For specified numbers of sets of cardinality one and cardinality two, an upper bound on the number of sets of cardinality three is established using shifting arguments.
This characterization is restated to determine the precise spectrum of possible numbers of sets of cardinality two for specified numbers of sets of cardinality one and three.
Given two cardinals $\kappa$ and $\lambda$, the sum $\kappa +\lambda$ is defined as the cardinality of the set consisting of the union of any two disjoint sets, one of cardinality $\kappa$ and one of cardinality $\lambda $
More precisely, 2 R is a good estimator of the multiset cardinality.
More explicitly, this theorem contains two parts: (1) If a theory has a model of infinite cardinality β, then, for each infinite cardinal α that is greater than β, the theory has a model of cardinality α. (2) If a theory has a model of infinite cardinality β, then, for each infinite cardinal α less than β, the theory has a model of cardinality α.
If we let the cardinality $\kappa$ of the non-logical vocabulary be greater than $\aleph_0$, which is the cardinality of the set of natural numbers, then if $\Gamma$ has an infinite model, then it has a model of cardinality $\kappa$.
It is well-known that coding tricks allow one to do classical mathematics without ever going above cardinality c: for example, the class of all functions from the reals to the reals, is too large to be even a proper class here, but the class of continuous functions is of cardinality c.
If λ is the supremum of all the κ, then, if a sentence of L has a model of cardinality λ, it has models of arbitrarily large cardinality.
For example, the L ω1,ω -sentence characterizing the standard model of arithmetic has a model of cardinality ℵ0 but no models of any other cardinality.
If κ and λ are cardinal numbers, the expression κλ represents the cardinality of the set of functions from any set of cardinality λ to any set of cardinality κ.
An attributed graph is defined as a directed multigraph with the set of vertices, the multiset of edges (there can be more than one edge between any two vertices).
