Set Theory: Separation Axiom and Garling's Theorem 1.2.2

In summary, the conversation discusses the importance of the Separation Axiom in proving Theorem 1.2.2 in D. J. H. Garling's "A Course in Mathematical Analysis: Volume I Foundations and Elementary Real Analysis". The Separation Axiom is necessary for the existence of ##b## in the proof of Theorem 1.2.2 and without it, the contradiction becomes meaningless. The conversation also mentions the need to define ##A## and ##Q(x)## in order to understand the usage of the Separation Axiom in the proof.
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I am reading D. J. H. Garling: "A Course in Mathematical Analysis: Volume I Foundations and Elementary Real Analysis" ... ...

At present I am focused on Chapter 1: The Axioms of Set Theory and need some help with Theorem 1.2.2 and its relationship to the Separation Axiom ... ...

The Separation Axiom and Theorem 1.2.2 read as follows:
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Garling argues that the Separation Axiom needs to be in place before we can prove Theorem 1.2.2 ... ... but I cannot see where the Separation Axiom is needed in the proof of Theorem 1.2.2 ...

Can someone give a clear explanation of exactly why we need the Separation Axiom in order to prove Theorem 1.2.2.

Help will be much appreciated ... ...

Peter
 

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He uses the separation axiom for the existence of ##b##.
Can you define ##A## and ##Q(x)## for the usage in the proof of Theorem 1.2.2? This is needed for otherwise ##b## simply couldn't exist, and then the contradiction became meaningless.
 
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1. What is the Separation Axiom in Set Theory?

The Separation Axiom in Set Theory is a principle that states that for any set A and any property P, there exists a subset of A that contains all elements of A that satisfy P. In other words, it allows for the creation of a new set from an existing set by selecting only elements that meet a certain condition.

2. What is Garling's Theorem 1.2.2 in Set Theory?

Garling's Theorem 1.2.2 is a result in Set Theory that states that if a set A has the Separation Axiom, then any subset of A also has the Separation Axiom. This theorem is useful in proving other results involving the Separation Axiom.

3. How is Garling's Theorem 1.2.2 useful in Set Theory?

Garling's Theorem 1.2.2 is useful in Set Theory because it allows for the extension of the Separation Axiom to subsets of a given set. This can be used to prove other results involving the Separation Axiom, as well as to ensure the consistency and completeness of the theory.

4. Can Garling's Theorem 1.2.2 be applied to any set?

Yes, Garling's Theorem 1.2.2 can be applied to any set that has the Separation Axiom. This includes both finite and infinite sets, as well as sets with any number of elements.

5. How does Garling's Theorem 1.2.2 relate to other theorems in Set Theory?

Garling's Theorem 1.2.2 is closely related to other theorems in Set Theory that involve the Separation Axiom, such as Zermelo's Theorem and the Axiom of Choice. It is also related to other results that deal with subsets and extensions of sets, making it a fundamental tool in Set Theory.

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