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In [[set theory]], '''L(R)''' (pronounced ''L of R'') is the smallest [[transitive class|transitive]] [[inner model]] of [[Zermelo–Fraenkel set theory|ZF]] containing all the [[ordinal number|ordinals]] and all the [[real number|real]]s.
==Construction==
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.
==Assumptions==
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.
==Results==
Given the assumptions above, some additional results of the theory are:
* Every [[projective set]] of reals – and therefore every [[analytic set]] and every [[Borel set]] 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 property]].
* L(R) does ''not'' satisfy the [[axiom of uniformization]] or the [[axiom of real determinacy]].
* R<sup>#</sup>, the [[sharp (set theory)|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 [[binary relation|relation]] on the reals in L(R) has a [[uniformization (set theory)|uniformization]] in L(R), every such relation ''does'' have a uniformization in L(R<sup>#</sup>).
* 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 [[forcing (mathematics)|forcing]].
* L(R) satisfies [[AD+|AD<sup>+</sup>]].
==References==
* {{cite journal|authorlink=W. Hugh Woodin|author=Woodin, W. Hugh|journal=[[Proceedings of the National Academy of Sciences of the United States of America]]|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}}
[[Category:Inner model theory]]
[[Category:Determinacy]]
[[Category:Descriptive set theory]]