Scott's trick
In set theory, Scott's trick is a method for giving a definition of equivalence classes for equivalence relations on a proper class (Jech 2003:65) by referring to levels of the cumulative hierarchy.
The method relies on the
Beyond the problem of defining set representatives for ordinal numbers, Scott's trick can be used to obtain representatives for
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
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, for some cardinal numbers it may not be possible to find such an ordinal number, and thus the cardinal numbers of those sets have no ordinal number as representatives.
Scott's trick assigns representatives differently, using the fact that for every set there is a least
Scott's trick in general
Let be an equivalence relation of sets. Let be a set and its equivalence class with respect to . If is non-empty, we can define a set, which represents , even if is a proper class. Namely, there exists a least ordinal , such that is non-empty. This intersection is a set, so we can take it as the representative of . We didn't use regularity for this construction.
The axiom of regularity is equivalent to for all sets (see Regularity, the cumulative hierarchy and types). So in particular, if we assume the axiom of regularity, then will be non-empty for all sets and equivalence relations , since . To summarize: given the axiom of regularity, we can find representatives of every equivalence class, for any equivalence relation.
References
- Thomas Forster (2003), Logic, Induction and Sets, Cambridge University Press. ISBN 0-521-53361-9
- Thomas Jech, Set Theory, 3rd millennium (revised) ed., 2003, Springer Monographs in Mathematics, Springer, ISBN 3-540-44085-2
- Akihiro Kanamori: The Higher Infinite. Large Cardinals in Set Theory from their Beginnings., Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1994. xxiv+536 pp.
- Scott, Dana (1955), "Definitions by abstraction in axiomatic set theory" (PDF), Bulletin of the American Mathematical Society, 61 (5): 442,