MHB Axioms of Set Theory: Separation Axiom and Garling Theorem 1.2.2 .... ....

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The discussion centers on the necessity of the Separation Axiom in proving Theorem 1.2.2 from Garling's text on set theory. Participants highlight that the Separation Axiom is essential for justifying the existence of a set in the proof, specifically relating set A in the axiom to set Ω in the theorem. Clarification is sought on how the axiom directly influences the proof's validity. The conversation emphasizes the foundational role of the Separation Axiom in set theory. Understanding this relationship is crucial for grasping the theorem's implications.
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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:
View attachment 6137

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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Hi Peter,

Peter said:
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 ...

The application of the Separation Axiom is what justifies the statement "and so there exists a set $b=\ldots$" The set $A$ in the axiom statement is $\Omega$ in the theorem. Does this help?
 
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