In
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
, 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
In mathematics, specifically set theory, a cumulative hierarchy is a family of sets W_\alpha indexed by ordinals \alpha such that
* W_\alpha \subseteq W_
* If \lambda is a limit ordinal, then W_\lambda = \bigcup_ W_
Some authors additionally r ...
.
The method relies on the
axiom of regularity
In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of Zermelo–Fraenkel set theory that states that every non-empty set ''A'' contains an element that is disjoint from ''A''. In first-order logic, the axi ...
but not on the
axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collectio ...
. It can be used to define
representative
Representative may refer to:
Politics
* Representative democracy, type of democracy in which elected officials represent a group of people
* House of Representatives, legislative body in various countries or sub-national entities
* Legislator, som ...
s for
ordinal number
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets.
A finite set can be enumerated by successively labeling each element with the least n ...
s in ZF,
Zermelo–Fraenkel set theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as ...
without the axiom of choice (Forster 2003:182). The method was introduced by .
Beyond the problem of defining set representatives for ordinal numbers, Scott's trick can be used to obtain representatives for
cardinal numbers
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. The ...
and more generally for
isomorphism types, for example,
order type
In mathematics, especially in set theory, two ordered sets and are said to have the same order type if they are order isomorphic, that is, if there exists a bijection (each element pairs with exactly one in the other set) f\colon X \to Y such ...
s of
linearly ordered sets (Jech 2003:65). It is credited to be indispensable (even in the presence of the axiom of choice) when taking
ultrapower
The ultraproduct is a mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direct product of a family of structures. All factor ...
s of proper classes in
model theory
In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the s ...
. (Kanamori 1994:47)
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
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
of sets, where two sets are equivalent if there is a
bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
between them. The difficulty is that almost every equivalence class of this relation is a
proper class
Proper may refer to:
Mathematics
* Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact
* Proper morphism, in algebraic geometry, an analogue of a proper map for ...
, 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
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collectio ...
, 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
rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* H ...
in the
cumulative hierarchy
In mathematics, specifically set theory, a cumulative hierarchy is a family of sets W_\alpha indexed by ordinals \alpha such that
* W_\alpha \subseteq W_
* If \lambda is a limit ordinal, then W_\lambda = \bigcup_ W_
Some authors additionally r ...
when some set of the same cardinality as
appears. Thus one may define the representative of the cardinal number of
to be the set of all sets of rank
that have the same cardinality as
. 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
In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of Zermelo–Fraenkel set theory that states that every non-empty set ''A'' contains an element that is disjoint from ''A''. In first-order logic, the axi ...
.
Scott's trick in general
Let
be an equivalence relation of sets. Let
be a set and