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I am currently focussed on Searcoid's treatment of ZFC in Chapter 1: Sets ...

I am trying to attain a full understanding of Searcoid's proof of the Pairing Principle ...

The Pairing Principle and its proof reads as follows:https://www.physicsforums.com/attachments/8285

In the above proof by Searcoid we read the following:

" ... ... By applying Axiom III to the set \(\displaystyle \mathcal{P} \mathcal{P} ( \emptyset )\) ... ... " What is \(\displaystyle \mathcal{P} \mathcal{P} ( \emptyset )\) ... what is its value and how (in detail) is it determined ... and further how exactly (in detail) do we apply Axiom III to it .. ?

Peter=========================================================================The above post refers to Axiom I and III ... so I am providing the text of these ... and for context/notation ... the rest of Searcoid's introduction to the ZFC Axioms up to the Pairing Principle ... as follows ...

https://www.physicsforums.com/attachments/8286

https://www.physicsforums.com/attachments/8287

View attachment 8288Hope that the provision of the above text helps ...

Peter