L(R)

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In set theory, L(R) (pronounced L of R) is the smallest transitive inner model of ZF containing all the ordinals and all the reals. 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.

One major difference between L(R) and L is that (assuming the existence of sufficient large cardinals), L(R) does not satisfy the axiom of choice, but rather the axiom of determinacy. L(R) does, however, satisfy the axiom of dependent choice.

Facts about L(R) (given large cardinals)

If the full von Neumann universe, V, satisfies the axiom of choice and contains sufficient large cardinals, then the following facts hold:

Every projective set of reals -- and therefore every analytic set and every Borel set of reals -- is an element of L(R).
L(R) satisfies ZF, AD (the axiom of determinacy), and DC (the axiom of dependent choice).
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^{#^{, 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 relation 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 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.
L(R) satisfies AD+.

References

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