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Regular cardinal

In set theory, a regular cardinal is an infinite cardinal number κ that equals its own cofinality, meaning cf(κ) = κ.^[1] Equivalently, κ is regular if no set of cardinality κ can be expressed as the union of fewer than κ many sets, each of cardinality strictly less than κ.^[2] The smallest regular cardinal is ℵ₀, the cardinality of the natural numbers, which is regular because any countable union of finite sets is countable.^[3] All successor cardinals are regular; for any infinite cardinal λ, the successor cardinal λ⁺ has cofinality λ⁺.^[1] In contrast, some limit cardinals are singular, such as ℵ_ω, the least upper bound of the sequence ℵ_n for n < ω, which has cofinality ω.^[4] Regular limit cardinals that are also strong limit cardinals—meaning that for every μ < κ, 2^μ < κ—are known as inaccessible cardinals, and their existence cannot be proved in ZFC set theory.^[2] Regular cardinals are fundamental in advanced set-theoretic constructions, including the definition of inaccessible cardinals, measurable cardinals, and supercompact cardinals, as well as in forcing techniques where they ensure closure properties.^[3] Beyond pure set theory, they underpin concepts in category theory, such as the accessibility of categories and the existence of filtered colimits in the category of sets bounded below a regular cardinal κ.^[4]

Definition

Cofinality definition

In set theory, the cofinality of an ordinal \kappa, denoted \mathrm{cf}(\kappa), is the smallest ordinal \alpha such that there exists an order-preserving map f: \alpha \to \kappa whose image is cofinal in \kappa, meaning that for every \beta < \kappa, there is some \gamma < \alpha with \beta \leq f(\gamma).^[5] Equivalently, \mathrm{cf}(\kappa) is the order type of the smallest cofinal subset of \kappa, where a subset S \subseteq \kappa is cofinal if every initial segment of \kappa intersects S.^[5] For an infinite cardinal \kappa, this notion extends to a measure of how \kappa can be "approached" by smaller structures: \mathrm{cf}(\kappa) is the smallest ordinal \alpha such that \kappa is the union of \alpha many sets, each of cardinality strictly less than \kappa.^[1] A cardinal \kappa is defined to be regular if \mathrm{cf}(\kappa) = \kappa.^[1] This condition captures the idea that \kappa is indivisible in the sense that it cannot be expressed as a union of fewer than \kappa many proper subcardinals; any decomposition into smaller pieces requires at least \kappa many components.^[1] To illustrate, consider the smallest infinite ordinal \omega, which has cofinality \omega because every cofinal subset of \omega must be unbounded and thus order-isomorphic to \omega itself, confirming its regularity.^[6] In contrast, the cardinal \omega_\omega = \sup_{n < \omega} \omega_n has cofinality \omega, as it arises as the union of the countable sequence \{\omega_n \mid n < \omega\} of strictly increasing smaller cardinals, making it singular.^[6] The concept of cofinality was introduced by Felix Hausdorff in 1906, initially for linearly ordered sets in the context of ordinal arithmetic and order types.^[7]

Set-theoretic definition

In set theory, an infinite cardinal \kappa is defined to be regular if it cannot be expressed as the union of fewer than \kappa many sets, each of cardinality less than \kappa. That is, for any family \{X_\alpha \mid \alpha < \lambda\} where \lambda < \kappa and |X_\alpha| < \kappa for each \alpha < \lambda, the cardinality of \bigcup_{\alpha < \lambda} X_\alpha is less than \kappa.^[8]^[9] Equivalently, \kappa cannot be written as a cardinal sum \sum_{i < \lambda} \mu_i with \lambda < \kappa and \mu_i < \kappa for each i < \lambda, where the sum denotes the cardinality of a disjoint union of sets of those sizes.^[8]^[9] This characterization captures the operational sense in which \kappa is "indecomposable" under small unions, reflecting its role as a foundational measure of size in the cumulative hierarchy. This union-based definition is equivalent to the cofinality condition \mathrm{cf}(\kappa) = \kappa, where \mathrm{cf}(\kappa) is the least ordinal \lambda such that there exists a cofinal function from \lambda into \kappa.^[8]^[9] To see one direction, suppose \mathrm{cf}(\kappa) = \lambda < \kappa; let (\alpha_\xi \mid \xi < \lambda) be a strictly increasing cofinal sequence of ordinals in \kappa with \sup_{\xi < \lambda} \alpha_\xi = \kappa. Then \kappa = \bigcup_{\xi < \lambda} \alpha_\xi, and each initial segment |\alpha_\xi| = \alpha_\xi < \kappa since \alpha_\xi < \kappa.^[8]^[9] For the converse, assume \kappa = \bigcup_{i < \lambda} A_i with \lambda < \kappa and |A_i| < \kappa for each i; without loss of generality (by a bijection between \kappa and the union), take the A_i \subseteq \kappa. For each i, if \sup A_i = \kappa, then A_i is unbounded in \kappa, so \mathrm{cf}(\kappa) \leq |A_i| < \kappa because any unbounded subset of \kappa of cardinality \mu < \kappa admits an increasing enumeration of length at most \mu whose supremum is \kappa. If instead \sup A_i < \kappa for all i, then the set \{\sup A_i \mid i < \lambda\} is cofinal in \kappa (since every \alpha < \kappa belongs to some A_i, hence \sup A_i \geq \alpha), and has cardinality at most \lambda < \kappa, so \mathrm{cf}(\kappa) \leq \lambda < \kappa.^[8]^[9] The Hartogs number of a set X, defined as the least ordinal not injectively embeddable into X, plays a role in formalizing such enumerations and ensuring that well-orderings of subsets of cardinality less than \kappa have order types below \kappa, thereby bounding the cofinal sequences in the proof.^[8]^[9] Regularity, via \mathrm{cf}(\kappa) = \kappa, represents the weakest nontrivial cofinality property among infinite cardinals, distinguishing it from stronger large cardinal notions like weak compactness, which require additional closure or embedding properties beyond mere regularity.^[8]^[9]

Equivalent characterizations

In terms of ordinal functions

A cardinal \kappa is regular if and only if every function f: \lambda \to \kappa for \lambda < \kappa has bounded range, meaning \sup \mathrm{ran}(f) < \kappa. This condition captures the ordinal-theoretic notion of regularity combinatorially, as it precludes the existence of any cofinal map from a smaller ordinal into \kappa, ensuring that \kappa cannot be approached cofinally by fewer than \kappa many steps. Equivalently, every subset of \kappa with cardinality less than \kappa is bounded below \kappa.^[10]^[11] The aleph function, defined by f(\alpha) = \aleph_\alpha, provides a canonical example of an ordinal enumeration function in set theory. This function is normal, meaning it is strictly increasing (f(\alpha) < f(\beta) for \alpha < \beta) and continuous at limit ordinals (f(\delta) = \sup_{\alpha < \delta} f(\alpha) for limit \delta). For limit ordinals \alpha, the cofinality satisfies \mathrm{cf}(\aleph_\alpha) = \mathrm{cf}(\alpha), reflecting the continuity of the enumeration. Thus, \aleph_\alpha is regular if and only if \mathrm{cf}(\alpha) = \aleph_\alpha, which occurs precisely when \alpha is a successor ordinal or a limit ordinal that is itself a fixed point of the aleph function with cofinality equal to its own value. The fixed-point property of normal functions like the aleph function ensures the existence of such points, but regularity imposes the additional condition that the index \alpha aligns the cofinality with the cardinal itself.^[11]^[10] Sierpiński's theorem provides another functional characterization: a cardinal \kappa is regular if and only if there does not exist a regressive function f: \kappa \to \kappa (i.e., f(\alpha) < \alpha for all limit \alpha < \kappa) that is constant on a stationary subset of \kappa. This equivalence highlights the combinatorial interplay between ordinal functions and stationary sets, where regressivity forces "pressing down" behavior incompatible with singularity. For singular \kappa, such a constant-on-stationary regressive function can exist, reflecting the lower cofinality.^[11]

In terms of cardinal arithmetic

For an infinite regular cardinal κ, the cardinal addition satisfies κ + λ = max(κ, λ) for any cardinal λ < κ.^[1] More generally, the sum of fewer than κ many cardinals, each of cardinality less than κ, has cardinality less than κ. This property serves as an equivalent characterization of regularity for infinite cardinals.^[1]^[2] Similarly, cardinal multiplication for an infinite regular cardinal κ satisfies κ · λ = max(κ, λ) for any cardinal λ < κ.^[1] The product of fewer than κ many cardinals, each less than κ, also has cardinality less than κ, paralleling the addition case.^[12] For exponentiation, König's theorem states that for any infinite cardinal κ, κ^{cf(κ)} > κ.^[1] For regular κ, where cf(κ) = κ, this specializes to κ^κ > κ, aligning with Cantor's theorem that the power set cardinality exceeds κ. The condition that κ^{cf(κ)} = κ cannot hold by König's theorem, as the exponentiation always exceeds κ; thus, regularity (cf(κ) = κ) is reinforced as the case where the cofinality matches the cardinal in this arithmetic context.^[1] Singular cardinals violate these arithmetic properties. For example, the singular cardinal ℵ_ω has cf(ℵ_ω) = ω < ℵ_ω, and ℵ_ω = ∑_{n < ω} ℵ_n, where each ℵ_n < ℵ_ω and there are ω < ℵ_ω terms in the sum.^[2]

Examples

Aleph fixed points

An aleph fixed point is a cardinal \kappa satisfying \kappa = \aleph_\kappa, meaning \kappa is the \kappa-th infinite cardinal.^[13] The aleph function \alpha \mapsto \aleph_\alpha is normal and continuous, so by standard results on normal functions, it has fixed points, which are necessarily cardinals.^[14] ZFC proves the existence of such fixed points via the axiom of replacement: starting from \kappa_0 = 0 and iterating \kappa_{n+1} = \aleph_{\kappa_n} for n < \omega, the supremum \kappa = \sup_{n < \omega} \kappa_n satisfies \kappa = \aleph_\kappa and has cofinality \omega, hence is singular.^[13] This construction yields arbitrarily large singular aleph fixed points.^[14] Most aleph fixed points are singular, typically with cofinality \omega.^[13] For instance, the least aleph fixed point greater than the continuum $2^{\aleph_0} is obtained by iterating the aleph function \omega many times starting above the continuum and thus has cofinality \omega, making it singular.^[13] Regular aleph fixed points are the uncountable regular limit cardinals. Those that are also strong limit cardinals are known as weakly inaccessible cardinals.^[15] Such cardinals are fixed points because their regularity and limit nature imply they equal \aleph_\kappa.^[16] However, the existence of regular aleph fixed points cannot be proved in ZFC and is equiconsistent with the existence of inaccessible cardinals, which are the first nontrivial large cardinals beyond those provable in ZFC.^[17]

Inaccessible cardinals

A strongly inaccessible cardinal, or simply inaccessible cardinal, is defined as an uncountable regular cardinal \kappa that is also a strong limit cardinal. This means that for every cardinal \mu < \kappa, the power set cardinality $2^\mu < \kappa.^[18] The regularity condition ensures that \kappa cannot be expressed as the supremum of fewer than \kappa many smaller cardinals, while the strong limit property prevents \kappa from being reached via exponentiation from below. This combination makes inaccessible cardinals the primary examples of large regular cardinals beyond the smaller infinite cardinals like \aleph_0 or \aleph_1. The least inaccessible cardinal \kappa, if it exists, exhibits significant model-theoretic properties. In particular, the cumulative hierarchy up to \kappa, denoted V_\kappa, is isomorphic to the class H_\kappa of all sets with transitive closure of cardinality less than \kappa. This equivalence holds because the strong limit condition bounds the sizes of power sets within V_\kappa, and regularity ensures the overall cardinality of V_\kappa is exactly \kappa. Moreover, V_\kappa forms a Grothendieck universe, a model closed under standard set operations sufficient for developing much of classical mathematics internally, including category theory and algebraic geometry.^[19] The existence of inaccessible cardinals has notable consistency strength relative to ZFC set theory. Their presence is independent of ZFC: ZFC neither proves nor refutes the existence of such cardinals, as models without inaccessibles can be constructed via forcing, while inner models under stronger assumptions yield them. Dana Scott first established in 1961 that the consistency of ZFC plus the existence of an inaccessible cardinal follows from the consistency of ZFC plus a measurable cardinal, marking a key step in understanding large cardinal hierarchies. In the broader hierarchy of large cardinals, inaccessible cardinals serve as a foundational level, with higher notions like Mahlo cardinals building upon them as inaccessible limits of sequences of inaccessibles. A Mahlo cardinal is an inaccessible \kappa such that the set of inaccessible cardinals below \kappa is stationary in \kappa, introduced by Paul Mahlo in his early work on transfinite numbers. However, the core significance of inaccessibility lies in its blend of regularity and limit properties, enabling robust models like V_\kappa \models ZFC.

Properties

Closure properties

Regular cardinals exhibit notable closure properties under various set-theoretic operations, which distinguish them from singular cardinals and underpin their role in infinitary combinatorics. A fundamental such property is closure under unions: if \kappa is a regular cardinal and \{A_\alpha \mid \alpha < \lambda\} is a family of sets with \lambda < \kappa and |A_\alpha| < \kappa for each \alpha < \lambda, then \left| \bigcup_{\alpha < \lambda} A_\alpha \right| < \kappa.^[1] This follows directly from the definition of regularity, as the cofinality of \kappa prevents any such union from reaching size \kappa.^[1] Successor cardinals provide a concrete class of regular cardinals with inherent closure characteristics. Every successor cardinal \kappa^+, such as \aleph_{\alpha+1} for any ordinal \alpha, is regular, meaning \mathrm{cf}(\kappa^+) = \kappa^+.^[20] This regularity ensures that operations like forming power sets or taking successors preserve the structure below \kappa^+ without introducing singularities at that level.^[20] Diagonal intersections further illustrate closure for structures on regular cardinals. For a regular cardinal \kappa > \omega, if \{C_\xi \mid \xi < \lambda\} is a family of club subsets of \kappa with \lambda < \kappa, the diagonal intersection \Delta_{\xi < \lambda} C_\xi = \{\beta < \kappa \mid \forall \xi < \beta \, (\beta \in C_\xi)\} is club in \kappa.^[21]^[22] The collection of stationary subsets of \kappa is similarly closed under such <\kappa-sized diagonal intersections, preserving stationarity.^[21]^[22] Reflection properties also arise naturally on regular cardinals, enabling the propagation of combinatorial structures to smaller ordinals. For a stationary set S \subseteq \kappa where \kappa > \omega is regular, Fodor's lemma (the pressing-down lemma) asserts that any regressive function f: S \to \kappa (with f(\alpha) < \alpha for \alpha \in S) is constant on some stationary subset of S.^[22] This facilitates reflection: stationary sets on \kappa can reflect to initial segments \alpha < \kappa with \mathrm{cf}(\alpha) > \omega, where S \cap \alpha is stationary in \alpha, under appropriate conditions tied to the regularity of \kappa.^[22]

Relation to singular cardinals

A singular cardinal \kappa is an infinite cardinal such that its cofinality \cf(\kappa) < \kappa, meaning \kappa can be expressed as the supremum of a sequence of length \cf(\kappa) consisting of fewer than \kappa many ordinals each strictly smaller than \kappa.^[23] This contrasts with regular cardinals, where \cf(\kappa) = \kappa, preventing such a decomposition into fewer than \kappa parts. For example, \aleph_\omega is the least singular cardinal, with \cf(\aleph_\omega) = \omega, as it is the union of the countable sequence \{\aleph_n : n < \omega\}.^[23] Every infinite cardinal is either regular or singular, as \cf(\kappa) \leq \kappa holds universally for infinite cardinals \kappa, with equality defining regularity and strict inequality defining singularity.^[23] Under the axiom of constructibility V=L, the least singular cardinal remains \aleph_\omega, consistent with the generalized continuum hypothesis implied by V=L.^[23] Singular cardinals permit decompositions into fewer than \kappa many smaller sets, which has implications for their role in set-theoretic embeddings and hierarchies, rendering their cardinalities relatively weaker in constraining certain ultrapower constructions compared to regular cardinals. A key arithmetic distinction arises from König's theorem, which states that for any infinite cardinal \kappa, \kappa^{\cf(\kappa)} > \kappa; for singular \kappa, this yields \kappa^{\cf(\kappa)} > \kappa without the full exponentiation scale of regular \kappa, where \cf(\kappa) = \kappa and the inequality follows from Cantor's theorem on power sets.^[23] Further implications for singular cardinals appear in exponentiation bounds, where regularity assumptions are absent. Shelah's PCF theory provides such a bound: if \aleph_\omega is a strong limit cardinal (i.e., $2^{\aleph_n} < \aleph_\omega for all n < \omega), then $2^{\aleph_\omega} < \aleph_{\omega^4}.^[23]

Applications

In forcing

In forcing, the regularity of uncountable cardinals is often preserved by certain posets, particularly those satisfying chain condition or closure properties that prevent the addition of short cofinal sequences. For instance, Cohen forcing, which adds real numbers via the poset of finite partial functions from \omega to $2, is countable chain complete (ccc) and thus preserves all uncountable cofinalities, including the regularity of any uncountable regular cardinal \kappa. Similarly, the Lévy collapse \mathrm{Col}(\mu, <\kappa), where \mu < \kappais regular and\kappais inaccessible, collapses all cardinals below\kappato have cardinality\muwhile preserving\kappaas a cardinal; due to its\mu-strategic closure and \kappa-chain condition, it maintains the regularity of \kappa$.^[24] Regularity can also be destroyed in forcing extensions, typically by adding a cofinal sequence to \kappa of length less than \kappa. Easton forcing, a class-sized product of Cohen forcings \mathrm{Add}(\kappa, F(\kappa)) over regular cardinals \kappa with Easton support (limited to fewer than \kappa coordinates below \kappa), generally preserves cofinalities but can be adapted to include components that add such sequences, singularizing a targeted regular \kappa without collapsing cardinals, subject to the Easton function F satisfying monotonicity and cofinality constraints.^[25] For measurable cardinals, which are regular, Prikry forcing provides a canonical example: starting from a measurable \kappa with normal measure U, the poset consists of finite stems and closed unbounded sets in the ultrapower, preserving all cardinals while forcing \mathrm{cf}(\kappa) = \omega and thus making \kappa singular.^[26] These preservation and destruction techniques find applications in consistency proofs within set theory. For example, iterated Cohen forcing with \aleph_1 many steps forces the continuum hypothesis ($2^{\aleph_0} = \aleph_1) while preserving the regularity of \aleph_1, as the ccc ensures no uncountable cofinalities are altered. Prikry forcing, in turn, is used to explore the behavior of large cardinals in extensions where regularity fails at specific points, aiding in models that test hypotheses like the singular cardinals problem.^[26] In generic extensions obtained by forcing, the regularity of successor cardinals such as \aleph_1^V typically remains intact under the axiom of choice, as standard forcing notions preserve the successor structure and uncountable cofinalities when no collapse occurs below them.^[27]

In inner model theory

In inner model theory, regular cardinals serve as foundational building blocks for analyzing the fine structure of canonical models like the constructible universe L and core models K. Fine structure theory, pioneered by Ronald Jensen, dissects these models through hierarchies such as J_\alpha and J_\alpha[A], where regular cardinals appear as projecta—the least ordinals admitting non-absolute definable subsets—and ensure the acceptability and solidity of premice. For an acceptable structure M, the \Sigma_1-projectum \rho = \rho_\omega^M is the least ordinal such that there exists a \Sigma_1-definable subset of \rho over M that is not absolute; this \rho is a cardinal in M, and embeddings \pi: M \to N preserve such cardinals above the critical point, as \pi(\rho) remains a cardinal if \crit(\pi) < \rho. Iterations of inner models, bounded by regular cardinals like \Theta (the least ordinal not surjectively onto from the reals in L(\mathbb{R})), terminate below these bounds due to their cofinality properties, enabling comparisons via the comparison lemma.^[28] In core model constructions, regular cardinals delineate the extent to which large cardinals are captured. For a measurable cardinal \kappa with normal measure \mu, the inner model L[\mu] has \kappa as its least measurable cardinal, with the same uncountable regular cardinals above \omega and below \kappa as in L, and GCH holding below it; more generally, the core model [K](/page/K)—the union of all iterable premice—incorporates sequences of measures or extenders up to the least "bad" regular cardinal where iterability fails. The Dodd-Jensen lemma guarantees that iterations along the main branch of length less than a regular \theta > \omega yield unique normal iteration maps, preserving solidity and soundness. If no inner model with a Woodin cardinal exists, the covering lemma implies that every uncountable set X \subseteq \mathrm{Ord} in V is covered by a set Y \in K with |Y|^K = |X| for regular cardinals bounding the strength. These properties underscore how regularity facilitates the minimality of core models relative to large cardinal assumptions.^[28] A pivotal application arises in models of determinacy, where the interplay between regularity and large cardinals reveals deep structural insights. In L(\mathbb{R}) under the axiom of determinacy (AD), John Steel established that every uncountable regular cardinal \kappa < \Theta—the supremum of ordinals constructible from reals—is measurable in \mathrm{HOD}^{L(\mathbb{R})}, the inner model of hereditarily ordinal-definable sets. This result, part of the HOD analysis, shows that \mathrm{HOD}^{L(\mathbb{R})} is a fine-structural core model satisfying GCH, with all such \kappa admitting a normal measure in its extender algebra; moreover, singular cardinals below \Theta are limits of measurables. The theorem links choiceless axioms like AD to large cardinal strength, as its consistency follows from the existence of a measurable cardinal above infinitely many Woodins in V, and it implies that projective determinacy holds without choice.^[29]

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### Definition
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Below is a merged summary of the definition of a regular cardinal based on the provided segments from Akihiro Kanamori's *The Higher Infinite* (2nd ed., Springer, 2009), consolidating all information into a concise yet comprehensive response. To retain all details efficiently, I will use a table in CSV format for key attributes (definition, equivalence, Hartogs number, page references, and URLs) across the various segments, followed by a narrative summary that ties everything together.
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Below is a merged response summarizing the definition of a regular cardinal from "Set Theory" by Thomas Jech, consolidating all information from the provided segments. To retain as much detail as possible, I will use a structured table format in CSV style for key attributes (e.g., definition, equivalence, page references, Hartogs number, and URLs), followed by additional context and notes where necessary. This ensures a dense, comprehensive representation while avoiding redundancy.
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