MHB ZFC and the Axiom of Power Sets ....

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The discussion centers on the Axiom of Power Sets in the context of ZFC set theory as presented in Michael Searcoid's book. A participant seeks clarification on why it is stated that we do not yet know if any set has a proper subset, despite examples suggesting otherwise. The conversation highlights that while subsets exist, the concept of proper subsets, particularly the empty set, has not been fully established at this stage in the text. This indicates a foundational aspect of set theory where the existence of proper subsets is contingent on the definitions being introduced. The dialogue emphasizes the importance of understanding these foundational principles in abstract analysis.
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I am reading Micheal Searcoid's book: Elements of Abstract Analysis ( Springer Undergraduate Mathematics Series) ...

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

I need help in order to fully understand the Axiom of Power Sets and Definition 1.1.1 ...

The relevant text from Searcoid is as follows:
Searcoid - The Axioms ... Page 6 .png

At the end of the above text we read the following:

" ... ... Thus every set is a subset of itself and no set is a proper subset of itself. But we do not yet now that any set has a proper subset. ... ... "My question is as follows:

Can someone explain exactly why/how it is that we do not yet now that any set has a proper subset. ... ..?

My thinking is that surely we do know that any set has a proper subset ... for example if $$b = \{ s, t, r \}$$ then $$a = \{ s, t \}$$ is a proper subset of $$b$$ ... and the existence of a is guaranteed by the Axiom of Power Sets ...

Help will be much appreciated ...

Peter
 
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The set \[ \{s\} \] has power set \[\{\varnothing, \{s\}\}.\] Empty set is proper by the definition, but it is not yet introduced. So we don’t know if \[\{s\}\] has any proper subsets.
 
Thanks for the help AndreI ...

... still reflecting on how what you have written answers my question ...

Thanks again ...

Peter
 

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