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Large cardinal

In set theory, large cardinals are infinite cardinal numbers that satisfy certain strong properties, and their existence is postulated by axioms extending the Zermelo–Fraenkel set theory with the axiom of choice (ZFC), providing a framework for exploring the limits of provability and consistency within mathematics.^[1] These axioms assert the presence of cardinals that are "unreachable" from smaller sets via standard operations like power sets and unions, forming a hierarchy of increasing strength that gauges the consistency of various mathematical theories.^[1] The hierarchy begins with relatively modest large cardinals, such as inaccessible cardinals, which are uncountable regular cardinals κ such that the power set of any smaller cardinal has cardinality less than κ, ensuring they model ZFC internally.^[1] Stronger examples include measurable cardinals, characterized by the existence of a non-principal ultrafilter on κ that defines an elementary embedding from the universe of sets V to an inner model M, with κ as the critical point.^[2] Even more potent are Woodin cardinals, which satisfy a strong reflection principle: for every set A ⊆ V_κ there exist arbitrarily many α < κ that are A-strong with respect to elementary embeddings into inner models, enabling proofs of determinacy for sets of reals beyond what ZFC alone can achieve.^[2] Large cardinals are significant because they address key independence results, such as showing the consistency of the negation of the Continuum Hypothesis relative to ZFC plus an inaccessible cardinal, and they underpin the study of inner models and forcing techniques to compare the interpretability of different axiom systems.^[1] Moreover, they imply axioms like projective determinacy (PD) and even the axiom of determinacy (AD) in certain contexts, resolving regularity properties for definable sets of real numbers, such as perfect set theorems and measurability.^[2] This hierarchy not only extends ZFC but also suggests a natural progression toward stronger foundations, influencing broader areas like descriptive set theory and the philosophy of mathematics.^[1]

Fundamentals

Definition

In set theory, cardinals are infinite cardinal numbers that measure the sizes of sets, extending the notion of finite cardinality to the infinite realm. Ordinals, on the other hand, are well-ordered sets that serve as the order types for well-orderings, providing the indexing structure for the cumulative hierarchy of sets. The von Neumann universe V is constructed as the union \bigcup_{\alpha \in \mathrm{Ord}} V_\alpha, where each stage V_\alpha is defined recursively: V_0 = \emptyset, V_{\alpha+1} = \mathcal{P}(V_\alpha) (the power set of V_\alpha), and for limit ordinals \lambda, V_\lambda = \bigcup_{\alpha < \lambda} V_\alpha.^[1] Large cardinals are uncountable infinite cardinals \kappa possessing strong extension properties that transcend the axioms of Zermelo-Fraenkel set theory with the axiom of choice (ZFC), often characterized by the property that V_\kappa—the \kappa-th level of the cumulative hierarchy—satisfies ZFC or even stronger axioms, thereby embodying a rich internal structure with "many" smaller cardinals below it.^[1] This informal notion highlights their role in extending the universe of sets beyond what ZFC alone can prove, asserting the existence of levels that are themselves models of significant portions of set theory.^[1] Unlike small cardinals, which are fully captured within the basic hierarchy of ZFC and do not introduce new foundational assumptions, large cardinals exhibit properties that form a proper class rather than a definable set, meaning there is no single set containing all of them, and their collection extends indefinitely across the ordinals. For instance, in certain hierarchies of large cardinals, the least inaccessible cardinal serves as the smallest such entity, marking the boundary where these extension properties first emerge.^[1] A formal partial characterization of large cardinals can be given through the concept of elementary embeddings, as exemplified by measurable cardinals: a cardinal \kappa is measurable if there exists a non-principal \kappa-complete ultrafilter U on \kappa yielding an ultrapower embedding j: V \to M, where M is a transitive and well-founded class, j(\mathrm{id}) = [\mathrm{id}]_U > \kappa (with \mathrm{id} the identity function on \kappa), and j(\xi) = \xi for all \xi < \kappa.^[3] This embedding witnesses the "largeness" of \kappa by mapping the universe into a proper extension while fixing all smaller ordinals, providing a precise measure of how \kappa transcends ordinary cardinal properties.^[3]

Historical Development

The concept of large cardinals emerged in the early 20th century as set theorists sought to extend the foundational axioms of set theory beyond the limitations of Zermelo-Fraenkel set theory with the axiom of choice (ZFC). A pivotal early development was Kurt Gödel's introduction of the constructible universe L in 1938, which demonstrated the relative consistency of the axiom of choice and the generalized continuum hypothesis with Zermelo–Fraenkel set theory (ZF) but also highlighted the restrictive nature of L in accommodating certain infinite structures, prompting interest in stronger axioms of infinity to describe larger transfinite hierarchies.^[4] In the 1930s and 1940s, Stanisław Ulam laid foundational groundwork for measurable cardinals in 1930 while exploring measure theory on infinite cardinals, defining them as cardinals admitting a non-trivial, countably additive measure that vanishes on singletons. This idea was formalized more rigorously by Dana Scott in 1961, who characterized measurability using \kappa-complete ultrafilters on a cardinal \kappa, establishing that the existence of a measurable cardinal implies V \neq L and thus transcends Gödel's constructible hierarchy.^[5] The 1960s marked a significant boom in large cardinal research, driven by advances in elementary embeddings and hierarchies of inaccessibility. Paul Mahlo had earlier introduced Mahlo cardinals in 1911 as fixed points of the aleph function among inaccessible cardinals, but their significance was popularized in this era through connections to reflection principles. Azriel Lévy contributed a classification of inaccessible cardinals in the early 1960s, delineating hierarchies based on reflection properties and strong limit cardinals. Kenneth Kunen advanced the field in 1970 by developing ultrapower constructions for elementary embeddings, providing a combinatorial framework for studying measurable and stronger cardinals. The 1970s and 1980s saw further innovations, including William Reinhardt's proposal of supercompact cardinals in the late 1960s, characterized by embeddings with large critical point closures, and his exploration of rank-to-rank embeddings in his 1968 dissertation. Kunen proved in 1971 the inconsistency of Reinhardt cardinals—non-trivial embeddings j: V \to V—with ZFC, establishing a boundary for choiceless large cardinal axioms. In the 1980s, W. Hugh Woodin introduced Woodin cardinals to address determinacy questions in descriptive set theory, showing that their existence implies projective determinacy and advancing inner model constructions. Post-2000 developments have extended these ideas through inner model theory and virtual large cardinals. Ralf Schindler and others in the 2010s explored rank-into-rank cardinals as embeddings from V to V_\lambda for limit ordinals \lambda, providing equiconsistency results with strong forcing axioms and resolving questions about the strength of HOD (hashing of definable sets). Key contributors like Alfred Tarski (early work on inaccessible limits in the 1930s), Robert Solovay (1965 connections between measurability and Lebesgue measurability of sets of reals), and William Mitchell (1970s inner models for measurable cardinals) have shaped the field's trajectory toward unifying consistency strengths.^[6]^[2]

Classification and Hierarchy

Inaccessible and Mahlo Cardinals

An inaccessible cardinal is defined as an uncountable regular strong limit cardinal \kappa, meaning \mathrm{cf}(\kappa) = \kappa and $2^\lambda < \kappa for all \lambda < \kappa.^[7] This condition ensures that \kappa cannot be reached from smaller cardinals via successor operations or power sets, positioning it as a natural boundary in the hierarchy of infinite cardinals.^[8] Equivalently, if \kappa is inaccessible, then the universe V_\kappa satisfies the axioms of Zermelo-Fraenkel set theory with the axiom of choice (ZFC), making \kappa a model of the full theory.^[7] Inaccessible cardinals are necessarily greater than the least uncountable cardinal \aleph_1, as any countable limit of smaller cardinals would violate regularity or the strong limit property.^[8] The distinction between weakly and strongly inaccessible cardinals arises from relaxing the strong limit condition. A weakly inaccessible cardinal \kappa is an uncountable regular limit cardinal, where \kappa is the supremum of smaller cardinals but without the power set closure.^[7] In contrast, strong inaccessibility imposes the additional requirement that \kappa remains closed under exponentiation, ensuring greater isolation from lower levels of the cardinal hierarchy.^[8] This stronger notion aligns with early axiomatic developments in set theory, where such cardinals serve as foundational points for constructing models.^[7] Building on inaccessibility, a Mahlo cardinal \kappa is an inaccessible cardinal such that the set of inaccessible cardinals below \kappa is stationary in \kappa.^[7] Stationarity here means that the set intersects every closed unbounded subset of \kappa, reflecting a dense distribution of inaccessibles up to \kappa.^[8] Introduced by Paul Mahlo in the early 20th century, this concept extends the hierarchy through fixed-point phenomena, where the Mahlo operation enumerates the inaccessibles below \kappa and iterates to form higher levels.^[7] A fundamental theorem states that if \kappa is Mahlo, then there are exactly \kappa-many inaccessible cardinals below it, emphasizing its role as a limit of such cardinals.^[8] Mahlo cardinals also exhibit reflection properties equivalent to the reflection of certain \Pi^1_1 sentences at \kappa.^[7] The set of inaccessible cardinals below a Mahlo \kappa is stationary and reflects in the sense that it intersects every club, but full stationary reflection (every stationary subset reflecting) is a property of weakly compact cardinals, which are stronger.^[8] The least inaccessible cardinal is not Mahlo, as the set of inaccessibles below it is empty and thus non-stationary.^[7] Iterations of the Mahlo operation yield hyper-Mahlo cardinals, which are fixed points where \kappa is a limit of Mahlo cardinals below it, further ascending the hierarchy of large cardinals.^[8]

Measurable Cardinals

A measurable cardinal is an uncountable cardinal \kappa such that there exists a \kappa-complete non-principal ultrafilter U on \kappa.^[3] Equivalently, \kappa is measurable if there is an elementary embedding j: V \to M into a transitive inner model M with critical point \kappa (so j(\xi) = \xi for all \xi < \kappa and j(\kappa) > \kappa) and M^\kappa \subseteq M.^[3]^[9] The ultrafilter characterization leads to an ultrapower construction that yields the embedding. Given such a U on \kappa, form the ultrapower V^\kappa / U consisting of equivalence classes _U for functions f: \kappa \to V, where f E_U g if and only if \{\xi < \kappa : f(\xi) \in g(\xi)\} \in U.^[3] The canonical embedding is j_U(a) = [c_a]_U, where c_a(\xi) = a is the constant function, and in particular [{\rm id}]_U = \kappa, the critical point.^[9] By Łoś's theorem, for any formula \phi(x_1, \dots, x_n) and f_1, \dots, f_n: \kappa \to V, we have

V^\kappa / U \models \phi([f_1]_U, \dots, [f_n]_U) \iff \{\xi < \kappa : V \models \phi(f_1(\xi), \dots, f_n(\xi))\} \in U.

^[3] Thus j_U: V \to V^\kappa / U is elementary. The relation E_U is well-founded (since U is \kappa-complete), and by the Mostowski collapse lemma, there is a transitive M and isomorphism \pi: (V^\kappa / U, E_U) \cong (M, \in) with j = \pi \circ j_U.^[9] Moreover, M^\kappa \subseteq M holds because any sequence of length < \kappa from M lifts to representatives in V^\kappa.^[3] Measurable cardinals enjoy strong reflection properties. Scott's theorem states that if \kappa is measurable, then V \neq L.^[3] Additionally, V_\kappa \models {\rm ZFC} + ``\kappa is measurable'', since the ultrapower restricted to V_\kappa yields an elementary embedding within V_\kappa.^[3] The least measurable cardinal greatly exceeds the first inaccessible cardinal, as it must be a limit of many inaccessibles (in fact, Mahlo in various inner models).^[10] A variant is the real-valued measurable cardinal, where \kappa admits a \kappa-additive probability measure \nu: {\cal P}(\kappa) \to [0,1] with \nu(\{\xi\}) = 0 for all \xi < \kappa and \nu(\kappa) = 1.^[11] All measurable cardinals are real-valued measurable (taking \nu(X) = 1 if X \in U and $0otherwise), but the converse fails; for example, it is consistent that2^{\aleph_0}$ is real-valued measurable but not measurable.^[11] Real-valued measurables come in atomless and two-valued types, with the former allowing measures without atoms of positive measure and implying weaker indescribability.^[11] On a measurable \kappa, the normal measures form a key structure: these are the \kappa-complete non-principal ultrafilters U on \kappa such that for any X \in U, the diagonal intersection \Delta(X) = \{\xi \in X : X \cap \xi \in U_\xi\} (where U_\xi = \{Y \subseteq \xi : Y \in j_U(Y)\}) also lies in U, or equivalently, every regressive function on a set in U is constant on a large set.^[12] Every measurable \kappa carries at least one normal measure, and the collection of all normal measures on \kappa generates a filter base for the club filter on {\cal P}_\kappa(\kappa).^[12] If \kappa is measurable, then \kappa is strongly inaccessible (regular and $2^\lambda < \kappa for all \lambda < \kappa).^[3] Moreover, with a normal measure U, the set of inaccessible cardinals below \kappa lies in U, so \kappa is a limit of inaccessibles (hence Mahlo); in fact, it is Mahlo of high order, as the inaccessibles below form a stationary set.^[3]

Strong and Woodin Cardinals

A strong cardinal is a large cardinal \kappa such that for every ordinal \lambda \geq \kappa, there exists a transitive inner model M of ZFC and an elementary embedding j: V \to M with critical point \kappa, j(\kappa) > \lambda, and V_\lambda \subseteq M.^[8] This closure condition V_\lambda \subseteq M ensures that the embedding reflects the entire initial segment of the universe up to \lambda, which is stronger than the closure under \kappa-sequences (i.e., ^\kappa M \subseteq M) characteristic of measurable cardinals.^[8] The strength of a strong cardinal \kappa is often measured by the least ordinal \lambda for which such an embedding exists with the given properties, allowing for a fine-grained hierarchy among strong cardinals.^[8] The embedding for a strong cardinal can be visualized as follows: the critical point \kappa is mapped to j(\kappa) > \lambda, with the model M containing all sets of rank less than or equal to \lambda from the universe V, thus capturing a significant portion of V's structure beyond what is achieved in measurable embeddings, where closure is limited to sequences of length less than \kappa.^[8] Strong cardinals were originally introduced in the context of exploring embeddings with extensive closure properties, building on earlier notions like measurability.^[8] A key property is that the existence of a strong cardinal implies the presence of many measurable cardinals below it, as the embeddings generate ultrapowers that yield measurables in inner models.^[8] Furthermore, if \kappa is strong, then $0^\dagger exists, meaning there is a non-trivial elementary embedding from L to itself, which has profound implications for the constructible universe L.^[13] Woodin cardinals represent a refinement of strong cardinals. A cardinal \kappa is Woodin if it is inaccessible and, for every A \subseteq V_\kappa, there are arbitrarily large \delta < \kappa such that \delta is A-strong: for every \lambda > \delta, there exists an elementary embedding j: V \to M with critical point \delta, j(\delta) > \lambda, V_\lambda \subseteq M, and V_\lambda \cap A = V_\lambda \cap j(A).^[14] This notion was isolated by W. Hugh Woodin in the early 1980s as part of investigations into determinacy and inner models.^[15] Woodin cardinals form a limit point in the hierarchy of strong cardinals, often referred to as the Woodin limit of Woodin cardinals, emphasizing their role as a cumulative strength below superstrong cardinals but above ordinary strong ones.^[8] A seminal result connecting Woodin cardinals to descriptive set theory is the Martin-Steel theorem, which states that if there are sufficiently many Woodin cardinals (specifically, n Woodin cardinals for projective levels, or infinitely many for full projective determinacy) below a measurable cardinal, then all projective sets of reals are determined.^[16] This theorem, established in the 1980s, demonstrates how Woodin cardinals provide the consistency strength necessary for projective determinacy (PD), bridging forcing techniques with embedding principles.^[16] In the large cardinal hierarchy, Woodin cardinals sit strictly between strong cardinals and superstrong cardinals, with their embeddings offering localized reflection properties that are crucial for applications in forcing and consistency proofs.^[8]

Supercompact and Larger Cardinals

A supercompact cardinal \kappa is defined as a cardinal such that for every \lambda \geq \kappa, there exists an elementary embedding j: V \to M with critical point \kappa, j(\kappa) > \lambda, and M^\lambda \subseteq M, where M is a transitive inner model.^[17] This closure condition provides stronger reflection properties than those for strong cardinals, as the model M captures all \lambda-sized subsets of its elements. Supercompact cardinals imply the existence of measurable, strong, and Woodin cardinals below them, with \kappa itself being a limit of such cardinals.^[18] They also exhibit significant stationary reflection: every stationary subset of \kappa^+ reflects to some \alpha < \kappa^+. The supercompactness index of \kappa measures the extent of this property, consisting of the class of all \lambda for which \kappa is \lambda-supercompact.^[19] Extendible cardinals represent an even stronger form of reflection. A cardinal \kappa is \lambda-extendible if there exists an elementary embedding j: V_\lambda \to V_{j(\lambda)} with critical point \kappa.^[18] \kappa is fully extendible if it is \lambda-extendible for every \lambda > \kappa. This notion captures near-universal embedding properties but leads to inconsistencies in ZFC: Kunen proved in 1971 that there is no nontrivial elementary embedding j: V \to V in ZFC, ruling out extendible embeddings from the full universe.^[20] Moreover, the existence of a supercompact cardinal implies V \neq L, as it entails the existence of a measurable cardinal, which contradicts the constructibility of all sets. Rank-into-rank cardinals push the boundaries toward potential ZFC-inconsistencies. The axiom I_0 (the strongest rank-into-rank axiom) asserts the existence of a nontrivial elementary embedding j: V_{\lambda+2} \to V_{\mu+2} for some limit ordinal \lambda of uncountable cofinality, with critical point \kappa < \lambda, j(\kappa) = \lambda, and V_{\eta+2} = M_{\eta+2} for all \eta \leq \lambda. Weaker axioms I_n for n \geq 1 involve iterated embeddings of decreasing strength, forming a hierarchy approaching the Kunen inconsistency from below.^[21]^[22] These cardinals are among the strongest known to be consistent with ZFC in certain models, with consistency results established relative to even larger assumptions like super-Reinhardt cardinals.^[23] Further extensions include Icarus sets, which are sets X \subseteq V_{\lambda+1} allowing elementary embeddings j: L(X, V_{\lambda+1}) \prec L(X, V_{\lambda+1}) with critical point below \lambda, pushing the boundaries slightly further beyond standard rank-into-rank embeddings.^[24] Berkeley cardinals extend this hierarchy further, defined as cardinals \kappa such that for every transitive set M with \kappa \in M and every \alpha < \kappa, there exists an elementary embedding j: M \to M with critical point \gamma satisfying \alpha < \gamma < \kappa. Introduced in the pre-2010s, they surpass rank-into-rank in strength but are inconsistent with V = \mathrm{HOD}, as their existence forces HOD to diverge significantly from V.

Properties and Implications

Reflection and Embedding Principles

Reflection principles in set theory postulate that the truth of certain formulas in the universe V of all sets is reflected in initial segments V_\alpha of the von Neumann cumulative hierarchy for sufficiently large ordinals \alpha. These principles generalize the classical Lévy reflection theorem, which is provable in ZFC and states that for any finite collection of formulas in the language of set theory, the class of ordinals \alpha such that V_\alpha is an elementary substructure of V with respect to those formulas is stationary. In the context of large cardinals, reflection principles take a stronger form known as \Pi^m_n-reflectivity, where a cardinal \kappa is \Pi^m_n-reflecting if, for every \Pi^m_n-formula \phi(v_1, \dots, v_k, X) and sets A_1, \dots, A_k \subseteq V_\kappa, whenever V_\kappa \models \phi(A_1, \dots, A_k, R) for some relation R \subseteq V_\kappa, there exists an ordinal \alpha < \kappa such that V_\alpha \models \phi(A_1 \cap V_\alpha, \dots, A_k \cap V_\alpha, R \cap V_\alpha^{<\omega}). This notion captures how properties true in the model V_\kappa "reflect downward" to smaller initial segments V_\alpha, providing a measure of how "indescribable" \kappa is with respect to the Lévy hierarchy of formulas.^[25] A related unifying mechanism for large cardinals is provided by elementary embeddings, which are class functions j: V \to M that preserve first-order properties, meaning for any formula \phi and sets x_1, \dots, x_n in V, V \models \phi(x_1, \dots, x_n) if and only if M \models \phi(j(x_1), \dots, j(x_n)). Here, M is a transitive inner model of ZFC, and the embedding is non-trivial, satisfying j(\alpha) = \alpha for all ordinals \alpha below some cardinal \kappa but j(\kappa) > \kappa. The critical point of j is defined as \kappa = \mathrm{crit}(j) = \min\{\alpha \in \mathrm{Ord} \mid j(\alpha) \neq \alpha\}, which is the smallest ordinal moved by j and must be a regular cardinal greater than all smaller cardinals in the hierarchy. Non-triviality is ensured by the existence of such a \kappa, distinguishing these embeddings from the identity map.^[26] These embeddings often arise from ultrapowers via normal measures on \kappa, and their properties imply reflection phenomena through elementarity. By Łoś's theorem, in the ultrapower construction \mathrm{Ult}(V, U) for a \kappa-complete ultrafilter U on \kappa, an element [f]_U satisfies \phi([f]_U) if and only if \{\xi < \kappa \mid V \models \phi(f(\xi))\} \in U, ensuring that the induced embedding j_U : V \to \mathrm{Ult}(V, U) is elementary. The generator of the embedding j is the set G(j) = \{\alpha < j(\kappa) \mid \alpha \neq j(\beta) \text{ for any } \beta < \kappa\}, consisting of the ordinals "moved" by j beyond \kappa; for extenders or stronger embeddings, this generator captures the "support" of the embedding. Well-foundedness of the ultrapower is guaranteed if U is \kappa-complete, allowing the Mostowski collapse to yield a transitive M with j: V \to M elementary, while absoluteness properties ensure that certain truths in V are preserved in M, particularly for bounded quantifiers or low-complexity formulas.^[26] Reflection principles and elementary embeddings interconnect to characterize large cardinals. For inaccessible cardinals \kappa, the model V_\kappa satisfies ZFC, reflecting all \Sigma_0-formulas from V in the absolute sense, as \Sigma_0-formulas with bounded quantifiers are preserved between transitive models containing the parameters. A measurable cardinal \kappa, via its elementary embedding j: V \to M with M closed under <\kappa-sequences, implies \Sigma_1-reflection, where \Sigma_1-formulas true in V reflect to truth in V_\kappa, since elementarity and closure ensure preservation of existential quantifiers over sets of size <\kappa. Supercompact cardinals \kappa yield even stronger \lambda-reflection for every \lambda > \kappa: there exists an elementary embedding j: V \to M with \mathrm{crit}(j) = \kappa, j(\kappa) > \lambda, and V^\lambda \subseteq M (or more precisely, M^{<\lambda} \subseteq M), implying that V_\lambda elementarily embeds into M and reflects a wide class of formulas from V to initial segments below \kappa.^[27] Variations of reflection principles distinguish weak and strong forms relevant to large cardinals. Weak reflection often involves stationary sets, such as the property that every stationary subset of \kappa reflects to a stationary subset of some \alpha < \kappa, which holds for weakly compact cardinals. Stronger forms require closure under power sets or higher definability, as in the case of indescribable cardinals, where \kappa is \Pi^1_m-indescribable (for m=1 or $2) if it reflects all \Pi^1_m-formulas with a single second-order parameter, generalizing \Pi^m_n-reflectivity to include relational structures on V_\kappa$. These notions unify the technical properties of large cardinals by linking syntactic reflection of formulas to semantic embeddings and structural preservation.^[25]

Consistency Strength and Independence

The consistency strength of a large cardinal axiom, asserting the existence of a cardinal of a particular type, is gauged relative to ZFC by identifying the minimal theory T such that \mathrm{Con}(T) implies \mathrm{Con}(\mathrm{ZFC} + ``\mathrm{there~is~a~}\kappa\mathrm{~of~type~}X") for the given type X.^[1] This establishes a hierarchy ordered by reverse logical implication: if \mathrm{Con}(T_1) \vdash \mathrm{Con}(T_2), then T_2 exceeds T_1 in consistency strength.^[1] Large cardinal axioms thus form a ladder of increasing provability power, where stronger axioms validate the consistency of weaker ones but not vice versa. In this hierarchy, the axiom of an inaccessible cardinal sits immediately above ZFC, as \mathrm{ZFC} + ``\mathrm{there~is~an~inaccessible~}\kappa" proves \mathrm{Con}(\mathrm{ZFC}) via the fact that V_\kappa \models \mathrm{ZFC} when \kappa is inaccessible.^[1] A measurable cardinal advances further, yielding \mathrm{Con}(\mathrm{ZFC} + ``\mathrm{there~is~an~inaccessible}") through ultrapower embeddings that produce inner models containing inaccessibles below the measurable.^[1] Strong cardinals extend this pattern, with a \kappa-strong cardinal implying consistency for all lower levels of strongness up to \lambda-strong for \lambda < \kappa, culminating in superstrong cardinals that assert even greater closure in elementary embeddings. Woodin cardinals provide the strength for determinacy principles, as the existence of infinitely many Woodin cardinals with a measurable above implies the axiom of determinacy (AD) in L(\mathbb{R}). Rank-into-rank axioms, such as I_0 (asserting an elementary embedding j: V_{\lambda+2} \prec V_{\lambda+2} with critical point \kappa and V_{\lambda+1} \in \mathrm{ran}(j)), exceed the consistency strength of V = L.^[21] The independence of large cardinals from ZFC follows directly from Gödel's second incompleteness theorem, which precludes ZFC from proving its own consistency and, by extension, the existence of any large cardinal whose assumption yields \mathrm{Con}(\mathrm{ZFC}).^[28] For instance, if a measurable cardinal \kappa exists, then \mathrm{Con}(\mathrm{ZFC} + ``\mathrm{there~is~a~measurable}") > \mathrm{Con}(\mathrm{ZFC}), as the former proves the latter while ZFC cannot prove the former without contradiction.^[1] Forcing techniques further underscore this independence by preserving large cardinals while altering cardinal structure below them; the Lévy collapse \mathrm{Col}(\mu, <\kappa), for inaccessible \kappa > \mu, collapses cardinals in (\mu, \kappa] to \mu^+ without destroying \kappa's inaccessibility.^[29] Supercompact cardinals amplify this, implying the consistency of diverse inner models, including those with sharps like $0^\sharp and iterable mice.^[1] Notable theorems highlight precise placements: the existence of a supercompact cardinal ensures \mathrm{Con}(\mathrm{ZFC} + 0^\sharp + ``\mathrm{there~are~proper~class~many~iterable~mice}"), bridging to fine-structural inner models.^[30] For I_0, its consistency strength lies strictly above the HOD conjecture (positing \mathrm{HOD} = L) but below ultimate L, the hypothetical ultimate inner model capturing all large cardinals.^[21] Gaps persist in the hierarchy, with the exact consistency strength of Vopěnka's principle—asserting that every proper class of structures has an elementary embedding between two members—remaining undetermined, though it exceeds supercompacts but falls short of rank-into-rank axioms like I_1.^[31] At the apex, Kunen's inconsistency theorem establishes an upper bound, proving that no nontrivial elementary embedding j: V \to V exists in ZFC, thereby refuting Reinhardt cardinals and limiting the hierarchy's reach.^[1]

Motivations and Status

Role in Axiomatic Set Theory

Large cardinals play a pivotal role in axiomatic set theory by providing tools to establish determinacy for infinite games on the reals beyond the Borel hierarchy. A key result is that the existence of infinitely many Woodin cardinals below a measurable cardinal implies the axiom of determinacy (AD) in the inner model L(ℝ), where ℝ denotes the set of real numbers.^[2] This theorem, developed through the methods of Martin and Steel and extended by Woodin, resolves the determinacy of all games played on reals in L(ℝ), yielding profound consequences such as the Lebesgue measurability of all sets of reals and the perfect set property.^[32] Such implications extend the scope of descriptive set theory, confirming determinacy for sets that are not projective, thus bridging large cardinal assumptions with analytic properties of the real line. In the context of forcing, large cardinals ensure the preservation of their properties under certain extensions, facilitating the study of axioms like the proper forcing axiom (PFA). Measurable cardinals are preserved by forcing notions of size less than the cardinal, such as ccc or κ-closed forcings, preventing the collapse of their status or the addition of destructive subsets.^[33] Furthermore, the existence of a supercompact cardinal allows forcing extensions where the singular cardinal hypothesis (SCH) fails, for instance, by making the power set of a singular cardinal larger than its successor.^[34] The consistency of PFA, which asserts that every proper poset has a filter meeting all dense sets of size at most the continuum, follows from a supercompact cardinal via iterated proper forcing, yielding models where the continuum is ℵ₂ and CH fails.^[35] Inner models constructed around large cardinals provide canonical structures for analyzing consistency strength and definability. For a measurable cardinal κ with normal measure U, the inner model L[U] is built by iterating the ultrapower by U, capturing the elementary embeddings derived from the measure and serving as a minimal model containing all ordinals up to the ultrapower of κ.^[36] For Woodin cardinals, core models extend this construction, incorporating extenders that witness Woodinness; these models, developed by Mitchell and Steel, compute the theory of sets up to the least Woodin cardinal and facilitate fine-structural analysis.^[37] Woodin's ultimate L conjecture, formulated in the 2010s, posits that there exists a canonical inner model Ultimate-L, akin to Gödel's L but incorporating all large cardinals via a hierarchy of extenders, which coincides with the hereditarily ordinal definable sets (HOD) under suitable assumptions.^[38] Large cardinals also underpin combinatorial principles realized through specialized forcing techniques. Stationary tower forcing, originating from strong cardinals, iterates stationary set preservations to singularize cardinals while maintaining reflection properties; for a strong cardinal δ, this forcing can collapse δ to ω₁ without destroying stationarity below it.^[39] Ramsey-like large cardinals, such as Ramsey cardinals κ, satisfy partition properties like \kappa \to (\kappa)^2_2, meaning that for any function f: [κ]^2 → 2, there exists a homogeneous set H ⊆ κ of order type κ. These properties generalize Ramsey's theorem to uncountable settings and are strengthened in the hierarchy, with measurable Ramsey cardinals ensuring such relations hold robustly.^[16] Beyond these, large cardinals resolve variants of the continuum hypothesis (CH) and establish generic absoluteness. For example, PFA implies CH fails and the continuum equals ℵ₂, while models from supercompacts can force 2^{ℵ₀} = ℵ₃ or other values. Extendible cardinals provide generic absoluteness for the structure of the universe up to their height, ensuring that truths about H_λ (λ extendible) hold in generic extensions by forcings of size less than λ, thus stabilizing set-theoretic truths across models.^[40]

Epistemic and Philosophical Considerations

Large cardinal axioms are motivated by the desire to extend the ZFC framework in a "natural" manner, serving as principles that maximize reflection principles and embody minimality in the iterative conception of sets. These axioms are seen as capturing the intuitive idea that the cumulative hierarchy continues indefinitely without abrupt terminations, thereby postulating a richer universe beyond the constructible sets L. Akihiro Kanamori articulates a maximality criterion for set theory, wherein large cardinals represent the strongest consistent extensions that preserve the core structure of ZFC while enhancing its expressive power, aligning with the field's pursuit of robust axioms of infinity. Epistemically, large cardinals occupy a position of relative consistency: the existence of weaker ones, such as inaccessible cardinals, is unprovable in ZFC but consistent relative to stronger hypotheses like measurable cardinals, forming a hierarchy where each level implies the consistency of the previous. This structure underscores their role in postulating the universe's inherent richness, transcending ZFC's limitations and enabling proofs of independence for key statements, such as the negation of the axiom of constructibility V = L. Philosophically, this raises debates between realism and formalism: realists, following Gödel's vision, view large cardinals as describing an objective "true" universe V, while formalists regard them as syntactic extensions that enrich the theory without ontological commitment. Penelope Maddy distinguishes "intrinsic" justifications, rooted in set-theoretic intuition like the iterative process, from "extrinsic" ones, based on fruitfulness in applications such as descriptive set theory, arguing that the latter currently predominates in justifying their adoption.^[1]^[2] Open questions persist regarding the existence of ultimate large cardinals and their boundaries, with the hierarchy suggesting no definitive endpoint, fueling the "large cardinal agenda" in modern set theory to explore escalating strengths and their implications for inner model theory. Joel David Hamkins' set-theoretic multiverse perspective posits multiple legitimate universes, each potentially realizing different large cardinals, challenging singular notions of truth and emphasizing pluralism over a unique V. Critiques portray large cardinals as potentially ad hoc, lacking direct intuitive grounding and relying on escalating assumptions that may overcommit ontologically, with speculative links to physics—such as analogies between infinite hierarchies and quantum field theory's infinities—highlighting risks of importing unverified commitments into foundational mathematics.^[41]^[42]^[43]

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