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In set theory, a cardinal number is called weakly inaccessible if it is an uncountable regular weak limit cardinal and strongly inaccessible, or just inaccessible, if it is an uncountable regular strong limit cardinal. Some authors do not require weakly and strongly inaccessible cardinals to be uncountable. Set theory is the mathematical theory of sets, which represent collections of abstract objects. ...
In linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to ordinal numbers, words that are used for order (first, second, third). ...
In set theory, a regular cardinal is an infinite well-orderable cardinal whose initial ordinal is regular, where a regular ordinal is an ordinal which is equal to its own cofinality. ...
In mathematics, limit cardinals are a type of cardinal number. ...
In mathematics, limit cardinals are a type of cardinal number. ...
In mathematics, an uncountable or nondenumerable set is a set which is not countable. ...
Every strongly inaccessible cardinal is also weakly inaccessible, as every strong limit cardinal is also a weak limit cardinal. If the Generalized Continuum Hypothesis holds, then a cardinal is strongly inaccessible if and only if it is weakly inaccessible. In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ...
 (aleph-null) is a regular strong limit cardinal. Assuming the axiom of choice, every other infinite cardinal number is either regular or a limit. However, only a rather large cardinal number can be both and thus weakly inaccessible. In the branch of mathematics known as set theory, the aleph numbers are a series of numbers used to represent the cardinality (or size) of infinite sets. ...
In mathematics, the axiom of choice, or AC, is an axiom of set theory. ...
An ordinal is a weakly inaccessible cardinal if and only if it is a regular ordinal and it is a limit of regular ordinals. (Zero, one, and are regular ordinals, but not limits of regular ordinals.) A cardinal which is weakly inaccessible and also a strong limit cardinal is strongly inaccessible. Commonly, ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ...
The assumption of the existence of a strongly inaccessible cardinal is sometimes applied in the form of the assumption that one can work inside a Grothendieck universe, the two ideas being intimately connected. In mathematics, a Grothendieck universe is a non-empty set U with the following properties: If x U and if y x, then y U. If x,y U, then {x,y} U. If x U, then P(x) U. (P(x) is the power set of x. ...
Models and consistency ZFC implies that the Vκ is a model of ZFC whenever κ is strongly inaccessible. And ZF implies that the Gödel universe Lκ is a model of ZFC whenever κ is weakly inaccessible. Thus ZF together with "there exists a weakly inaccessible cardinal" implies that ZFC is consistent. Therefore, inaccessible cardinals are a type of large cardinal. The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based in modern formulations. ...
In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the models which underlie mathematical systems. ...
In mathematics, the constructible universe (or Gödels constructible universe), denoted L, is a particular class of sets which can be described entirely in terms of simpler sets. ...
In mathematics, a cardinal is called a large cardinal if it belongs to a class of cardinals, the existence of which provably cannot be proved within the standard axiomatic set theory ZFC, if one assumes ZFC itself is consistent. ...
Suppose V is a model of ZFC. Either V contains no strong inaccessible or, taking κ to be the smallest strong inaccessible in V, Vκ is a standard model of ZFC which contains no strong inaccessibles. So consistency of ZFC implies consistency of ZFC+"there are no strong inaccessibles". Similarly, either V contains no weak inaccessible or, taking κ to be the smallest ordinal which is weakly inaccessible relative to any standard sub-model of V, then Lκ is a standard model of ZFC which contains no weak inaccessibles. So consistency of ZFC implies consistency of ZFC+"there are no weak inaccessibles". Zermelo-Fraenkel set theory, commonly abbreviated ZFC, is the most common form of axiomatic set theory, and as such is the most common foundation of mathematics. ...
If V is a standard model of ZFC and κ is an inaccessible in V, then: Vκ is one of the intended models of Zermelo–Fraenkel set theory; and Def (Vκ) is one of the intended models of Von Neumann–Bernays–Gödel set theory; and Vκ+1 is one of the intended models of Morse–Kelley set theory. Here Def (X) is the Δ0 definable subsets of X (see constructible universe). Zermelo–Fraenkel set theory, with the axiom of choice, commonly abbreviated ZFC, is the most common form of axiomatic set theory, and as such is the most common foundation of mathematics. ...
In foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiom system for set theory designed to yield the same results as Zermelo-Fraenkel set theory, together with the axiom of choice (ZFC), but with only a finite number of axioms, that is without axiom schemas. ...
Morse–Kelley set theory or Kelley–Morse set theory (MK or KM) is a set theory with proper classes properly extending the usual set theory ZF. It is a first order theory (though it can be confused with second-order ZF). ...
In mathematics, the constructible universe (or Gödels constructible universe), denoted L, is a particular class of sets which can be described entirely in terms of simpler sets. ...
Existence of a proper class of inaccessibles There are many important axioms in set theory which assert the existence of a proper class of cardinals which satisfy a predicate of interest. In the case of inaccessibility, the corresponding axiom is the assertion that for every cardinal μ, there is an inaccessible cardinal κ which is strictly larger, μ < κ. Thus this axiom guarantees the existence of an infinite tower of inaccessible cardinals (and may occasionally be referred to as the inaccessible cardinal axiom). As is the case for the existence of any inaccessible cardinal, the inaccessible cardinal axiom is independent of the axioms of ZFC. Assuming ZFC, the inaccessible cardinal axiom is equivalent to the universe axiom of Grothendieck and Verdier: every set is contained in a Grothendieck universe. The axioms of ZFC along with the universe axiom (or equivalently the inaccessible cardinal axiom) are denoted ZFCU (which could be confused with ZFC with urelements). This axiomatic system is useful to prove for example that every category has an appropriate Yoneda embedding. Alexander Grothendieck (born March 28, 1928, Berlin) is one of the greatest mathematicians of the 20th century, with major contributions to algebraic geometry, homological algebra, and functional analysis. ...
In mathematics, a Grothendieck universe is a non-empty set U with the following properties: If x U and if y x, then y U. If x,y U, then {x,y} U. If x U, then P(x) U. (P(x) is the power set of x. ...
In mathematics, categories allow one to formalize notions involving abstract structure and processes which preserve structure. ...
In mathematics, the Yoneda lemma in category theory is an abstract result on functors of the type morphisms into a fixed object. ...
This is a relatively weak large cardinal axiom since it amounts to saying that ∞ is 1-inaccessible in the language of the next section, where ∞ denotes the least ordinal not in V, i.e. the class of all ordinals in your model.
α-inaccessible cardinals and hyper-inaccessible cardinals A cardinal κ is α-inaccessible, for α any ordinal, if and only if κ is inaccessible and for every ordinal β < α, the set of β-inaccessibles less than κ is unbounded in κ (and thus of cardinality κ, since κ is regular).
The inaccessible cardinals can be equivalently described as fixed points of functions which count the lower inaccessibles. For example, denote by ψ0(λ) the λth inaccessible cardinal, then the fixed points of ψ0 are the 1-inaccessible cardinals. Then letting ψβ(λ) be the λth β-inaccessible cardinal, the fixed points of ψβ are the (β+1)-inaccessible cardinals (the values ψβ+1(λ)). If α is a limit ordinal, an α-inaccessible is a fixed point of every ψβ for β < α (the value ψα(λ) is the λth such cardinal). This process of taking fixed points of functions generating successively larger cardinals is commonly encountered in the study of large cardinal numbers. This page is a list of some types of cardinals; it is arranged roughly in order of the consistency strength of the axiom asserting the existence of cardinals with the given property. ...
A cardinal κ is hyper-inaccessible if and only if κ is κ-inaccessible. (It can never be κ+1-inaccessible.)
For any ordinal α, a cardinal κ is α-hyper-inaccessible if and only if κ is hyper-inaccessible and for every ordinal β < α, the set of β-hyper-inaccessibles less than κ is unbounded in κ.
Using "weakly inaccessible" instead of "inaccessible", similar definitions can be made for "weakly α-inaccessible", "weakly hyper-inaccessible", and "weakly α-hyper-inaccessible".
See Mahlo cardinals. In mathematics, a Mahlo cardinal is a certain kind of large cardinal number. ...
See also A club set is a subset of a limit ordinal which is closed under the order topology, and is unbounded. ...
In mathematical logic, suppose T is a theory in the language . If M is a model of describing a set theory and N is a class of M such that is a model of T then we say that N is an inner model of T (in M). ...
In axiomatic set theory and related branches of mathematics, the Von Neumann universe, or Von Neumann hierarchy of sets is the class of all sets, divided into a transfinite hierarchy of individual sets. ...
In mathematics, the constructible universe (or Gödels constructible universe), denoted L, is a particular class of sets which can be described entirely in terms of simpler sets. ...
References - Drake, F. R. (1974). Set Theory: An Introduction to Large Cardinals (Studies in Logic and the Foundations of Mathematics ; V. 76). Elsevier Science Ltd. ISBN 0444105352.
- Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings, 2nd ed, Springer. ISBN 3540003843.
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