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 real. 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:


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