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L(R)
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. 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniformization (set Theory)
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 to Y, and whose domain (the Set (mathematics), set of all x such that f(x) exists) equals : \\, Such a function is called a uniformizing function for R, or a uniformization of R. To see the relationship with the axiom of choice, observe that R can be thought of as associating, to each element of X, a subset of Y. A uniformization of R then picks exactly one element from each such subset, whenever the subset is Empty set, non-empty. Thus, allowing arbitrary sets ''X'' and ''Y'' (rather than just Polish spaces) would make the axiom of uniformization equivalent to the axiom of choice. A pointclass \boldsymbol is said to have the uniformization property if every Binary relation, relation R in \boldsymbol can be uniformized by a partial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |