ZFC and the Pairing Principle .... Searcoid Theorem 1.1.5 ....

[Math Amateur]

I am reading Micheal Searcoid's book: Elements of Abstract Nalysis ( Springer Undergraduate Mathematics Series) ...

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 $$\mathcal{P} \mathcal{P} ( \emptyset )$$ ... ... " What is $$\mathcal{P} \mathcal{P} ( \emptyset )$$ ... 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

 

[Evgeny.Makarov]

In the above proof by Searcoid we read the following:

" ... ... By applying Axiom III to the set $$\mathcal{P} \mathcal{P} ( \emptyset )$$ ... ... "

What is $$\mathcal{P} \mathcal{P} ( \emptyset )$$

The notation $\mathcal{P}(x)$ is introduced after Axiom II. The notation $\mathcal{P}\mathcal{P}(\emptyset)$ means $\mathcal{P}(\mathcal{P}(\emptyset))$.

how exactly (in detail) do we apply Axiom III to it .. ?

The axiom of replacement (Axiom III) says that the image of a set under a function is a set. Here we apply the function that maps $\emptyset$ to $a$ and $\{\emptyset\}$ to $b$ (more precisely, the corresponding functional relation) to the set $\mathcal{P}(\mathcal{P}(\emptyset))=\{\emptyset,\{\emptyset\}\}$.