L(R)
Encyclopedia
In set theory
, L(R) is the smallest transitive inner model
of ZF containing all the ordinals
and all the real
s. It can be constructed in a manner analogous to the construction of L (that is, Gödel's constructible universe), by adding in all the reals at the start, and then iterating the definable powerset operation through all the ordinals.
In general, the study of L(R) assumes a wide array of large cardinal 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, but rather the axiom of determinacy
. However, L(R) will still satisfy the axiom of dependent choice
, given only that the von Neumann universe
, V, also satisfies that axiom.
Some additional results of the theory are:
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
, L(R) is the smallest transitive inner model
Inner model
In mathematical logic, suppose T is a theory in the languageL = \langle \in \rangleof set theory.If M is a model of L describing a set theory and N is a class of M such that \langle N, \in_M, \ldots \rangle...
of ZF containing all the ordinals
Ordinal number
In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
and all the real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s. It can be constructed in a manner analogous to the construction of L (that is, Gödel's constructible universe), by adding in all the reals at the start, and then iterating the definable powerset operation through all the ordinals.
In general, the study of L(R) assumes a wide array of large cardinal 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, but rather the axiom of determinacy
Axiom of determinacy
The axiom of determinacy is a possible axiom for set theory introduced by Jan Mycielski and Hugo Steinhaus in 1962. It refers to certain two-person games of length ω with perfect information...
. However, L(R) will still satisfy the axiom of dependent choice
Axiom of dependent choice
In mathematics, the axiom of dependent choices, denoted DC, is a weak form of the axiom of choice which is still sufficient to develop most of real analysis...
, given only that the von Neumann universe
Von Neumann universe
In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted V, is the class of hereditary well-founded sets...
, V, also satisfies that axiom.
Some additional results of the theory are:
- Every projective set of reals -- and therefore every analytic setAnalytic setIn 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 definitions of analytic set...
and every Borel setBorel setIn mathematics, a Borel set is any set in a topological space that can be formed from open sets through the operations of countable union, countable intersection, and relative complement...
of reals -- is an element of L(R). - Every set of reals in L(R) is Lebesgue measurable (in fact, universally measurable) and has the property of Baire and the perfect set propertyPerfect set propertyIn 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...
. - L(R) does not satisfy the axiom of uniformization or the axiom of real determinacyAxiom of real determinacyIn mathematics, the axiom of real determinacy is an axiom in set theory. It states the following:The axiom of real determinacy is a stronger version of the axiom of determinacy, which makes the same statement about games where both players choose integers; it is inconsistent with the axiom of choice...
. - R#, the sharp of the set of all reals, has the smallest Wadge degree of any set of reals not contained in L(R).
- While not every relationBinary relationIn mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = . More generally, a binary relation between two sets A and B is a subset of...
on the reals in L(R) has a uniformizationUniformization (set theory)In set theory, the axiom of uniformization, a weak form of the axiom of choice, states that if R is a subset of X\times Y, where X and Y are Polish spaces,...
in L(R), every such relation does have a uniformization in L(R#). - Given any (set-size) generic extension V[G] of V, L(R) is an elementary submodel of L(R) as calculated in V[G]. Thus the theory of L(R) cannot be changed by forcingForcing (mathematics)In the mathematical discipline of set theory, forcing is a technique invented by Paul Cohen for proving consistency and independence results. It was first used, in 1963, to prove the independence of the axiom of choice and the continuum hypothesis from Zermelo–Fraenkel set theory...
. - L(R) satisfies AD+AD plusIn set theory, AD+ is an extension, proposed by W. Hugh Woodin, to the axiom of determinacy. The axiom, which is to be understood in the context of ZF plus DCR , states two things:# Every set of reals is ∞-Borel....
.