Axiom of Plenitude in ZFU

• Axiom of Plenitude is a width-maximality principle in ZFU asserting every ordinal is equinumerous with a set of urelements.
• Plenitude⁺ strengthens this by requiring every set, not just ordinals, has a matching urelement set, linking Collection and Reflection under definability conditions.
• The principle reveals subtle interplay between urelements, weak choice assumptions, and maximal set-theoretic behaviors within non-standard frameworks.

The Axiom of Plenitude, in the technical sense developed for ZF set theory with urelements, is the assertion that every ordinal is equinumerous with a set of urelements; its strengthened form, Plenitude$^+$, extends this requirement from ordinals to all sets. In this setting, the axiom functions as a width-maximality principle for the class of urelements $A$, calibrating how rich the universe is in non-set objects and how much of the familiar structural behavior of ZF or ZFC can be recovered without full choice. The central results show a sharp asymmetry: assuming definability of cardinality, Plenitude$^+$ unifies the Collection Principle and the Reflection Principle, whereas Plenitude is substantially weaker and does not, even together with Small Violations of Choice (SVC), recover the same consequences (Yao, 28 Aug 2025).

1. ZFU as the ambient framework

In $ZFU$, Zermelo-Fraenkel set theory is extended by allowing a proper class of urelements. Urelements are objects that are not sets but can be elements of sets, and $A$ denotes the class of all urelements. This universe has a richer automorphism structure due to the permutability of urelements. Within this framework, standard maximality axioms such as the Collection Principle and the Reflection Principle are not provable, and their relation to the maximality of the class of urelements becomes a central structural problem (Yao, 28 Aug 2025).

The specific Plenitude axioms belong to a broader family of set-theoretic maximality or plenitude principles. In a wider foundational usage, plenitude principles assert the existence of sufficiently large, rich, or unconstrained sets or universes, often in support of generation schemes and closure phenomena. A distinct example is Broad Infinity, introduced as a new set-theoretic axiom scheme based on the slogan “Every time we construct a new element, we gain a new arity,” and studied as a principle going beyond ZFC under weak choice assumptions (Levy, 2021). This broader context is useful because it shows that “plenitude” can refer either to width-maximality for urelements, as in $ZFU$, or to stronger generative axioms about the existence of large set-like closures.

2. Formal statements of Plenitude and Plenitude$^+$

The Axiom of Plenitude is formulated as follows:

$$(\mathrm{Plenitude})\qquad \forall \alpha \in \mathrm{Ord}\; \exists A_\alpha \subseteq A\, (A_\alpha \sim \alpha).$$

It states that for every ordinal $\alpha$, there is a set of urelements equinumerous with $\alpha$. The strengthened principle is:

$A$0

This asserts that for every set $A$1, there is a set of urelements equinumerous with $A$2 (Yao, 28 Aug 2025).

The distinction is exact. Plenitude requires matching cardinalities only for ordinals; Plenitude$A$3 requires matching cardinalities for arbitrary sets. In the terminology of the paper, Plenitude is a width-maximality principle for urelements, but only ordinal-indexed, whereas Plenitude$A$4 requires a richer duplication capacity across the entire universe. With AC, Plenitude is strong enough to imply Collection and Reflection, but without AC it is significantly weaker (Yao, 28 Aug 2025).

A useful comparison comes from another maximality principle studied in a different formal environment. Dzhafarov and Mummert analyze the principle that every set has a maximal subset satisfying a property of finite character, and note that in the context of set theory this variant of Tukey’s lemma is equivalent to the axiom of choice (Dzhafarov et al., 2011). This comparison does not identify Plenitude with Tukey-style maximality, but it situates Plenitude among principles whose strength depends strongly on the surrounding choice theory.

3. Cardinality, representability, and weak choice

A recurring notion is that cardinality is definable: there is a formula assigning to every set $A$5 a cardinal $A$6, or parametrically definable if the formula uses extra parameters. In $A$7, the definability of cardinality acts as a weak form of the Axiom of Choice and can support arguments that, in $A$8, would ordinarily use well-orderings. With enough structure, such as well-orderable urelements or SVC, cardinality can be definable (Yao, 28 Aug 2025).

The paper also distinguishes representability of cardinality from mere definability. Parametric representability is stronger: the assignment actually picks a representative, not just an equivalence class. This stronger hypothesis is enough, together with Plenitude$A$9, to imply Reflection. SVC likewise functions as a mild choice principle; in the formulation used here, it is the principle that the Axiom of Choice holds in some forcing extension (Yao, 28 Aug 2025).

Several technical statements clarify why these hypotheses matter. If cardinality is parametrically definable, there is always a set of urelements $^+$0 so large that every cardinal is in $^+$1. If cardinality is parametrically representable, or if SVC holds, then universal sets of urelements exist. These facts make it possible to construct large subuniverses $^+$2 that simulate ZF-like behavior sufficiently well for reflection arguments (Yao, 28 Aug 2025).

A common misconception is that SVC itself should settle all relevant cardinality questions in $^+$3. The paper explicitly notes that SVC does not guarantee definability of cardinality without parameters. This is one of the reasons why Plenitude$^+$4 is paired, in different theorems, with either parametrically definable cardinality, parametrically representable cardinality, or SVC, rather than with a single undifferentiated “weak choice” assumption (Yao, 28 Aug 2025).

4. Collection, Reflection, and the unifying role of Plenitude$^+$5

The Collection Principle over $^+$6 is given by

$^+$7

The Lévy-Montague Reflection Principle says that for every set $^+$8 and formula $^+$9, there is a transitive set $ZFU$0 that reflects $ZFU$1 down to $ZFU$2. In $ZFU$3 and $ZFU$4, Collection and Reflection are equivalent. In $ZFU$5 without choice, their relationship is subtle and they are generally not provably equivalent (Yao, 28 Aug 2025).

The main structural results may be summarized as follows.

Assumptions	Consequence	Remark
Plenitude$ZFU$6 + cardinality parametrically definable	Collection $ZFU$7 Reflection	Theorem 1
Plenitude$ZFU$8 + cardinality parametrically representable	Reflection	Theorem 2
Plenitude$ZFU$9 + SVC	Reflection	Theorem 3
Plenitude + SVC	does not prove Collection	independence result
Plenitude + SVC + Reflection	does not prove Plenitude$A$0	independence result

These results isolate the exact role of the strengthened axiom. Plenitude$A$1 serves as a bridge between size-theoretic control over arbitrary sets and the existence of reflecting transitive sets. Under parametrically definable cardinality, it restores the equivalence between Collection and Reflection that holds in choice-based settings. Under parametrically representable cardinality or SVC, it still yields Reflection even without establishing the full equivalence (Yao, 28 Aug 2025).

The significance is not merely that Plenitude$A$2 implies familiar principles. More precisely, it unifies two principles that are otherwise conjectured to be non-equivalent in $A$3. This makes Plenitude$A$4 a “choiceless unifier” for structural principles that split in the presence of urelements and the absence of full choice. By contrast, Plenitude alone does not provide enough duplication or width to force the same outcome.

5. Duplication, universal urelements, and separation results

A key lemma is the Duplication Principle, $A$5. Plenitude$A$6 implies that for any set of urelements $A$7, and any set $A$8 disjoint from $A$9, there is always a duplicate of $ZFU$0, also disjoint from $ZFU$1. This is used to build large, structurally homogeneous configurations of urelements and underpins the automorphism arguments needed in reflection proofs (Yao, 28 Aug 2025).

Another important device is the existence of universal sets of urelements. Under SVC or cardinality representability, such sets exist, enabling the universe to simulate ZF-like behavior within $ZFU$2. Combined with parametrically definable cardinality, this gives a way to control the ranks and sizes of sets when constructing reflecting sets from collection hypotheses. The availability of large urelement domains is thus not merely combinatorial; it is a mechanism for transferring structural properties into internal subuniverses (Yao, 28 Aug 2025).

The weakness of Plenitude is established by model construction. The paper constructs permutation models and small-kernel models, modeled on Lévy and Blass, in which the set of urelements is large enough for Plenitude but not rich enough for Plenitude$ZFU$3. In these models, SVC can ensure every set is “almost” surjected onto by a fixed set-up to size, but this is not enough for full Collection or for the maximal duplication required in Plenitude$ZFU$4 (Yao, 28 Aug 2025).

These independence results block two natural but incorrect inferences. First, Plenitude together with SVC does not imply Collection. Second, even Plenitude together with SVC and Reflection does not imply Plenitude$ZFU$5. The strengthened axiom is therefore not a routine reformulation of the weaker one; it marks a genuine increase in width-maximality.

6. Broader plenitude principles and foundational comparisons

The term “plenitude” is used more broadly in the foundations literature for axioms asserting the existence of rich generative or closure structures. Broad Infinity is one such example. It is introduced as an axiom scheme saying that three-dimensional trees whose growth is controlled by a specified class function form a set, and under AC or WISC it is equivalent to Mahlo’s principle. Under the same assumptions, it yields Broad Set Generation and thereby the existence of Grothendieck universes; without choice, it yields Broad Derivation Set and the existence of Tarski-style universes (Levy, 2021).

This comparison shows that the specific Axiom of Plenitude in $ZFU$6 is one instance of a wider methodological pattern: maximality principles can be formulated either as width conditions on existing objects, such as urelements, or as generation principles for new sets and universes. A plausible implication is that these principles should not be compared solely by syntactic strength; their operative content depends on whether they regulate cardinal duplication, closure under constructions, or the existence of maximal extensions.

A related comparison arises from reverse mathematics. Dzhafarov and Mummert show that, in set theory, the principle that every set has a maximal subset satisfying a property of finite character is equivalent to the axiom of choice, while in second-order arithmetic its strength depends on the quantifier complexity of the defining formula and can be substantially amplified by finitary or nondeterministic closure operators (Dzhafarov et al., 2011). This suggests that maximality principles are highly framework-sensitive: principles that coincide under full choice can diverge sharply in weaker settings.

Within that broader landscape, the $ZFU$7 Axiom of Plenitude is best understood not as a generic maximization slogan but as a sharply formulated width principle about urelements. Its strengthened form, Plenitude$ZFU$8, has precise consequences for Collection and Reflection under mild choice-like or cardinality-definability assumptions. Its weaker form does not. That distinction is the principal conceptual contribution of the recent analysis (Yao, 28 Aug 2025).