ZF Set Theory and Law of the Excluded Middle

AI Thread Summary
The discussion centers on whether the law of excluded middle (LEM) is implied in ZF set theory without the axiom of choice, particularly for infinite sets. The original poster, HJ Farnsworth, seeks clarification and proof regarding this implication. They mention that the question has also been posed on Math Stack Exchange, where it is being addressed. The conversation highlights the relationship between ZF axioms and the law of excluded middle. Overall, the inquiry emphasizes the foundational aspects of set theory and logical principles.
HJ Farnsworth
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Hello,

I know that the law of the excluded middle is implied in ZFC set theory, since it is implied by the axiom of choice. Taking away the axiom of choice, does ZF set theory (with axioms as stated in the Wikipedia article http://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory), imply the law of the excluded middle [for infinite sets]?

If LEM does follow from ZF, could you please provide the proof if you know it, or point me to the proof if you know where it is, or tell me what ZF axioms it follows from if you don't know of theproof or its location?

Thanks very much.

-HJ Farnsworth
 
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