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Mathematical logic, Set theory, Peano arithmetic, Model theory, Proof theory, Recursion theory, Computability theory, Univalent foundations, Reverse mathematics, Frege foundation of arithmetic, Goedel's incompleteness and Mathematics, Structural set theory, Category theory, Type theory.

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Why can't an explicit well-ordering of the reals be ruled out in ZF?

Here is the error in your reasoning "If B were true, then A would be provable in ZF, which is impossible since A is independent of ZF, so B must be false." The statement "B is independent of ZF" means …
Cody Dance's user avatar