Logic Seminar

Please note that the time is PST

The axiom of Dependent Choices (DC) and the axiom of Countable Choices (CC) play an important role in set theory. Both axioms admit a "local" version denoted by DC(X) and CC(X), meaning that the choices are made in a given set X. It is well known that DC implies CC, but does DC(X) imply CC(X) for any non-empty set X? (The question is obviously formulated in the choice-less context of ZF.) The answer is affirmative for many X, e.g. if X is in bijection with its square, but it is not true in general.
Theorem 1: Assume CC(R), where R is the set of real numbers. Then DC(X) implies CC(X) for all non-empty X.
Theorem 2: There is a model of ZF in which there is a subset X of R such that DC(X) holds and CC(X) fails.
The model of Theorem 2 is a symmetric extension using a forcing notion P that is hybrid between a product and an iteration.
This is joint work with Lorenzo Notaro.