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Inaccessible cardinal

In set theory, an inaccessible cardinal is an uncountable cardinal number \kappa that is both regular—meaning its cofinality equals \kappa itself, so it cannot be expressed as the union of fewer than \kappa many sets each of cardinality less than \kappa—and a strong limit cardinal, satisfying $2^\lambda < \kappa for every \lambda < \kappa.^[1] This dual condition ensures that \kappa cannot be "accessed" or constructed from smaller cardinals using standard set-theoretic operations like exponentiation (powersets) or summation (unions).^[2] The existence of such cardinals transcends the axioms of Zermelo-Fraenkel set theory with the axiom of choice (ZFC), as their presence cannot be proven within ZFC and requires additional large cardinal axioms.^[1] A key property of an inaccessible cardinal \kappa is that the cumulative hierarchy up to \kappa, denoted V_\kappa, forms a model of ZFC, making \kappa a fixed point of the cumulative hierarchy where the universe "restarts" with full set-theoretic strength.^[2] This implies that if \kappa is inaccessible, then V_\kappa satisfies the axiom of infinity, replacement, and power set without collapse, and under the generalized continuum hypothesis (GCH), every weakly inaccessible cardinal (regular limit but not necessarily strong limit) coincides with a strongly inaccessible one.^[2] The concept originated in the early 20th century, with Felix Hausdorff considering weakly inaccessible cardinals (uncountable regular limit cardinals) in 1908, though the modern formulation emphasizing "unreachability" was developed by Alfred Tarski in his 1938 paper "Über unerreichbare Kardinalzahlen," where he characterized them via the condition \kappa^{<\kappa} = \kappa.^[3] Tarski proved that if \kappa is an uncountable limit cardinal satisfying \kappa^{<\kappa} = \kappa, then \kappa is strongly inaccessible, linking it to the power set operation's behavior.^[3] Subsequent work by Paul Erdős and Alfred Tarski in 1961 explored implications for measure theory, showing that inaccessible cardinals imply the consistency of ZFC and properties related to λ-additive measures and ideals.^[1] In the hierarchy of large cardinals, inaccessible cardinals are the weakest nontrivial ones, serving as a foundation for stronger notions like measurable or supercompact cardinals, and their assumption underlies alternative set theories like Tarski-Grothendieck set theory.^[2]

Definition and Properties

Formal Definition

In set theory within the framework of Zermelo–Fraenkel set theory with the axiom of choice (ZFC), cardinal numbers measure the size of sets and extend the natural numbers to transfinite ordinals, often indexed as aleph fixed points \aleph_\alpha where \aleph_0 denotes the cardinality of the natural numbers and successors are defined via initial ordinals. The power set \mathcal{P}(X) of a set X has cardinality $2^{|X|}, which strictly exceeds |X| by Cantor's theorem, and this exponential operation plays a central role in bounding cardinal growth. A cardinal \kappa is a strong limit cardinal if the power set operation cannot reach \kappa from below, formally expressed as \forall \lambda < \kappa \, (2^\lambda < \kappa). This condition ensures that \kappa is closed under exponentiation in a robust sense, preventing any smaller cardinal's power set from equaling or exceeding \kappa. A cardinal \kappa is regular if it cannot be expressed as the union of fewer than \kappa many sets each of cardinality less than \kappa; equivalently, its cofinality \mathrm{cf}(\kappa) = \kappa, meaning the least ordinal \alpha such that there exists a cofinal function f: \alpha \to \kappa (with \sup \mathrm{range}(f) = \kappa) satisfies \alpha = \kappa. This is captured by the condition that for every ordinal \beta < \kappa and every function f: \beta \to \kappa, the range of f has cardinality strictly less than \kappa. An inaccessible cardinal \kappa is an uncountable cardinal that is both regular and a strong limit cardinal, i.e., \kappa > \aleph_0, \mathrm{cf}(\kappa) = \kappa, and \forall \lambda < \kappa \, (2^\lambda < \kappa). This notion was introduced by Wacław Sierpiński and Alfred Tarski in 1930 as a generalization of \aleph_0, which satisfies the regularity and strong limit conditions but is countable.

Key Properties

An inaccessible cardinal \kappa exhibits strong closure properties derived directly from its definition as a regular strong limit cardinal. For any cardinal \lambda < \kappa, the successor cardinal \lambda^+ satisfies \lambda^+ < \kappa, ensuring that \kappa cannot be reached by taking successors from below. Similarly, \kappa is closed under cardinal exponentiation: for any cardinals \mu, \nu < \kappa, the power \mu^\nu < \kappa. This follows from the strong limit condition, as \mu^\nu \leq 2^{\max(\mu, \nu)^{|I|}} for some index set I with |I| \leq \nu < \kappa, and thus $2^\alpha < \kappa for all \alpha < \kappa implies the bound.^[4]^[5] A key consequence is that the von Neumann hierarchy initial segment V_\kappa = \bigcup_{\alpha < \kappa} V_\alpha forms a model of ZFC. Since \kappa is an ordinal, V_\kappa is transitive, so extensionality, foundation, pairing, union, and infinity hold as they do in V. Separation and replacement are preserved because sets in V_\kappa have rank below \kappa, and comprehension formulas are absolute for transitive models. The power set axiom holds internally due to the strong limit property: for any x \in V_\alpha with \alpha < \kappa, |x| < \kappa, so $2^{|x|} < \kappa, ensuring \mathcal{P}(x) \in V_{\alpha+1} \subseteq V_\kappa. Replacement is secured by regularity: for A \in V_\kappa with |A| < \kappa and a class function F: A \to V definable such that F''A \subseteq V_\kappa, the range F''A has cardinality less than \kappa (as \kappa is regular), so its rank is below \kappa, placing F''A \in V_\kappa. The axiom of choice holds in V_\kappa because well-orderings in V of sets in V_\kappa remain within V_\kappa.^[4]^[6]^[5] Inaccessible cardinals also possess a Mahlo-like quality in that they serve as fixed points in the hierarchy of cardinals, acting as least upper bounds for the smaller cardinals below them under the operations of successor and exponentiation. This closure makes them natural boundaries unreachable from below via standard cardinal arithmetic. Worldly cardinals, which satisfy V_\kappa \models ZFC but may be singular, represent a weaker notion, with inaccessible cardinals being precisely the regular worldly ones.^[4]

Extensions and Variations

α-Inaccessible Cardinals

A cardinal \kappa is defined to be \alpha-inaccessible, for an ordinal \alpha, if \kappa is inaccessible and, for every ordinal \beta < \alpha, \kappa is a limit of \beta-inaccessible cardinals; in particular, the $0-inaccessible cardinals coincide with the ordinary inaccessible cardinals. This recursive definition builds a hierarchy of large cardinals by iterating the notion of inaccessibility along the ordinals.^[7] The concept of \alpha-inaccessible cardinals was introduced by William Hanf and Dana Scott in their 1961 work to classify and organize the hierarchy of inaccessible cardinals and related notions, providing a framework for studying stronger forms of large cardinals through ordinal iteration.^[8] For successor ordinals, the definition simplifies: a cardinal \kappa is (\alpha+1)-inaccessible if it is \alpha-inaccessible and a limit of \alpha-inaccessible cardinals below it. At limit ordinals \lambda, \kappa must be inaccessible and serve as the least upper bound of \beta-inaccessible cardinals for all \beta < \lambda, ensuring closure under the iterative operation. For example, a $1-inaccessible cardinal is an inaccessible cardinal that is the supremum of a cofinal sequence of ordinary inaccessible cardinals. The hierarchy is constructed via enumerating functions: let I_0(\gamma) denote the \gamma-th inaccessible cardinal, and recursively define I_{\alpha+1}(\gamma) as the \gamma-th \alpha-inaccessible cardinal; the fixed points of I_\alpha yield the (\alpha+1)-inaccessible cardinals. If an \alpha-inaccessible cardinal exists for some \alpha > 0, then the class of all \alpha-inaccessible cardinals forms a proper class, as the hierarchy extends unboundedly. Moreover, for a \kappa that is \alpha-inaccessible, the model V_\kappa satisfies ZFC together with the assertion that there exist \beta-inaccessible cardinals for all \beta < \alpha. Hyper-inaccessible cardinals correspond to the specific case of \omega-inaccessible cardinals.

Hyper-Inaccessible Cardinals

A hyper-inaccessible cardinal \kappa is defined as an inaccessible cardinal that is \alpha-inaccessible for every finite ordinal \alpha < \omega. This means \kappa arises as the limit stage in the finite iterations of the \alpha-inaccessible hierarchy, where the hierarchy is constructed inductively starting from 0-inaccessible (ordinary inaccessible) cardinals: a cardinal is (\alpha + 1)-inaccessible if it is inaccessible and the supremum of \alpha-inaccessibles below it; and \gamma-inaccessible for limit \gamma if it is \alpha-inaccessible for all \alpha < \gamma. Thus, hyper-inaccessibility captures the \omega-limit of these successive strengthenings within the inaccessible cardinal hierarchy. Equivalently, \kappa is hyper-inaccessible if it is inaccessible and there is a cofinal sequence of order type \omega in the inaccessible cardinals below \kappa, where each level consists of limits of the prior levels in the finite hierarchy. This formulation emphasizes the iterative closure under limits at each finite step leading up to \kappa. If \kappa is hyper-inaccessible, then the model V_\kappa satisfies ZFC together with the assertion that there exists a proper class of inaccessible cardinals. Moreover, hyper-inaccessibles exhibit enhanced closure properties under certain elementary embeddings, reflecting the layered structure of the underlying hierarchy. Hyper-inaccessible cardinals differ from Mahlo cardinals in that the former are unbounded limits along the specific inaccessible hierarchy at the \omega-level, whereas Mahlo cardinals are inaccessible limits of regular cardinals (with the set of regulars below forming a stationary set). Every Mahlo cardinal is hyper-inaccessible, but the converse does not hold. In applications, hyper-inaccessible cardinals facilitate the construction of set-theoretic models V_\kappa that incorporate multiple levels of large cardinals below \kappa, aiding in the analysis of consistency strengths for axioms positing hierarchies of inaccessibles.

Consistency and Models

Consistency Implications

The theory ZFC augmented with the axiom asserting the existence of an inaccessible cardinal, denoted ZFC + I, has strictly greater consistency strength than ZFC. Specifically, if κ is inaccessible, then V_κ is a transitive model of ZFC, so ZFC + I proves Con(ZFC).^[9] However, by Gödel's second incompleteness theorem, ZFC cannot prove its own consistency and thus cannot prove Con(ZFC + I). The consistency of ZFC + I is instead relative to stronger theories, such as ZFC plus the existence of a Mahlo cardinal, which yields a model of ZFC + I via a suitable initial segment.^[9] In Gödel's constructible universe L, the existence of inaccessible cardinals is possible, as the property of inaccessibility is absolute between transitive models containing the relevant ordinals. Thus, if an inaccessible cardinal exists in the universe V, it also exists in the inner model L, and ZFC + I + V = L is consistent relative to ZFC plus a stronger large cardinal axiom.^[6] This compatibility contrasts with stronger large cardinals like measurable ones, which cannot exist in L and thus imply V ≠ L. The existence of inaccessible cardinals has foundational implications, particularly regarding the independence of the continuum hypothesis (CH) from ZFC. Since V_κ models ZFC for inaccessible κ, it serves as a ground model for forcing extensions that can violate CH (e.g., by adding many Cohen reals), showing Con(ZFC + ¬CH) relative to ZFC + I, while Gödel's L models ZFC + CH.^[10] A key historical development concerning the consistency of large cardinals is Paul Cohen's invention of forcing in 1963, which proved the independence of CH from ZFC and provided methods to explore the independence of inaccessible cardinals. While ZFC alone cannot disprove the existence of inaccessibles (as Con(ZFC) implies Con(ZFC + ¬I) via models like L without assuming I), forcing preserves the non-existence in certain extensions but cannot create genuine inaccessibles from below; upward consistency relies on assuming stronger cardinals.^[11] There is no transitive model of ZFC + I whose height is smaller than the least inaccessible cardinal κ. Suppose such a model M existed with height λ < κ; then λ would satisfy the definition of an inaccessible cardinal (as M would witness the required regularity and strong limit properties up to λ), contradicting the minimality of κ.^[12]

Existence in Set-Theoretic Models

In Gödel's constructible universe L, inaccessible cardinals exist precisely when they exist in V, due to the absoluteness of the inaccessibility property.^[12] Forcing techniques allow the addition or preservation of inaccessible cardinals in set-theoretic models. Easton's theorem establishes that, starting from a ground model satisfying GCH, one can use a class forcing with Easton support—an iteration over the class of regular cardinals—to arbitrarily prescribe the continuum function $2^\alpha = F(\alpha) for regular \alpha, as long as F is non-decreasing, F(\alpha) > \alpha, and \mathrm{cf}(F(\alpha)) > \alpha.^[13] This enables the creation of new inaccessible cardinals by forcing a regular cardinal \kappa to become a strong limit (e.g., by setting $2^\lambda = \lambda^+ for all \lambda < \kappa) while preserving existing inaccessibles above \kappa and maintaining regularity. Class forcing can also collapse inaccessibles; for instance, the Lévy collapse \mathrm{Col}(\omega, \langle \kappa) over an inaccessible \kappa adds surjections from \omega onto every ordinal below \kappa, making \kappa = \aleph_1 in the extension without affecting cardinals above \kappa, due to the \kappa^+-chain condition preserved by inaccessibility.^[14] In inner models, the presence of an inaccessible cardinal in V can lead to its appearance in extensions like L[U], where U is a normal measure on a measurable cardinal \mu > \kappa. The model L[U] is constructed by adjoining the measure to the constructible hierarchy, and under the embedding j_U : L[U] \to L[U^*] induced by U, inaccessible cardinals below \mu are often preserved as inaccessibles in the target model if they satisfy closure properties relative to the ultrapower.^[15] Specifically, if \kappa < \mu is inaccessible in V, then \kappa remains inaccessible in L[U] because the fine-structural properties of the model, including absoluteness of power sets below \mu, ensure that regularity and the strong limit condition hold internally.^[16] Reflection principles highlight the model-theoretic existence of inaccessibles despite limitations in V. Kunen's inconsistency theorem proves that there is no nontrivial elementary embedding j : V \to V, which implies that no measurable cardinal \kappa can satisfy V = \mathrm{Ult}(V, U) for an ultrafilter U on \kappa, as such an embedding would contradict the theorem.^[17] However, inaccessible cardinals do not rely on such embeddings for their definition and can consistently exist in V. By the Skolem paradox—arising from the Löwenheim-Skolem theorem—every consistent extension of ZFC, including ZFC + "there exists an inaccessible cardinal," has countable transitive models, in which the purported inaccessible cardinal appears as a countable ordinal externally, yet satisfies the internal properties of uncountable regularity and strong limit status.^[18] If \kappa is the least inaccessible cardinal, then V_\kappa serves as the smallest transitive model of ZFC. In this model, all axioms of ZFC hold due to the closure of V_\kappa under replacement and comprehension, as \kappa's regularity ensures that images under set functions remain below \kappa, and its strong limit property guarantees that power sets of smaller cardinals stay within V_\kappa. Moreover, V_\kappa satisfies "there are no inaccessible cardinals," since any potential inaccessible in V_\kappa would be below \kappa, contradicting minimality.^[9]

Proper Classes of Inaccessibles

A proper class of inaccessible cardinals refers to the scenario in which the class of all inaccessible cardinals is unbounded within the class of ordinals, meaning that inaccessible cardinals exist arbitrarily high in the ordinal hierarchy.^[19] This condition asserts that for every ordinal α, there is an inaccessible cardinal κ > α, ensuring that the collection of such cardinals cannot be bounded by any single ordinal.^[19] The assumption of a proper class of inaccessible cardinals has significant implications for the consistency strength of set-theoretic theories. Specifically, the theory ZFC augmented with the axiom "there exists a proper class of inaccessible cardinals" is consistent relative to ZFC plus the existence of stronger large cardinal axioms, such as a hyper-inaccessible cardinal.^[19] This is stronger in consistency strength than the existence of merely a single inaccessible cardinal, as the latter can be modeled by V_κ for inaccessible κ, whereas the former requires an unbounded chain.^[20] In the hierarchy of large cardinals, the existence of a proper class of inaccessible cardinals is equivalent to the existence of a hyper-inaccessible cardinal (also known as a 1-inaccessible or ω-inaccessible cardinal), which is itself an inaccessible cardinal that is a limit of inaccessible cardinals.^[19] More generally, this fits into the α-inaccessible hierarchy, where higher levels (such as 2-inaccessible cardinals) require proper classes of lower-level inaccessibles below them.^[19] A key model-theoretic observation is that if κ is a hyper-inaccessible cardinal, then the model V_κ satisfies ZFC together with the internal assertion of a proper class of inaccessible cardinals, since the inaccessible cardinals below κ form an unbounded class within V_κ.^[19]

Characterizations

Model-Theoretic Characterizations

A cardinal \kappa is inaccessible if and only if the cumulative hierarchy up to \kappa, denoted V_\kappa, is a transitive model of ZFC. This equivalence provides a foundational model-theoretic characterization of inaccessibility. To see that inaccessibility implies V_\kappa \models \mathrm{ZFC}, note that axioms such as extensionality, pairing, union, foundation, infinity, and choice are absolute between the universe V and the transitive inner model V_\kappa, which contains all ordinals below \kappa. The power set axiom holds in V_\kappa because \kappa is a strong limit: for any \lambda < \kappa, the power set \mathcal{P}(V_\lambda) has cardinality $2^\lambda < \kappa, so \mathcal{P}(V_\lambda) \in V_\kappa. For the replacement schema, suppose a \in V_\kappa and a formula \phi(x,y) such that V_\kappa \models \forall x \in a \, \exists! y \, \phi(x,y). In V, there exists a unique function F: a \to V satisfying \phi, with range b = F''a \subseteq V_\kappa. The ranks of elements of b are each less than \kappa, and there are at most |a| < \kappa such ranks; by regularity of \kappa, their supremum is less than \kappa, so \mathrm{rank}(b) < \kappa and thus b \in V_\kappa, confirming replacement in V_\kappa. Conversely, if V_\kappa \models \mathrm{ZFC}, then \kappa must be inaccessible. Absoluteness of cofinality between V and V_\kappa implies that \kappa is regular in V, as V_\kappa satisfies that \kappa (its height) is a regular cardinal. For the strong limit property, consider any \lambda < \kappa: the power set axiom in V_\kappa ensures \mathcal{P}(V_\lambda) \in V_\kappa, so |\mathcal{P}(V_\lambda)| = 2^\lambda < \kappa. This bidirectional link highlights how the structural properties of V_\kappa encode the combinatorial features of inaccessibility. A proof sketch leveraging the reflection principle further illuminates regularity: the principle guarantees that for any formula, there are arbitrarily large \alpha < \kappa where V_\alpha reflects it, but to avoid cofinal subsets of \kappa of smaller cardinality, regularity ensures no such proper cofinal sequence exists within V_\kappa. Similarly, the strong limit prevents power set "overflow," as reflection combined with the limit hypothesis bounds cardinalities below \kappa. An alternative model-theoretic perspective identifies inaccessible cardinals as the ordinals \kappa such that V_\kappa \models \mathrm{ZFC} and no smaller ordinal \beta < \kappa satisfies V_\beta \models \mathrm{ZFC} in a collapsing manner; however, since the property V_\alpha \models \mathrm{ZFC} holds precisely at inaccessible \alpha, each such \kappa is the least upper bound in its segment without prior satisfaction, with the strong limit ensuring the hierarchy does not collapse prematurely below \kappa. This leastness follows from the fact that if there were a \beta < \kappa with V_\beta \models \mathrm{ZFC}, then \beta would be inaccessible, contradicting the minimality in the local context unless \kappa is the smallest overall. Connected to this is Tarski's insight on fixed points: inaccessible cardinals are fixed points of the aleph function, satisfying \kappa = \aleph_\kappa, as regularity ensures \kappa is the \kappa-th infinite cardinal and the strong limit bounds intermediate exponentiations. This characterization extends naturally to \alpha-inaccessible cardinals. If \kappa is \alpha-inaccessible for some ordinal \alpha, then V_\kappa \models \mathrm{ZFC} + ``\mathrm{there \, are \,} \alpha\mathrm{-many \, inaccessible \, cardinals}", because the hierarchy of lower inaccessibles below \kappa is reflected into V_\kappa via the inductive definition and the model's satisfaction of ZFC, ensuring the existence statement holds internally without exceeding the height \kappa.

Equivalent Formulations

An inaccessible cardinal \kappa can be characterized order-theoretically as an uncountable regular cardinal that is a fixed point of the beth function, satisfying \beth_\kappa = \kappa. The beth function is defined recursively by \beth_0 = \aleph_0, \beth_{\alpha+1} = 2^{\beth_\alpha} for successor ordinals, and \beth_\lambda = \sup_{\alpha < \lambda} \beth_\alpha for limit ordinals \lambda. This fixed-point condition captures the strong limit property intrinsically through the iteration of power sets.^[21] The equivalence between being a strong limit cardinal and a beth fixed point follows from transfinite induction on the beth hierarchy. If \kappa is a strong limit cardinal, then for every \alpha < \kappa, $2^{\beth_\alpha} < \kappa by the definition of strong limit (since \beth_\alpha < \kappa), so \beth_{\alpha+1} < \kappa; at limit stages below \kappa, the supremum remains below \kappa. Thus, \beth_\kappa = \sup_{\alpha < \kappa} \beth_\alpha < \kappa. But since \beth_\kappa \geq \kappa (as the hierarchy includes all alephs up to at least \kappa if \kappa is a limit cardinal), equality holds. Conversely, if \beth_\kappa = \kappa, then for any \lambda < \kappa, \beth_{\lambda+1} = 2^\lambda \leq 2^{\beth_\lambda} = \beth_{\lambda+1} < \kappa by the fixed-point property, establishing the strong limit condition.^[22] Regularity admits an order-theoretic equivalent: \kappa is regular if and only if there exists no ordinal \lambda < \kappa and strictly increasing cofinal function f: \lambda \to \kappa. If such an f existed, the image would be a cofinal subset of order type \lambda < \kappa, contradicting regularity. Conversely, if \kappa is singular with \mathrm{cof}(\kappa) = \lambda < \kappa, then a strictly increasing enumeration of a cofinal sequence witnesses the cofinal map. Combining this with the beth fixed-point condition yields the full order-theoretic formulation of inaccessibility.^[23] A combinatorial equivalent of inaccessibility is that \kappa is uncountable and satisfies: for every collection of fewer than \kappa sets, each of cardinality less than \kappa, their union has cardinality less than \kappa; additionally, $2^\lambda < \kappa for all \lambda < \kappa. The union condition is equivalent to regularity, as a counterexample would provide a cover by fewer than \kappa smaller sets reaching size \kappa, implying singular cofinality. The power set condition directly encodes the strong limit property in combinatorial terms, avoiding explicit reference to the beth hierarchy.^[24] Inaccessible cardinals relate to weakly compact cardinals in the large cardinal hierarchy: every weakly compact cardinal is inaccessible, but the converse fails, as weak compactness requires additional combinatorial principles like the tree property or partition relations (e.g., \kappa \to (\kappa)^2_2). Inaccessibles form a proper initial segment below weakly compacts, with the least weakly compact (if existent) exceeding the least inaccessible.^[25] Historical variants include characterizations using generalized indescribability properties on structures like \mathcal{P}_\kappa \lambda, where inaccessibility emerges as a base case for the \Pi^1_0-indescribability hierarchy, though full indescribability typically strengthens to weakly compact or beyond. These formulations, developed in the 1970s, emphasize descriptive set-theoretic analogs for higher cardinals.^[26]

References

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