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

AI Thread Summary
The discussion centers on understanding Michael Searcoid's proof of the Pairing Principle in his book "Elements of Abstract Analysis." The key focus is on the notation $$\mathcal{P} \mathcal{P} ( \emptyset )$$, which represents the power set of the power set of the empty set, specifically $$\mathcal{P}(\mathcal{P}(\emptyset))$$. Participants clarify that Axiom III, the axiom of replacement, is applied by mapping elements of the power set to new sets based on a defined function. The proof illustrates how Axiom III ensures that the image of a set under this function remains a set. This detailed examination aims to deepen the understanding of set theory principles as presented by Searcoid.
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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
 
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Peter said:
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 $\mathcal{P}(\mathcal{P}(\emptyset))$.

Peter said:
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 $\mathcal{P}(\mathcal{P}(\emptyset))=\{\emptyset,\{\emptyset\}\}$.
 
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