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Cofinality

In set theory, the cofinality of a limit ordinal \alpha, denoted \operatorname{cf}(\alpha), is defined as the smallest ordinal \beta such that there exists a cofinal function f: \beta \to \alpha, where the image of f is cofinal in \alpha (meaning \sup f[\beta] = \alpha).^[1] Equivalently, \operatorname{cf}(\alpha) is the least cardinality of a cofinal subset of \alpha, and this value is always a regular cardinal.^[2]^[3] Cofinality measures how "approachable" a limit ordinal is by smaller ordinals and plays a central role in distinguishing regular and singular cardinals: an infinite cardinal \kappa is regular if \operatorname{cf}(\kappa) = \kappa, and singular otherwise.^[1] For example, the smallest infinite ordinal \omega has \operatorname{cf}(\omega) = \omega, making it regular, while \omega_\omega = \sup\{\omega_n \mid n < \omega\} has \operatorname{cf}(\omega_\omega) = \omega < \omega_\omega, rendering it singular.^[3] Key properties include \operatorname{cf}(\alpha) \leq |\alpha| for any ordinal \alpha, and for limit ordinals, cofinal subsets are unbounded, ensuring \alpha = \bigcup_{\beta \in S} \beta if S \subset \alpha is cofinal.^[1]^[2] The concept extends to partially ordered sets and is foundational in advanced topics such as the study of large cardinals, forcing, and the continuum hypothesis, where cofinality constraints influence cardinal arithmetic and the existence of certain embeddings.^[1] Under the axiom of choice, cofinality coincides across equivalent definitions (e.g., via functions, monotone maps, or order types of cofinal subsets), but without it, distinctions may arise.^[1]

Definition and Fundamentals

Definition

In a partially ordered set (poset) (A, \leq), a subset B \subseteq A is said to be cofinal in A if for every element x \in A, there exists an element y \in B such that x \leq y. The cofinality of the poset A, denoted \cf(A), is defined as the least cardinality of any cofinal subset of A.^[4] This cardinality-based definition requires the axiom of choice to guarantee that cardinals are well-ordered and that the infimum over cofinal subsets corresponds to an actual minimal cardinality. An alternative formulation, particularly in the context of ordinal theory, defines the cofinality of A as the smallest ordinal \delta such that there exists a strictly increasing function f: \delta \to A whose image is cofinal in A. The notation for cofinality is commonly expressed as

\cf(A) = \inf \{ |B| : B \subseteq A \text{ is cofinal in } A \}.

This captures the minimal "size" needed to reach all elements of the poset from above.^[4]

Cofinal Subsets

In a partially ordered set (poset) (A, \leq), a subset B \subseteq A is cofinal in A if for every a \in A, there exists b \in B such that a \leq b.^[2] This property ensures that B "reaches" all upper levels of the poset, making it a fundamental structure for analyzing the order's "end behavior." Cofinal subsets need not possess any particular internal structure, such as being a chain (totally ordered) or an antichain (pairwise incomparable); their elements can interrelate in arbitrary ways under the induced order from A. In the special case of totally ordered sets, the axiom of choice guarantees the existence of a well-ordered cofinal subset.^[5] Under the axiom of choice, every linear order admits a cofinal subset that is well-ordered, allowing the cofinality to be realized as the order type of such a subset. This contrasts with general posets, where cofinal subsets may lack such regularity. Minimal cofinal subsets, those of cardinality equal to the cofinality \mathrm{cf}(A), exist in any poset under the axiom of choice, as the well-ordering of cardinals permits selecting a cofinal subset of the smallest possible size.^[6] However, these minimal cofinal subsets are not necessarily unique up to order isomorphism; distinct posets or even isomorphic posets can admit minimal cofinal subsets with non-isomorphic induced orders, reflecting the flexibility of the structure. In directed posets—those where every pair of elements has an upper bound—the cofinality \mathrm{cf}(A) can equivalently be characterized as the minimal cardinality of the domain I of an unbounded map f: I \to A, where the image f(I) is cofinal in A.^[7] This perspective emphasizes the role of cofinal subsets as images of functions that "unbound" the poset by covering its upper extents without a global maximum.

Examples

Examples in General Posets

In finite posets that possess a maximum element, the cofinality is 1, as the singleton consisting of that maximum element forms a cofinal subset.^[8] Any cofinal subset must include all maximal elements, and in the presence of a greatest element, this reduces the minimal cardinality of such a subset to 1.^[9] The poset of natural numbers (\mathbb{N}, \leq) under the standard ordering provides a simple infinite example. Its cofinality is \aleph_0, the smallest infinite cardinal, because every cofinal subset must be unbounded above, and no finite subset can achieve this, while countable unbounded subsets exist.^[8] Representative countable cofinal subsets include the even natural numbers \{2k \mid k \in \mathbb{N}\}, since for any n \in \mathbb{N} there exists k such that $2k \geq n, or the prime numbers, which are unbounded by Euclid's theorem.^[8] The poset of real numbers (\mathbb{R}, \leq) under the standard ordering similarly exhibits cofinality \aleph_0. A countable cofinal subset is the natural numbers \mathbb{N}, as for every real x there is some n \in \mathbb{N} with n \geq x.^[10] The rational numbers \mathbb{Q} also serve as a countable cofinal subset, owing to their density in \mathbb{R} and lack of upper bound.^[10] Consider the poset of all finite subsets of \mathbb{N}, denoted [\mathbb{N}]^{<\omega}, ordered by inclusion \subseteq. This poset has cofinality \aleph_0, as it admits a countable cofinal chain but no finite cofinal subset. The chain \{ \{1, 2, \dots, n\} \mid n \in \mathbb{N} \} is cofinal, since for any finite F \subseteq \mathbb{N}, choosing n > \max F ensures F \subseteq \{1, 2, \dots, n\}.^[11]

Examples for Ordinals and Cardinals

For successor ordinals, which are of the form \alpha = \beta + 1 for some ordinal \beta, the cofinality is 1. This follows from the fact that the singleton set \{\beta\} is cofinal in \alpha, as \beta is the unique immediate predecessor, and no smaller cofinal subset exists since cofinality is defined as the least order type of a cofinal increasing sequence.^[12] The first infinite ordinal \omega, which is the order type of the natural numbers, has cofinality \omega = \aleph_0. Any cofinal subset of \omega must be unbounded and thus infinite, requiring at least countably many elements to approach the supremum, while the identity sequence on \omega itself provides a cofinal map of order type \omega. This illustrates how the cofinality of a limit ordinal can equal its own order type when no smaller unbounded sequence suffices.^[12] Consider \omega^2, the ordinal obtained as the supremum of \omega \cdot n for finite n < \omega. Its cofinality is \omega, achieved by the increasing sequence \langle \omega \cdot n \mid n < \omega \rangle, which is cofinal since \sup_{n < \omega} \omega \cdot n = \omega^2. No finite sequence can be cofinal, as \omega^2 is a limit ordinal, but the countable length suffices, demonstrating how limit ordinals larger than \omega can still have cofinality \omega.^[12] The first uncountable cardinal \aleph_1, which is also the smallest uncountable ordinal \omega_1, has cofinality \aleph_1 under the standard assumptions of ZFC set theory, as it is a regular cardinal. This means there is no cofinal sequence of length less than \aleph_1, such as countable or smaller; any attempt to bound it with fewer than \aleph_1 many ordinals below \omega_1 fails to reach the supremum. Regularity here highlights a contrast to singular limits, where cofinality is strictly smaller.^[12] Finally, \aleph_\omega, the least upper bound of the sequence \langle \aleph_n \mid n < \omega \rangle of the first \omega infinite cardinals, is a singular cardinal with cofinality \aleph_0 = \omega. The increasing enumeration \langle \aleph_n \mid n < \omega \rangle forms a countable cofinal sequence in \aleph_\omega, and no smaller (finite) length works since it is a limit cardinal. This example underscores how fixed points in the aleph function can exhibit countable cofinality, influencing behaviors in cardinal arithmetic and forcing extensions.^[12]

Properties

Basic Properties

The cofinality of a partially ordered set A, denoted \mathrm{cf}(A), is defined as the least cardinality of a cofinal subset of A. For any non-empty poset A, \mathrm{cf}(A) \geq 1, since A itself serves as a cofinal subset.^[4] If A possesses a maximum element m, then \mathrm{cf}(A) = 1, as the singleton \{m\} is cofinal in A.^[4] The cofinal relation exhibits transitivity: if B is a cofinal subset of A and C is a cofinal subset of B, then C is cofinal in A. This property arises directly from the transitivity of the partial order on A.^[4] Cofinality is preserved under order-isomorphisms: if A and B are order-isomorphic, then \mathrm{cf}(A) = \mathrm{cf}(B).^[4]

Monotonicity and Preservation

For ordinal sums, consider limit ordinals \alpha and \beta. The cofinality of the sum \alpha + \beta equals \mathrm{cf}(\beta). To see this, note that \beta embeds order-preservingly as the terminal segment of \alpha + \beta, and this embedding is cofinal, so \mathrm{cf}(\beta) \leq \mathrm{cf}(\alpha + \beta). Conversely, any cofinal subset of \alpha + \beta must eventually lie in the terminal segment \beta, implying \mathrm{cf}(\alpha + \beta) \leq \mathrm{cf}(\beta).^[13] Ordinal products under lexicographic order also exhibit structured cofinality behavior. For ordinals \alpha > 0 and limit ordinal \beta, the cofinality of the lexicographic product \alpha \times \beta—ordered by (a_1, b_1) < (a_2, b_2) if b_1 < b_2 or (b_1 = b_2 and a_1 < a_2)—is \mathrm{cf}(\beta). This arises because the order prioritizes the \beta-coordinate, requiring a cofinal subset to be unbounded in \beta while the \alpha-copies contribute to the structure within each level.^[13]

Cofinality in Well-Ordered Sets

Ordinals

In ordinal arithmetic, the cofinality of an ordinal \alpha, denoted \mathrm{cf}(\alpha), is defined as the smallest ordinal \delta such that there exists a strictly increasing function f: \delta \to \alpha whose range is cofinal in \alpha, meaning \sup \mathrm{ran}(f) = \alpha.^[14]^[15] This definition captures the minimal "length" required to approach \alpha from below via an increasing sequence of ordinals less than \alpha. For successor ordinals, \mathrm{cf}(\alpha + 1) = 1, as the singleton sequence consisting of \alpha itself is cofinal in \alpha + 1.^[14] In contrast, for limit ordinals \alpha, \mathrm{cf}(\alpha) is the order type of the shortest strictly increasing cofinal sequence in \alpha, ensuring that the supremum of the sequence equals \alpha.^[15] This distinguishes limit ordinals by requiring an infinite approach, with \mathrm{cf}(\alpha) always a regular cardinal less than or equal to \alpha. Closed unbounded (club) sets in ordinals provide a key framework for studying cofinality. A subset C \subseteq \alpha is club if it is closed under limits (containing all limit points of its subsets) and unbounded in \alpha (intersecting every initial segment). For a regular limit ordinal \alpha, the intersection of fewer than \alpha club subsets of \alpha is itself a club subset, preserving cofinality \mathrm{cf}(\alpha) = \alpha in the sense that its order type is at least \alpha and cofinal in \alpha.^[15]

Other Well-Ordered Sets

In set theory, every well-ordered set W is order-isomorphic to a unique ordinal \alpha, known as its order type, and consequently, the cofinality of W, denoted \operatorname{cf}(W), equals \operatorname{cf}(\alpha).^[16] This isomorphism ensures that properties of cofinality transfer directly from the ordinal to the set, allowing the study of well-ordered sets to leverage ordinal arithmetic and limits.^[2] Cofinal subsets of a well-ordered set W inherit the well-ordering from the induced subspace topology, making them well-ordered themselves.^[2] The order type of any such cofinal subset is at least \operatorname{cf}(W), and \operatorname{cf}(W) is precisely the least ordinal \beta admitting an order-preserving cofinal function f: \beta \to W.^[17] This minimal \beta characterizes the "length" of the shortest unbounded increasing sequence approaching the end of W. Unlike in general partially ordered sets, where cofinal subsets may lack any chain structure, in well-ordered sets, cofinality aligns with the existence of normal functions when W has ordinal type \alpha. A normal function on \alpha is a strictly increasing, continuous function f: \operatorname{cf}(\alpha) \to \alpha whose range is cofinal in \alpha.^[18] Such functions generate the club filter on \alpha, consisting of closed unbounded subsets, which forms a filter base for studying stationary sets and reflection principles in well-orders.^[19] For concrete illustrations beyond pure ordinals, consider well-ordered structures arising in ordered field extensions or scattered linear orders restricted to well-ordered components. In non-archimedean ordered fields like Hahn series over well-ordered supports, the cofinality of the value group (itself well-ordered) matches that of its terminal ordinal segment, determining the overall approach to "infinity" in the field.^[20] Similarly, in scattered linear orders—those without dense rational subcopies—the well-ordered terminal segments have cofinality equal to the supremum of their preceding ranks in the Hausdorff derivative hierarchy.^[21] These examples highlight how cofinality captures the unbounded ascent in well-ordered tails of broader ordered structures.

Regularity and Singularity

Regular Ordinals and Cardinals

In set theory, a regular ordinal \alpha is defined as an ordinal equal to its own cofinality, i.e., \mathrm{cf}(\alpha) = \alpha. This means that there is no cofinal subset of \alpha with order type strictly smaller than \alpha, capturing the idea that \alpha cannot be "approached" by a shorter increasing sequence of ordinals below it.^[22] Successor ordinals are regular by definition, as they are not limit ordinals.^[1] Examples of regular ordinals include all successor ordinals and certain limit ordinals. The smallest infinite ordinal \omega is regular, as any cofinal sequence in \omega must itself have order type \omega, with no finite or smaller subsequence unbounded in the natural numbers.^[23] Similarly, the first uncountable ordinal \omega_1 is regular in ZFC set theory, meaning its cofinality is itself, regardless of whether the continuum hypothesis holds; any countable cofinal subset would contradict the uncountability of \omega_1.^[22] A regular cardinal \kappa is an infinite cardinal satisfying \mathrm{cf}(\kappa) = \kappa, implying that \kappa cannot be expressed as the union of fewer than \kappa many sets each of cardinality less than \kappa.^[24] This property ensures that regular cardinals are "indivisible" in terms of smaller cardinal sums. Examples include the smallest infinite cardinal \aleph_0 = |\omega|, which is regular since the union of finitely many finite sets remains finite, and all successor cardinals \aleph_{\alpha + 1}, which inherit regularity from their ordinal structure.^[25] All inaccessible cardinals are regular, as their definition requires uncountable regularity alongside limit and strong limit properties.^[25] Strong limit regular cardinals, where \kappa is regular and $2^\lambda < \kappa for all \lambda < \kappa, coincide with strongly inaccessible cardinals, which are also weakly inaccessible (uncountable regular limit cardinals).^[22]

Singular Ordinals and Cardinals

A singular ordinal is defined as a limit ordinal \alpha for which the cofinality \mathrm{cf}(\alpha) < \alpha.^[22] Such ordinals arise as the supremum of a sequence of smaller ordinals of length strictly less than \alpha itself, distinguishing them from regular limit ordinals where the cofinality equals the ordinal. Singular ordinals are defined only for limit ordinals; successor ordinals are always regular.^[22] A representative example of a singular ordinal is \omega \cdot \omega = \sup\{\omega \cdot n : n < \omega\}, which has cofinality \omega since it is the least upper bound of the countable sequence \omega, \omega \cdot 2, \omega \cdot 3, \dots.^[22] Another key example is the ordinal \aleph_\omega, the least upper bound of the sequence \aleph_0 < \aleph_1 < \aleph_2 < \dots < \aleph_n < \dots, which also has cofinality \aleph_0 = \omega.^[22] Turning to cardinals, a singular cardinal \kappa is an infinite cardinal satisfying \mathrm{cf}(\kappa) < \kappa.^[22] In this case, the cofinality \mathrm{cf}(\kappa) is itself a regular cardinal, ensuring that the "approach" to \kappa cannot be further singularized in a trivial way. Examples include \aleph_\omega, the first fixed point of the aleph function and the smallest singular cardinal, which can be expressed as the union of countably many smaller cardinals \aleph_n for n < \omega.^[22] Similarly, \beth_\omega = \sup\{\beth_n : n < \omega\}, where \beth_n denotes the n-th beth cardinal starting from \beth_0 = \aleph_0, is a singular cardinal of cofinality \omega.^[22] A significant conjecture related to singular cardinals is Shelah's singular cardinal hypothesis (SCH), which addresses the behavior of cardinal exponentiation at singular points. For a singular strong limit cardinal \kappa of uncountable cofinality, SCH asserts that $2^\kappa = \kappa^+, while more generally, for singular \kappa with \mathrm{cf}(\kappa) = \mu, it states that \kappa^\mu = \max(\kappa^+, 2^\mu).^[22] This hypothesis, which follows from the generalized continuum hypothesis (GCH) and holds in certain models involving supercompact cardinals, provides bounds on power sets and products involving singular cardinals, influencing much of modern cardinal arithmetic.^[22]

Advanced Applications

König's Theorem

König's theorem is a fundamental result in cardinal arithmetic that relates the cofinality of a cardinal to the size of its powers. For an infinite cardinal \kappa, it asserts that \kappa^{\mathrm{cf}(\kappa)} > \kappa.^[26] More generally, if \delta = \mathrm{cf}(\kappa) and \{\kappa_i \mid i < \delta\} is a family of cardinals with \kappa_i < \kappa for each i < \delta and \sum_{i < \delta} \kappa_i = \kappa, then \prod_{i < \delta} \kappa_i > \kappa.^[27] The proof of the specific form proceeds without the axiom of choice via a diagonal argument. Let \lambda = \mathrm{cf}(\kappa) and fix a cofinal function g: \lambda \to \kappa. To show there is no surjection from \kappa onto the set of all functions from \lambda to \kappa, suppose for contradiction that f: \kappa \to {}^\lambda \kappa is such a surjection. Define a diagonal function d: \lambda \to \kappa by d(\xi) = f(g(\xi))(\xi) + 1. Then d differs from f(\alpha) at \xi where g(\xi) > \alpha, ensuring d is not in the range of f, a contradiction.^[26] The general form follows similarly by considering injections between disjoint unions and products.^[27] A key corollary is that \mathrm{cf}(2^\kappa) > \kappa for any infinite cardinal \kappa. This follows because $2^\kappa \leq (2^\kappa)^\kappa = 2^{\kappa \cdot \kappa} = 2^\kappa, so if \mathrm{cf}(2^\kappa) \leq \kappa, then by the theorem applied to $2^\kappa, we would have (2^\kappa)^{\mathrm{cf}(2^\kappa)} > 2^\kappa, contradicting the equality.^[28] In particular, for \kappa = \aleph_0, the cofinality of the continuum exceeds \aleph_0. This is consistent with the continuum hypothesis (where $2^{\aleph_0} = \aleph_1 and \mathrm{cf}(\aleph_1) = \aleph_1 > \aleph_0) and implies that if CH fails, the continuum cannot have countable cofinality.^[28] This theorem also implies that no singular strong limit cardinal of cofinality \aleph_0 can exist below the first inaccessible cardinal. Suppose \kappa is such a cardinal; then \kappa^\omega = \kappa since for any \mu < \kappa, \mu^\omega \leq 2^\mu < \kappa by strong limit property, and the union over countably many such bounds remains below \kappa. However, König's theorem yields \kappa^\omega > \kappa, a contradiction. Since the first inaccessible is the least regular strong limit, all strong limits below it are singular, reinforcing the absence of such examples.^[27]

Implications for Large Cardinals

Measurable cardinals represent a foundational large cardinal notion where cofinality plays a critical role in their definition and properties. A measurable cardinal κ is a regular uncountable cardinal equipped with a non-principal κ-complete ultrafilter, implying that cf(κ) = κ, as any smaller cofinality would contradict the completeness of the ultrafilter.^[29] Moreover, measurable cardinals are strong limit cardinals, meaning that for every λ < κ, 2^λ < κ; while singular strong limit cardinals are consistent with ZFC, the regularity of measurables distinguishes them.^[30] This regularity ensures that measurable cardinals cannot be collapsed to singular ones without significant forcing interventions, such as Prikry forcing, which preserves measurability but alters cofinality in extensions.^[31] The singular cardinals problem investigates bounds on 2^κ for singular strong limit cardinals κ, particularly those with uncountable cofinality, where the singular cardinals hypothesis (SCH) posits that 2^κ = κ^+ under the generalized continuum hypothesis below κ. Post-2000 developments have partially resolved aspects of SCH by demonstrating its consistency failure without relying on excessively strong large cardinal assumptions in some cases. For instance, Gitik and Koepke (2012) constructed a model where ℵ_ω, a singular cardinal of countable cofinality, is strong limit but 2^{ℵ_ω} > ℵ_{ω+1} via choiceless forcing, without large cardinals; failures at singular cardinals of uncountable cofinality typically require stronger assumptions like measurable cardinals.^[32] These constructions highlight how countable cofinality allows milder violations of SCH while preserving ZFC consistency. As of 2023, further results show the consistency of SCH failure at ℵ_ω together with the reflection of all stationary subsets of ℵ_ω.^[33] Shelah's PCF theory provides a framework for analyzing how cofinality influences the structure of power sets via the set of possible cofinalities, pcf(A), for a set A of regular cardinals, which enumerates cofinalities of reduced products ∏_{a ∈ A} a / D for ideals D on A. In this theory, the cofinality of singular cardinals determines the spectrum of possible cofinalities for ultrapowers of the power set, restricting the cardinality of 2^κ to lie below certain bounds derived from max pcf(A) when |A| < cf(κ). For singular κ of uncountable cofinality, PCF theory implies that the cofinality spectrum of subsets of κ narrows, yielding upper bounds on 2^κ that align with SCH in many cases but allow controlled failures in forcing extensions.^[34] This approach has been pivotal in proving that cofinalities dictate the arithmetic of power sets without invoking choice principles fully.^[35] Recent advances in singular cardinal combinatorics, particularly through inner model theory and forcing, have explored implications for cf(ℵ_ω) in extensions where large cardinals are preserved. For example, iteration schemes using Σ-Prikry forcings have produced models where a singular cardinal κ of uncountable cofinality violates SCH while maintaining reflection principles at κ^+, demonstrating that inner models can embed such behaviors without collapsing cardinals.^[36] These developments, building on core model induction, show that forcing extensions can alter cf(ℵ_ω) to countable while keeping ℵ_ω strong limit, with implications for the consistency strength of SCH failures at higher singulars.^[37] Such results refine our understanding of how cofinality interacts with inner models to bound power set growth in post-2020 constructions.

References

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Jun 11, 2015 · Definition 8 (Cofinal and unbounded subsets) A subset X of an ordered set Y is cofinal (resp. unbounded) if, for all y ∈ Y , there is x ∈ X such ...
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The cofinality of an ordinal is always a cardinal. Hence alternatively we can define the cofinality of \alpha as the least cardinality of a cofinal subset of \ ...
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Todorcevic [8] studies such 'cofinality types' for posets of cofinality ℵ1; it turns out that, depending on the axioms of set theory assumed, there can be as ...
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... Every totally ordered set has a well-ordered cofinal subset (§2,. Exercise 2). The least of the ordinals Ord (M) of the well-ordered cofinal subsets M of E ...
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Feb 8, 2003 · We study which infinite posets have simple cofinal subsets such as chains, or decompose canonically into such subsets. The posets of countable ...
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[PDF] Cofinal types of ultrafilters - Dilip Raghavan
We say that a poset hD, ≤i is directed if any two members of D have an upper bound in D. A set X ⊂ D is unbounded in D if it doesn't have an upper bound in. D.
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Mar 22, 2013 · Let (P,≤) ( P , ≤ ) be a poset. A subset A⊆P A ⊆ P is said to be cofinal in P P if for every x∈P x ∈ P there is a y∈A y ∈ A such that x≤y x ...
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... define the cofinality of (P,≤), cf(P,≤), to be the smallest ... Set Theory · Functional Analysis · Order, Lattices, Ordered Algebraic Structures ...
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Rating 3.0 (2) ... cofinality of the reals is therefore . We have therefore the following invariants defining the nature of a real closed field F: The cardinality of F. The ...
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May 1, 2020 · A collection A of finite subsets of N is cofinal with respect to inclusion if it has the following property: for each finite F⊆N there is an A∈A such that F⊆A.Intuitive approach to cofinality - set theory - Math Stack ExchangeShow that the set of all finite subsets of N is countable.More results from math.stackexchange.com
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**Summary of Cofinality Definitions and Examples**
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Axiom of Choice. Every family of nonempty sets has a choice func- tion. The theory with axioms 1.1–1.8 is the Zermelo-Fraenkel axiomatic set.
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Below is a merged summary of cofinality for ordinals from Enderton’s *Elements of Set Theory* (1977), consolidating all the information from the provided segments into a single, comprehensive response. To maximize detail and clarity, I will use a table in CSV format for key information (definitions, properties, page references, and URLs), followed by a narrative summary that ties everything together. Since the system has a "no thinking token" limit, I’ll focus on directly synthesizing the provided data without additional inference.
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None
Below is a merged summary of cofinality, club sets, and related concepts from Jech's *Set Theory*, consolidating all information from the provided segments into a comprehensive response. To maximize detail and clarity, I will use a structured format with tables where appropriate, followed by additional narrative explanations for concepts that are less tabular in nature. The response retains all mentioned information, organized by key topics: cofinality, club sets, and regular ordinals, with page references, definitions, properties, and relations.
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Subsection 4.7.1 (03PV): Ordinals and Well-Orderings—Kerodon
We refer to \mathrm{cf}(T) as the cofinality of the linearly ordered set T. If \beta is an ordinal, let \mathrm{cf}(\beta ) denote the cofinality \mathrm{cf}( ...
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If α is a limit ordinal of uncountable cofinality, then the intersection of fewer than cf(α) clubs of α is a club of α. Example 8. If, in Theorem 7, we drop ...
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Here cf is the cofinality of an ordinal under the canonical well-ordering. We prove Theorem 3 in Section 5. The key element of the proof is a new reflection.
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[PDF] Equimorphism invariants for scattered linear orderings
The cofinality of P, cf(P), is the least cardinal κ such that there is a cofinal set of P of size κ. A regular cardinal is one whose cofinality is itself.
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Oct 8, 2014 · Set theory is the mathematical theory of well-determined collections, called sets, of objects that are called members, or elements, of the set.
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regular cardinal in nLab
### Summary of Regular Cardinal from nLab
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Finally, we refine the distinction between regular and singular cardinals by introducing the cofinality of a cardinal κ ; as it will turn out, a cardinal is ...<|separator|>
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[PDF] Handout: K˝onig's Lemma - Hugo Nobrega
Theorem (K˝onig's Lemma). For every cardinal κ, we have κcf(κ) > κ. Proof. We write λ := cf(κ) ≤ κ. Fix some cofinal function g : λ → κ. If λ = κ, the ...
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### Summary of König's Theorem from nLab
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If ν is a measurable cardinal, then Prikry forcing converts ν to a singular cardinal of countable cofinality. By Dodd and Jensen, if ν is a regular cardinal and ...
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Aug 5, 2025 · 2 MOTI GITIK AND PETER KOEPKE Isr. J. Math. Note that in the presence of the axiom of choice (AC) the latter theory for λ>ℵω+2 has large.
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The purpose of this paper is to give a self-contained survey of the basic elements of the theory of possible cofinalities and some of the main applications. The ...
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Jun 19, 2019 · ... cofinal set of cardinality κ; provided there is at least one true cofinal set,. 4. ≺-true cofinality of X, denoted tcf≺(X): The least ...
[35]
[PDF] A new iteration scheme with applications to singular cardinals ...
Suppose V ⊆ W are two inner models where a cardinal κ is a V -inaccessible but. W-singular with cfW (κ) = ω. If moreover (κ+)V = (κ+)W then W |= Dκ,ω. Page ...Missing: advances 2020-2025
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[PDF] a brief account of recent developments in inner model theory
The goal of this survey paper is to give an overview of recent developments in inner model theory. We discuss several most important questions in the field ...Missing: 2020-2025 ℵ_ω)