f(\alpha ), u)$ for every $\alpha <\kappa $ .

By a standard argument, $\text {ZFCU}_{\text {R}}$ + $\mathscr {A}$ is a set proves that the DC $_\kappa $ -scheme for every $\kappa $ [Reference Yao24, lemma 23]. If $\lambda < \kappa $ , the DC $_\kappa $ -scheme implies the DC $_\lambda $ -scheme; and the DC $_\omega $ -scheme is a reformulation of the DC-scheme. Moreover, the DC $_\kappa $ -scheme is equivalent to the following [Reference Yao24, proposition 13].

• For every definable class X, if for every $s \in X^{<\kappa }$ , there is some $y \in X$ with $\varphi (x, y, u)$ , then there is some function $f : \kappa \rightarrow X$ such that $\varphi (f\mathord {\upharpoonright } \alpha , f(\alpha ), u)$ for every $\alpha < \kappa $ .

Proof. The first step is to build a permutation model that violates AC. We assume some familiarity with permutation models and only provide some necessary details of the construction. For a detailed presentation of this topic, see [Reference Jech12]. Fix some infinite cardinal $\kappa $ . We start with a model U of $\text {ZFCU}_{\text {R}}$ in which $\mathscr {A}$ is a set of size $\kappa ^{+}$ . Enumerate $\mathscr {A}$ with $\kappa ^{+} \times \kappa ^{+}$ , i.e., $\mathscr {A} = \bigcup _{\alpha < \kappa ^{+}} A_\alpha $ , where each row $A_\alpha $ has size $\kappa ^{+}$ . Let $\mathcal {G}$ be the group of permutations of $\mathscr {A}$ such that for every $\pi \in \mathcal {G}$ and $\alpha < \kappa ^{+}$ , $\pi A_\alpha = A_\alpha $ . Let $I = \{E \subseteq \mathscr {A} : |E \cap A_\alpha | < \kappa ^{+} \text { for each } \alpha < \kappa ^{+}\}$ be an ideal of $\mathscr {A}$ . For every x, define $sym(x) = \{\pi \in \mathcal {G}_A : \pi x = x\}$ ; if x is a set, define $fix(x) = \{\pi \in \mathcal {G}_A : \pi y = y \text { for all } y\in x \}$ . We say an object is symmetric if there is some $E \in I$ , called a support of x, such that $fix(E) \subseteq sym(x)$ . The permutation model W is the class of all hereditarily symmetric objects, i.e., $W = \{x \in U : x \text { is symmetric} \land x \subseteq W\}$ , which is a model of $\text {ZFU}_{\text {R}}$ (see [Reference Jech12, theorem 4.1] or [Reference Yao24, theorem 33] for a more general proof).

Note that $\mathscr {A} \in W$ so there is only a set of urelements in W and consequently Hume’s Principle still holds in W. The next step is to produce a model with proper-class many urelements in which Hume’s Principle fails. In W, define a class of sets $\mathscr {I}$ of urelements such that

$$ \begin{align*} \mathscr{I} = \{B \subseteq \mathscr{A} : B \text{ is a subset of } \kappa\text{-many } A_\alpha\}. \end{align*} $$

Let $W^{\mathscr {I}} = \{ x \in W : ker(x) \in \mathscr {I}\}$ . $\mathscr {I}$ is a class ideal of $\mathscr {A}$ which picks out some “small” sets of urelements, while $W^{\mathscr {I}}$ is the class of all objects in W whose kernel is small in the sense of $\mathscr {I}$ . By [Reference Yao24, theorem 26], $W^{\mathscr {I}} \models \text {ZFU}_{\text {R}}$ and there is a proper class of urelements in $W^{\mathscr {I}}$ .

Lemma 11. $W^{\mathscr {I}} \models $ DC $_\kappa $ -scheme.

Proof. First observe that $W^{\mathscr {I}}$ is closed-under $\kappa $ -sequences. Consider any $s : \kappa \rightarrow W^{\mathscr {I}}$ . In U, for each $\alpha < \kappa ,$ choose some $E_\alpha \in I$ to be a support of $s_\alpha $ . Then $\bigcup _{\alpha < \kappa }E_\alpha \in I$ , which is a support of s. And since $ker(s) = \bigcup _{\alpha < \kappa } ker(s_\alpha )$ , it follows that $ker(s)$ is also contained in a $\kappa $ -block of $A_\alpha $ . Thus, $s \in W^{\mathscr {I}}$ . Now suppose that $W^{\mathscr {I}} \models \forall x \exists y \varphi (x, y, u)$ , where $u \in W^{\mathscr {I}}$ . Then, for every $s \in (W^{\mathscr {I}})^{<\kappa }$ , there is some $y \in W^{\mathscr {I}}$ such that $\varphi ^{W^{\mathscr {I}}}(s, y, u)$ . By the DC $_\kappa $ -scheme in U, there is an $f: \kappa \rightarrow W^{\mathscr {I}}$ with $\varphi ^{W^{\mathscr {I}}}(f\mathord {\upharpoonright }\alpha , f_\alpha , u)$ for every $\alpha < \kappa $ . Since $f \in W^{\mathscr {I}}$ , it follows that $W^{\mathscr {I}} \models $ DC $_\kappa $ -scheme.

Next we show that RP holds in $W^{\mathscr {I}}$ . By Theorem 9 and Lemma 11, to show $W^{\mathscr {I}} \models $ RP it suffices to check that $W^{\mathscr {I}} \models $ Collection. A permutation $\sigma $ of $\mathscr {A}$ in U is said to be row-swapping if for every $\alpha < \kappa ^{+}$ , $\sigma A_{\alpha } = A_{\beta }$ for some $\beta < \kappa ^{+}$ (consequently, for every $\alpha $ , $A_{\alpha } = \sigma A_{\beta }$ from some $\beta $ ).

Lemma 12. Every row-swapping $\sigma $ in U is an automorphism of $W^{\mathscr {I}}$ .

Proof. Let $\sigma $ be a row-swapping permutation of $\mathscr {A}$ . It suffices to show that $\mathcal {G}_{\mathscr {A}}, \ I$ , and $\mathscr {I}$ are all fixed by $\sigma $ . If $\pi \in \mathcal {G}_{\mathscr {A}}$ , then for every $A_\alpha $ , since $A_\alpha = \sigma A_\beta $ for some $\beta $ and $\pi A_\beta = A_\beta $ , by automorphism, it follows that $(\sigma \pi ) (\sigma A_\beta ) = \sigma A_\beta $ ; so $(\sigma \pi )A_\alpha = A_\alpha $ and hence $\sigma \pi \in \mathcal {G}_{\mathscr {A}}$ (note here $\sigma \pi $ is not $\sigma \circ \pi $ but $\{\langle \sigma a, \sigma (\pi a) \rangle : a \in \mathscr {A} \rangle \})$ . This shows that $\sigma \mathcal {G}_{\mathscr {A}} = \mathcal {G}_{\mathscr {A}}$ . If $E \in I$ , then for every $A_\alpha $ , since $A_\alpha = \sigma A_\beta $ for some $\beta $ and $E \cap A_\beta $ has size $< \kappa ^{+}$ , $\sigma E \cap A_\alpha $ has size $< \kappa ^{+}$ and hence $\sigma E \in I$ . Therefore, $\sigma I= I$ . Similarly, if B is contained in $\kappa $ -many $A_\alpha $ , then so is $\sigma B $ . Hence, $\sigma \mathscr {I}= \mathscr {I}$ and the lemma is proved.

Lemma 13. $W^{\mathscr {I}} \models $ RP.

Proof. We show that $W^{\mathscr {I}} \models $ Collection. Suppose that $W^{\mathscr {I}}\models \forall x \in w \exists y \varphi (x, y, u)$ for some $w, u \in W^{\mathscr {I}}$ . Let $A=\bigcup _{\alpha < \kappa } A_{\alpha }$ be a $\kappa $ -block containing $ker(w) \cup ker(u)$ and $B = \bigcup _{\alpha < \kappa } B_{\alpha }$ be another $\kappa $ -block that is disjoint from A. It is enough to show that $W^{\mathscr {I}}\models \forall x \in w \exists y \in V(A \cup B) \varphi (x, y, u)$ because then a sufficiently tall $V_\alpha (A \cup B)$ will be a desired collection set.

Consider any $x \in w$ and $y \in W^{\mathscr {I}}$ such that $W^{\mathscr {I}} \models \varphi (x, y, u)$ . Let $C = \bigcup _{\alpha < \kappa } C_{\alpha }$ be another $\kappa $ -block containing $ker(y)-A$ .

Claim 13.1. In $U,$ there is a row-swapping $\sigma $ such that $\sigma (B \cup C) = B$ .

Proof of the Claim

Fix a $\kappa $ -block $D = \bigcup _{\alpha < \kappa } D_{\alpha }$ disjoint from $A \cup B \cup C$ . Split B into two pair-wise disjoint $\kappa $ -blocks $B^1= \bigcup _{\alpha < \kappa } B^1_{\alpha }$ and $B^2= \bigcup _{\alpha < \kappa } B^2_{\alpha }$ , where each $B^1_{\alpha }$ (and $B^2_{\alpha }$ ) is some row $A_\alpha $ . Then split D into two pair-wise disjoint $\kappa $ -blocks $D^1= \bigcup _{\alpha < \kappa } D^1_{\alpha }$ and $D^2= \bigcup _{\alpha < \kappa } D^2_{\alpha }$ in the same way. In $U,$ we can define a row-swapping permutation $\sigma $ of $\mathscr {A}$ as follows. For each $\alpha < \kappa ^{+}$ , let $\sigma B_\alpha = B^1_\alpha $ , $\sigma C_\alpha = B^2_\alpha $ , $\sigma D^1_\alpha = C_\alpha $ and $\sigma D^2_\alpha = D_\alpha $ . It is clear that $\sigma (B \cup C) = B$ .

By Lemma 12, $\sigma $ is an automorphism of $W^{\mathscr {I}}$ fixing x and u, so $W^{\mathscr {I}} \models \varphi (x, \sigma y, u) \land \sigma y \in V(A \cup B)$ , which proves the lemma.

Finally, we turn to the failure of Hume’s Principle in $W^{\mathscr {I}}$ . Suppose for reductio that $\varphi (x, \# x, u)$ defines a cardinality-assignment map with some parameter $u \in W^{\mathscr {I}}$ . It follows that there must be two $A_\alpha $ and $A_\beta $ that are disjoint from $ker(u)$ .

Claim 13.2. $W^{\mathscr {I}} \models A_\alpha \nsim A_\beta $ .

Proof of the Claim

Suppose for reductio that f is an injection from $A_\alpha $ to $A_\beta $ in $W^{\mathscr {I}}$ . Let $E \in I$ be a support of f. Then there are two urelements $a, b \in A_\alpha - E$ . Let $\pi \in fix(E)$ swap only a and b. Then $f(b) = \pi f(a) = f(a)$ , contradicting the assumption that f is an injection.

By Lemma 12, in $U,$ there is an automorphism $\sigma $ of $W^{\mathscr {I}}$ that swaps only $A_\alpha $ and $A_\beta $ . Since $\sigma $ point-wise fixes $ker(u)$ and $ker(\# A_\alpha ) \subseteq ker(u)$ by the proof of Proposition 7, $\sigma $ point-wise fixes $ker(\# A_\alpha )$ . It follows that $W^{\mathscr {I}} \models \varphi (A_\beta , \# A_\alpha , u)$ so $\# A_\beta = \# A_\alpha $ . Hence, $W^{\mathscr {I}} \models A_\alpha \sim A_\beta $ , contradicting Claim 13.2. This completes the proof of the theorem.

We now consider the philosophical implications of Theorem 10 for Hume’s Principle. Since we are working in ZF with urelements, we shall assume a predicativist view of classes—that is, all classes are first-order definable. Hume’s Principle in impredicative class theories will be discussed in §5. As noted in the introduction, one desideratum for Hume’s Principle to be regarded as a fundamental principle of cardinality is its universality, meaning that the opening quantifiers in Hume’s Principle must range over concepts of any kind. Given this requirement, models of urelement set theory with definable classes are precisely the context in which Hume’s Principle should be tested. Consequently, the model $W^{\mathscr {I}}$ in Theorem 10 provides evidence against the universality of Hume’s Principle, assuming it is regarded as a natural model of set theory with urelements.

One possible objection to the naturalness of $W^{\mathscr {I}}$ comes from proponents of the Axiom of Choice, who may argue that $W^{\mathscr {I}}$ is unnatural because mathematics without AC can exhibit various pathologies. While it is well known that AC itself leads to certain paradoxical consequences, let us grant, for the sake of argument, that AC is an overall desirable axiom in pure set theory. In response, we note that AC does hold in the class of pure sets (V) of $W^{\mathscr {I}}$ , as no pure sets were eliminated in the process of constructing the model. Therefore, since all mathematical objects can be reconstructed within the universe of pure sets, no mathematical pathologies that would typically arise in the absence of AC occur in $W^{\mathscr {I}}$ . Moreover, as we observed earlier, AC asserts more than what is necessary for Hume’s Principle to hold: the existence of a universal set of urelements suffices. Thus, there seems to be no compelling reason to consider the claim “every set is well-orderable” a natural axiom in urelement set theory.

5 Hume’s Principle in urelement class theory

A map $X \mapsto \# X$ from classes to first-order objects fulfills Hume’s Principle if for any classes X and Y, $X \sim Y$ just in case $\# X = \# Y$ . When a map as such exists, we say that Hume’s Principle holds for classes. Unlike Basic Law V, Hume’s Principle for classes is consistent with impredicative class theories, such as the Kelley–Morse class theory (KM). In particular, Hume’s Principle will hold for classes if von Neumann’s Limitation of Size holds, which states that all proper classes are equinumerous. This is because Limitation of Size implies the Axiom of Global Well-Ordering, which asserts the existence of a well-ordering of the universe. So with Limitation of Size, we can define, without using parameters, the following map $X \mapsto \# X$ that fulfills Hume’s Principle.

And unlike Hume’s Principle for sets, without assuming AC, Hume’s Principle can fail for classes even when there are no urelements (see [Reference Roberts and Shapiro17, theorem 2]).

Let Kelley-Morse Class Theory with Urelements with Global Well-Ordering (KMCU)Footnote ³ be KMU plus Global Well-Ordering. In KMCU, let W be a global well-ordering on U. Then we can use W as a class parameter to define a map $X \mapsto \# X$ as followsFootnote ⁴ :

where $W_y = \{x \mid x W y\}$ is the initial segment of W below y. This is a well-defined map due to the fact that for every class X, the well-order $W\mathord {\upharpoonright } X$ is either isomorphic to W or isomorphic to $W_x$ for some x.Footnote ⁵ Thus, if $X \not \sim U$ , it must be isomorphic and hence equinumerous to some $W_x$ . It is then routine to check that $X \mapsto \# X$ so defined fulfills Hume’s Principle.

A natural question is whether such a map $X \mapsto \# X$ is definable without parameters in KMCU. A standard argument shows that Global Well-Ordering implies Limitation of Size with the assumption that the urelements form a set [Reference Yao24, proposition 98], so the answer to the question is positive if the urelements are “few” compared to sets. It is shown in [Reference Howard, Rubin and Rubin11] that KMCU by itself does not prove Limitation of Size. The next theorem shows that Global Well-Ordering is not sufficient for a parameter-free definable map that fulfills Hume’s Principle when the urelements are “many” compared to the pure sets.

Proof. Let V be a model of ZFC $+$ an inaccessible cardinal $\kappa $ . We first construct a model U of $\text {ZFCU}_{\text {R}}$ + Plenitude by treating copies of ordinals as urelements. In particular, we define in V by recursion as follows:

• 

For every ,

• $\bar {x} \ \bar {\in } \ \bar {y} \text { if and only if } \exists y (\bar {y} = \left \langle 1, y\right \rangle \land \bar {x} \in y);$

• $\bar {\mathscr {A}}(\bar {x}) \text { if and only if } \bar {x} \in \{0\} \times Ord.$

Let U denote the model for the language of urelement set theory. By [Reference Yao24, theorems 7 and 9], $U \models \text {ZFCU}_{\text {R}}$ + Plenitude. Moreover, the class of pure sets in U is isomorphic to V [Reference Yao24, lemma 8] so U also contains (a copy of) $\kappa $ as an inaccessible cardinal.

In U, let $\lambda = \aleph _{\kappa ^{+}}$ and A be a set of urelements of size $\lambda $ . Define

$$ \begin{align*} U_\kappa(A) = \bigcup_{B \in P_\kappa (A)} V_\kappa (B), \end{align*} $$

where $P_\kappa (A)$ is the set of subsets of A with size $< \kappa $ . By [Reference Yao23, lemma 4.2], $U_\kappa (A)$ , equipped with all of its subsets, is a model of KMCU. Now suppose for reductio that $\varphi (X, \# X)$ defines a cardinality-assignment function in $U_\kappa (A)$ .

Claim 14.1. For every class X of $U_\kappa (A)$ , $\# X$ is a pure set.

Proof of the Claim

Suppose $\# X$ is not pure for some class X of $U_\kappa (A)$ . Then, in $U_\kappa (A),$ there will be an automorphism $\pi $ swapping some $a \in ker(\#X)$ with some $b \notin ker(\#X)$ , and so $\pi \# X \neq \# X$ . Since $\pi X \sim X$ , it follows that $\# X = \# \pi X = \pi \# X$ , yielding a contradiction.

Since there are at least $\kappa ^{+}$ -many cardinalities for the proper classes of $U_\kappa (A)$ , $\varphi $ thus defines an injection from $\kappa ^{+}$ into $V^{U_\kappa (A)}$ , namely, $V_\kappa $ . This is impossible as $V_\kappa $ has size  $\kappa $ .

In fact, this argument shows that in $U_\kappa (A)$ there is no definable cardinality-assignment map which only uses first-order objects in $U_\kappa (A)$ as parameters. For, if $\varphi (X, \# X, u)$ defines a cardinality-assignment map, where $u \in U_\kappa (A)$ , then the same argument would show that $ker(\# X) \subseteq \ker (u)$ for every class X. This means that we could inject $\kappa ^{+}$ -many cardinalities into $V_\kappa (ker(u))$ , but $V_\kappa (ker(u))$ only has size $\kappa $ as $|ker(u)| < \kappa $ —contradiction.