In the latter case, there is an infinite subset of indecomposable modules K i in K all of which have the same Gabriel Roiter measure γ 1 < γ 0. First, assume that K = add K for some module K, let C = τ - K. For any natural number t, we may choose a subset J of I of cardinality t.
In fact if T is a first-order theory with infinite models, then the strongest kind of categoricity we can hope for in T is that for certain infinite cardinals κ, T has exactly one model of cardinality κ, up to isomorphism.
N B t induced by the genes in G ~ t and compute the sets à t and B ~ t of genes that are contained in connected components of Ñ A t and Ñ B t with cardinality greater or equal 3.
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.
If a theory has any infinite model, then, for any infinite cardinality α, that theory has a model of cardinality α.
In set theory, the power-set operation assigns to each set of cardinality ℵα its set of all subsets, which has cardinality 2ℵα.
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 $
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.
