In mathematics, specifically set theory, a cumulative hierarchy is a family of sets W α {\displaystyle W_{\alpha }} indexed by ordinals α {\displaystyle...

divided into the transfinite hierarchy Vα , called the cumulative hierarchy, based on their rank. The cumulative hierarchy is a collection of sets Vα indexed...

Tarski. One can describe the cumulative hierarchy into which Zermelo developed his models as the universe of a cumulative TT in which transfinite types...

collection of all sets in the von Neumann universe (also known as the cumulative hierarchy). The metamathematics of Zermelo–Fraenkel set theory has been extensively...

organized into a cumulative hierarchy, based on how deeply their members, members of members, etc. are nested. Each set in this hierarchy is assigned (by...

Universe: In mathematics, the Von Neumann universe (denoted by V) is the cumulative hierarchy of all sets in Zermelo–Fraenkel set theory, structured in layers:...

that an axiomatic system based on the inherent properties of the cumulative hierarchy turns out to be equivalent to ZF, including regularity. The concept...

relations on a proper class (Jech 2003:65) by referring to levels of the cumulative hierarchy. The method relies on the axiom of regularity but not on the axiom...

{\displaystyle V_{\alpha }} is the rank- α {\displaystyle \alpha } set in the cumulative hierarchy—forms a model of second-order Zermelo set theory within ZFC whenever...

union Axiom schema of specification Boltzmann brain Choice function Cumulative hierarchy Pairwise comparison Von Neumann universe 14990 Zermelo, asteroid...

critical point of something CTM Countable transitive model cumulative hierarchy A cumulative hierarchy is a sequence of sets indexed by ordinals that satisfies...

Foundations (NF). Of these, ZF, NBG, and MK are similar in describing a cumulative hierarchy of sets. New Foundations takes a different approach; it allows objects...

According to Gagne, the cumulative learning theory is better than the maturational model because of the focus on the hierarchies of capabilities. In this...

and MK have models describable in terms of the cumulative hierarchy Vα  and the constructible hierarchy Lα . Let V include an inaccessible cardinal κ,...

can naturally be considered as proper classes by cutting off the cumulative hierarchy of the universe one stage above the cardinal, and Alling accordingly...

i } {\displaystyle i=\{i\}} for each i in I. In ZFC, define the cumulative hierarchy as the ordinal-indexed sequence of sets satisfying the following...

), then there is a level V α {\displaystyle V_{\alpha }} of the cumulative hierarchy such that V α ⊨ ϕ ( x 1 , … , x n ) {\displaystyle V_{\alpha }\vDash...

foundations of set theory after Skolem's paper led to his discovery of the cumulative hierarchy and formalization of infinitary logic. The surprise with which set...

model) which moves a rank V α {\displaystyle V_{\alpha }} of the cumulative hierarchy of sets. We may suppose without loss of generality that j ( α ) <...

ISBN 978-1-139-17313-1, OCLC 817922080 Viale, Matteo (November 2003), "The cumulative hierarchy and the constructible universe of ZFA", Mathematical Logic Quarterly...

of limitation of size. These models are built in ZFC by using the cumulative hierarchy Vα, which is defined by transfinite recursion: V0 = ∅. Vα+1 = Vα ∪ P(Vα)...

In logic, an important one is Tarski's hierarchy. In set theory, an important one is the cumulative hierarchy. higher-order logic A form of logic that...

arbitrary infinite set. In particular, Vω + ω, a particular level of the cumulative hierarchy with countable rank, is a model of Zermelo set theory. The axiom...

related to the cumulative distribution function of a normal distribution; another is the arctan function, which is related to the cumulative distribution...

introduced, though not as explicitly as John von Neumann later, the cumulative hierarchy of sets and the notion of von Neumann ordinals; although he introduced...

{\displaystyle \kappa } satisfying the above version based on the cumulative hierarchy were called strongly Q-indescribable. This property has also been...

Boolean-valued set, they are defined inductively in a hierarchy similar to the cumulative hierarchy. For each ordinal α of V, the set VBα is defined as...

"Learning and cognitive development: Representative samples, cumulative-hierarchical learning, and experimental-longitudinal learning. Monographs of...

well-founded or non-well-founded, ...), the various constructions of the cumulative hierarchy of sets, forcing models, sheaf models and realisability models. Instead...

probabilistic interpretation considers the activation nonlinearity as a cumulative distribution function. The probabilistic interpretation led to the introduction...