Cumulative hierarchy information

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...

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...

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

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...

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

{\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...

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

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

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...

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 ) <...

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)...

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

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

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...

set. In particular, Vω + ω, a particular countable level of the cumulative hierarchy, is a model of Zermelo set theory. The axiom of replacement, on the...

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...

individual psyches in a cumulative form up to the present day––not merely as capitalism but as the vast history of hierarchical society from its inception...

such processes. They are also referred to under the names Yule process, cumulative advantage, the rich get richer, and the Matthew effect. They are also...

culture progresses and develops to the point where civilization develops hierarchies. The concept behind unilinear theory is that the steady accumulation...

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

embedding then j U(ƒ)(κ) = x. (Here Vκ denotes the κ-th level of the cumulative hierarchy, TC(x) is the transitive closure of x) The original application of...

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