In
set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), 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 mathema ...
, L(R) (pronounced ''L of R'') is the smallest
transitive inner model
In set theory, a branch of mathematical logic, an inner model for a theory ''T'' is a substructure of a model ''M'' of a set theory that is both a model for ''T'' and contains all the ordinals of ''M''.
Definition
Let ''L'' = ⟨∈� ...
of
ZF containing all the
ordinals and all the
reals.
Construction
L(R) can be constructed in a manner analogous to the construction of
Gödel's constructible universe
In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted by L, is a particular class of sets that can be described entirely in terms of simpler sets. L is the union of the constructible hierarchy L_\ ...
, L, by adding in all the reals at the start, and then iterating the definable powerset operation through all the ordinals.
Assumptions
In general, the study of L(R) assumes a wide array of
large cardinal
In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers. Cardinals with such properties are, as the name suggests, generally very "large" (for example, bigger than the least ...
axioms, since without these axioms one cannot show even that L(R) is distinct from L. But given that sufficient large cardinals exist, L(R) does not satisfy the
axiom of choice
In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from e ...
, but rather the
axiom of determinacy
In mathematics, the axiom of determinacy (abbreviated as AD) is a possible axiom for set theory introduced by Jan Mycielski and Hugo Steinhaus in 1962. It refers to certain two-person topological games of length ω. AD states that every game o ...
. However, L(R) will still satisfy the
axiom of dependent choice
In mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. T ...
, given only that the
von Neumann universe
In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted by ''V'', is the class of hereditary well-founded sets. This collection, which is formalized by Zermelo–Fraenkel set theory ( ...
, V, also satisfies that axiom.
Results
Given the assumptions above, some additional results of the theory are:
* Every
projective set
In the mathematical field of descriptive set theory, a subset A of a Polish space X is projective if it is \boldsymbol^1_n for some positive integer n. Here A is
* \boldsymbol^1_1 if A is analytic
* \boldsymbol^1_n if the complement of A, X\se ...
of reals – and therefore every
analytic set
In the mathematical field of descriptive set theory, a subset of a Polish space X is an analytic set if it is a continuous image of a Polish space. These sets were first defined by and his student .
Definition
There are several equivalent ...
and every
Borel set
In mathematics, a Borel set is any subset of a topological space that can be formed from its open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets ...
of reals – is an element of L(R).
* Every set of reals in L(R) is
Lebesgue measurable
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it coin ...
(in fact,
universally measurable In mathematics, a subset A of a Polish space X is universally measurable if it is measurable with respect to every complete probability measure on X that measures all Borel subsets of X. In particular, a universally measurable set of reals is ne ...
) and has the
property of Baire
A subset A of a topological space X has the property of Baire (Baire property, named after René-Louis Baire), or is called an almost open set, if it differs from an open set by a meager set; that is, if there is an open set U\subseteq X such tha ...
and the
perfect set property In the mathematical field of descriptive set theory, a subset of a Polish space has the perfect set property if it is either countable or has a nonempty perfect subset (Kechris 1995, p. 150). Note that having the perfect set property is ...
.
* L(R) does ''not'' satisfy the
axiom of uniformization In set theory, a branch of mathematics, the axiom of uniformization is a weak form of the axiom of choice. It states that if R is a subset of X\times Y, where X and Y are Polish spaces, then there is a subset f of R that is a partial function from X ...
or the
axiom of real determinacy
In mathematics, the axiom of real determinacy (abbreviated as ADR) is an axiom in set theory. It states the following:
The axiom of real determinacy is a stronger version of the axiom of determinacy (AD), which makes the same statement about ga ...
.
* R
#, the
sharp of the set of all reals, has the smallest
Wadge degree In descriptive set theory, within mathematics, Wadge degrees are levels of complexity for sets of reals. Sets are compared by continuous reductions. The Wadge hierarchy is the structure of Wadge degrees. These concepts are named after William W. W ...
of any set of reals ''not'' contained in L(R).
* While not every
relation
Relation or relations may refer to:
General uses
* International relations, the study of interconnection of politics, economics, and law on a global level
* Interpersonal relationship, association or acquaintance between two or more people
* ...
on the reals in L(R) has a
uniformization in L(R), every such relation ''does'' have a uniformization in L(R
#).
* Given any (set-size)
generic extension
In the mathematical discipline of set theory, forcing is a technique for proving consistency and independence results. Intuitively, forcing can be thought of as a technique to expand the set theoretical universe V to a larger universe V /math> by ...
V
of V, L(R) is an
elementary submodel
In model theory, a branch of mathematical logic, two structures ''M'' and ''N'' of the same signature ''σ'' are called elementarily equivalent if they satisfy the same first-order ''σ''-sentences.
If ''N'' is a substructure of ''M'', one oft ...
of L(R) as calculated in V
Thus the theory of L(R) cannot be changed by
forcing.
* L(R) satisfies
AD+.
References
* {{cite journal, authorlink=W. Hugh Woodin, author=Woodin, W. Hugh, journal=
, year=1988, title=Supercompact cardinals, sets of reals, and weakly homogeneous trees, volume=85, issue=18, pages=6587–6591, doi=10.1073/pnas.85.18.6587, pmid=16593979, pmc=282022, doi-access=free
Inner model theory
Determinacy
Descriptive set theory