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Size of subsets in order theory
of cofinality relies on the axiom of choice, as it uses the fact that every non-empty set of cardinal numbers has a least member. The cofinality of a
Cofinality
Mathematical property of subsets in order theory
where the minimum possible cardinality of a cofinal subset of A {\displaystyle A} is referred to as the cofinality of A . {\displaystyle A.} Let ≤ {\displaystyle
Cofinal_(mathematics)
Generalization of "n-th" to infinite cases
The cofinality of any ordinal α is a regular ordinal, i.e. the cofinality of the cofinality of α is the same as the cofinality of α. So the cofinality operation
Ordinal_number
Infinite cardinal number
cardinals with cofinality ℵ 0 {\displaystyle \aleph _{0}} . An uncountably infinite cardinal κ {\displaystyle \kappa } having cofinality ℵ 0 {\displaystyle
Aleph_number
Topics referred to by the same term
there is a "larger element" in B Cofinality (mathematics), the least cardinality of a cofinal subset in this sense Cofinal (music), a part of some Gregorian
Cofinal
System of pitch organization in Gregorian chant
below, C, remained the lower limit. In addition to the range, the tenor (cofinal, or dominant, corresponding to the "reciting tone" of the psalm tones)
Gregorian_mode
Field in mathematics similar to the real numbers
Archimedean property is related to the concept of cofinality. A set X contained in an ordered set F is cofinal in F if for every y in F there is an x in X such
Real_closed_field
Proposition in mathematical logic
disproof and established Kőnig's theorem, which by using the concept of cofinality introduced in 1908 by Felix Hausdorff, shows that result that 2 ℵ 0 {\displaystyle
Continuum_hypothesis
mathematical theory, introduced by Saharon Shelah (1978), that deals with the cofinality of the ultraproducts of ordered sets. It gives strong upper bounds on
Pcf_theory
has cofinality ω in V. If β has cofinality 0, 1, or ω and α is any ordinal, then the least β-Silver ordinal strictly greater than α has cofinality ω in
Silver_cardinal
Type of large transfinite number
an initial subsequence of the cf(κ)-sequence. Thus its cofinality is less than the cofinality of κ and greater than it at the same time; which is a contradiction
Mahlo_cardinal
Mathematical theorem in set theory
{\displaystyle \kappa <\operatorname {cf} (2^{\kappa })} (where cf(α) is the cofinality of α) and if κ < λ then 2 κ ≤ 2 λ . {\displaystyle {\text{if }}\kappa
Easton's_theorem
Theorem in set theory
the theorem. Kőnig's theorem has also important consequences for the cofinality of cardinal numbers. If κ ≥ ℵ 0 {\displaystyle \kappa \geq \aleph _{0}}
Kőnig's_theorem_(set_theory)
Large cardinal property in set theory
into itself then α {\displaystyle \alpha } is either a limit ordinal of cofinality ω {\displaystyle \omega } or the successor of such an ordinal. The axioms
Rank-into-rank
Set-theoretic concept
restriction to uncountable cofinality is in order to avoid trivialities: Suppose κ {\displaystyle \kappa } has countable cofinality. Then S ⊆ κ {\displaystyle
Stationary_set
Set theory concept
following statement: 2cf(κ) < κ implies κcf(κ) = κ+, where cf denotes the cofinality function. Note that κcf(κ)= 2κ for all singular strong limit cardinals
Singular_cardinals_hypothesis
Generalization of the concept of subsequence to the case of nets
h ( I ) {\displaystyle h(I)} is cofinal in A . {\displaystyle A.} The set h ( I ) {\displaystyle h(I)} being cofinal in A {\displaystyle A} means that
Subnet_(mathematics)
Large cardinal number
worldly κ of cofinality ω1 (corresponds to the extension of the above item to a chain of length ω1). The least worldly κ of cofinality ω2 (and so on)
Worldly_cardinal
Well-quasi-ordering of finite trees
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Kruskal's_tree_theorem
continuum has size at least κ. Here, there is no restriction. If κ has cofinality ω, the cardinality of the reals ends up bigger than κ. Grigorieff forcing
List_of_forcing_notions
Mathematical result on order relations
consequence of the axiom of choice, the principle that every total order has a cofinal well-order, can be combined to prove the full axiom of choice. With these
Szpilrajn_extension_theorem
Class of cardinal numbers
\ldots \}=\bigcup _{n<\omega }\beth _{n}} is a strong limit cardinal of cofinality ω. More generally, given any ordinal α, the cardinal ℶ α + ω = ⋃ n < ω
Limit_cardinal
Theorem in axiomatic set theory
\kappa \mapsto \kappa ^{\mathrm {cf} (\kappa )}} where cf denotes the cofinality function; the gimel function is used for studying the continuum function
Gimel_function
Generalization of a category
a final map. Also, a map f : X → Y {\displaystyle f:X\to Y} is called cofinal if f : X o p → Y o p {\displaystyle f:X^{op}\to Y^{op}} is final. Presheaf
Quasi-category
Type of cardinal number in mathematics
theory, a regular cardinal is a cardinal number that is equal to its own cofinality. More explicitly, this means that κ {\displaystyle \kappa } is a regular
Regular_cardinal
American mathematician
group theory, semigroups, and cofinality in universal algebra. Her final publication, published posthumously, was "Cofinality of algebras" (1986).[F] During
Anne_C._Morel
Special subset of a partially ordered set
case where μ is counting measure. Given an ordinal a with uncountable cofinality, a subset of a is called a club if it is closed in the order topology
Filter_(mathematics)
Order-preserving mathematical function
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Monotonic_function
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Absolutely and completely monotonic functions and sequences
Absolutely_and_completely_monotonic_functions_and_sequences
Generalised alphabetical order
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Lexicographic_order
Set theory concept
set). Let κ {\displaystyle \kappa \,} be a limit ordinal of uncountable cofinality λ . {\displaystyle \lambda .} For some α < λ {\displaystyle \alpha <\lambda
Club_set
Mathematical ordering of a partial order
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Linear_extension
Reversal of the order of elements of a binary relation
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Converse_relation
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Locally_finite_poset
superior and limit inferior Irreducible element Prime element Compact element Cofinal and coinitial set, sometimes also called dense Meet-dense set and join-dense
List_of_order_theory_topics
Four mathematical theorems
category of free modules and D is the category of projective modules. Cofinality theorem—Let ( A , v ) {\displaystyle (A,v)} be a Waldhausen category that
Basic theorems in algebraic K-theory
Basic_theorems_in_algebraic_K-theory
Subset of incomparable elements
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Antichain
Mathematical function on ordinals
_{\alpha }(\beta )<\delta } ). The fundamental sequence for an ordinal with cofinality ω is a distinguished strictly increasing ω-sequence that has the ordinal
Veblen_function
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Reflexive_closure
Topics referred to by the same term
Partition regularity Regular cardinal, a cardinal number that is equal to its cofinality Regular modal logic Regular conditional probability, a concept that has
Regular
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Specialization_preorder
Characterizes the height of any finite partially ordered set
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Mirsky's_theorem
Term in the mathematical area of order theory
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Duality_(order_theory)
Infinite Cardinal number
indexed by ℶ {\displaystyle \beth } . On the other hand, beth numbers are cofinal (every cardinal number is less than a beth number) in plain Zermelo-Fraenkel
Beth_number
Glossary of terms used in branch of mathematics
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Glossary_of_order_theory
Cardinality of the set of real numbers
cases, equality can be ruled out by König's theorem on the grounds of cofinality (e.g. c ≠ ℵ ω {\displaystyle {\mathfrak {c}}\neq \aleph _{\omega }} )
Cardinality_of_the_continuum
There are equally many countable order types and real numbers
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Cantor–Bernstein_theorem
Subset of a preorder that contains all larger elements
isomorphism) in this way as the lattice of lower sets of a unique finite poset. Cofinal set – a subset U {\displaystyle U} of a partially ordered set ( X , ≤ )
Upper_and_lower_sets
Mathematical set with an ordering
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Partially_ordered_set
Mathematical ordering with upper bounds
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Directed_set
Mathematical relation inside orderings
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Covering_relation
Order whose elements are all comparable
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Total_order
On chains and antichains in partial orders
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Dilworth's_theorem
Type of monotone function
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Order_embedding
Visual depiction of a partially ordered set
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Hasse_diagram
Partially ordered set in which all subsets have both a supremum and infimum
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Complete_lattice
Ideals in a Boolean algebra can be extended to prime ideals
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Boolean_prime_ideal_theorem
Certain topology in mathematics
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Order_topology
Type of logical relation
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Total_relation
Mathematical result or axiom on order relations
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Hausdorff_maximal_principle
Concept in set theory
_{2}} to an ordinal of cofinality ω {\displaystyle \omega } . Let G {\displaystyle G} be an ω {\displaystyle \omega } -sequence cofinal on ω 2 L {\displaystyle
Zero_sharp
Type of topology in mathematics
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Alexandrov_topology
Mathematical concept
closed and unbounded subset of κ, so that for every λ in C of uncountable cofinality, there is an unbounded subset of λ that is homogenous for f; slightly
Ramsey_cardinal
Property of elements related by inequalities
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Comparability
set cofinal A subset of a poset is called cofinal if every element of the poset is at most some element of the subset. cof cofinality cofinality 1. The
Glossary_of_set_theory
theory, the notion of final functor is a generalization of the notion of cofinal set from order theory. A functor F : C → D {\displaystyle F:C\to D} is
Final_functor
Partially ordered vector space, ordered as a lattice
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Riesz_space
Construction in order theory
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Star_product
I)(\exists A\in {\mathcal {A}})(B\subseteq A){\big \}}.} The "cofinality" of I is the cofinality of the partial order (I, ⊆). It is easy to see that we must
Cichoń's_diagram
Alternative mathematical ordering
quotient L / Z, where L is a linearly ordered group and Z is a cyclic cofinal subgroup of L. Every cyclically ordered group can also be expressed as
Cyclic_order
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Better-quasi-ordering
Non-empty family of sets that is closed under finite unions and subsets
numbers. If λ {\displaystyle \lambda } is an ordinal number of uncountable cofinality, the nonstationary ideal on λ {\displaystyle \lambda } is the collection
Ideal_on_a_set
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Series-parallel_partial_order
Set whose pairs have minima and maxima
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Lattice_(order)
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
List of order structures in mathematics
List_of_order_structures_in_mathematics
with the order topology. These continuous functions are often used in cofinalities and cardinal numbers. A normal function is a function that is both continuous
Continuous function (set theory)
Continuous_function_(set_theory)
Generalization of the real numbers
class of ordinal numbers, and because O n {\textstyle \mathbb {On} } is cofinal in N o {\textstyle \mathbb {No} } we have { N o ∣ } = { O n ∣ } = O n {\textstyle
Surreal_number
Type of ordering of a set
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Dense_order
Collection of mathematical objects of finite size
Euclidean distance. A class of ordinal numbers is said to be unbounded, or cofinal, when given any ordinal, there is always some element of the class greater
Bounded_set
Reflexive and transitive binary relation
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Preorder
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Prefix_order
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Symmetric_closure
Mathematical proposition equivalent to the axiom of choice
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Zorn's_lemma
Nonempty, upper-bounded, downward-closed subset
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Ideal_(order_theory)
Construction in order theory
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Product_order
Concept in order theory
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Join_and_meet
Generalization of a sequence of points
x_{\bullet }=\left(x_{a}\right)_{a\in A}} is said to be frequently or cofinally in S {\displaystyle S} if for every a ∈ A {\displaystyle a\in A} there
Net_(mathematics)
Israeli mathematician
Arrow property". arXiv:math/0112213. Malliaris, M.; Shelah, S. (2016). "Cofinality spectrum theorems in model theory, set theory, and general topology".
Saharon_Shelah
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Eulerian_poset
Algebraic object with an ordered structure
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Ordered_field
Partially ordered set equipped with a rank function
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Graded_poset
Extreme element of a preorder
{\displaystyle Q} of a partially ordered set P {\displaystyle P} is said to be cofinal if for every x ∈ P {\displaystyle x\in P} there exists some y ∈ Q {\displaystyle
Maximal_and_minimal_elements
Branch of mathematics
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Order_theory
Equivalence of partially ordered sets
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Order_isomorphism
Mathematical operation
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Composition_of_relations
Size of a possibly infinite set
κcf(κ) and κ < cf(2κ) for any infinite cardinal κ, where cf(κ) is the cofinality of κ. Assuming the axiom of choice and, given an infinite cardinal κ and
Cardinal_number
Sequence of points that get progressively closer to each other
multiples of p r . {\displaystyle p_{r}.} If H {\displaystyle H} is a cofinal sequence (that is, any normal subgroup of finite index contains some H
Cauchy_sequence
Graph linking pairs of comparable elements in a partial order
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Comparability_graph
Algebraic structure modeling logical operations
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Boolean_algebra_(structure)
American mathematician (1942–2016)
Problem", Silver proved that if a cardinal κ is singular with uncountable cofinality and 2λ = λ+ for all infinite cardinals λ < κ, then 2κ = κ+. Prior to Silver's
Jack_Silver
COFINALITY
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COFINALITY
Girl/Female
Arabic, Muslim
Precious Stone; Expensive Jewel
Girl/Female
Tamil
Grace, Beauty
Boy/Male
Hindu, Indian
Lotus (Water Lily)
Male
Greek
(Τελαμών) Greek myth name of the father of Ajax, possibly TELAMON means "support."
Male
Native American
Native American Algonquin name MUKKI means "child."
Boy/Male
Muslim/Islamic
The First
Girl/Female
Indian
Matchless, Alone, Unique, Goddess Durga
Girl/Female
Indian
Wishes, Aspirations
Girl/Female
Hindu, Indian, Traditional
Crest Jewel
Boy/Male
Hindu
COFINALITY
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COFINALITY
COFINALITY
COFINALITY