ZF Set Theory and Law of the Excluded Middle

In summary, the conversation discusses the relationship between the law of the excluded middle and ZF set theory. It is implied that the law of the excluded middle follows from ZFC set theory, but the question is whether it also follows from ZF set theory without the axiom of choice. The speaker also mentions a link to a math stack exchange post where the question is being answered.
  • #1
HJ Farnsworth
128
1
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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  • #2

Related to ZF Set Theory and Law of the Excluded Middle

1. What is ZF Set Theory?

ZF Set Theory is a foundational theory of mathematics that deals with the construction and manipulation of sets. It was developed in the early 20th century by mathematician Ernst Zermelo, and is based on the principles of set theory proposed by mathematician Georg Cantor.

2. What is the Law of the Excluded Middle?

The Law of the Excluded Middle is a fundamental principle in logic that states that for any given statement, either that statement or its negation must be true. In other words, there is no middle ground or third option.

3. How does ZF Set Theory relate to the Law of the Excluded Middle?

ZF Set Theory is built on the principle of the Law of the Excluded Middle. In order to prove theorems within ZF Set Theory, one must use the Law of the Excluded Middle as a logical rule.

4. Are there any criticisms of the Law of the Excluded Middle?

Yes, there have been some criticisms of the Law of the Excluded Middle, particularly in the field of intuitionistic logic. Some argue that there are statements for which it is not clear whether they are true or false, and therefore the Law of the Excluded Middle cannot be applied.

5. How is ZF Set Theory used in other fields of science?

ZF Set Theory has applications in various fields of science, such as computer science, physics, and linguistics. It provides a foundation for mathematical reasoning and allows for the construction of rigorous proofs in these fields.

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