Scott's Trick - Application To Cardinalities

Application To Cardinalities

The use of Scott's trick for cardinal numbers shows how the method is typically employed. The initial definition of a cardinal number is an equivalence class of sets, where two sets are equivalent if there is a bijection between them. The difficulty is that every equivalence class of this relation is a proper class, and so the equivalence classes themselves cannot be directly manipulated in set theories, such as Zermelo–Fraenkel set theory, that only deal with sets. It is often desirable in the context of set theory to have sets that are representatives for the equivalence classes. These sets are then taken to "be" cardinal numbers, by definition.

In Zermelo–Fraenkel set theory with the axiom of choice, one way of assigning representatives to cardinal numbers is to associate each cardinal number with the least ordinal number of the same cardinality. These special ordinals are the ℵ numbers. But if the axiom of choice is not assumed, it is possible that some sets do not have the same cardinality as any ordinal number, and thus the cardinal numbers of those sets have no ordinal number as representative.

Scott's trick assigns representatives differently, using the fact that for every set A there is a least rank γ_A in the cumulative hierarchy when some set of the same cardinality as A appears. Thus one may define the representative of the cardinal number of A to be the set of all sets of rank γ_A that have the same cardinality as A. This definition assigns a representative to every cardinal number even when not every set can be well-ordered (an assumption equivalent to the axiom of choice). It can be carried out in Zermelo–Fraenkel set theory, without using the axiom of choice, but making essential use of the axiom of regularity.