ZFC and the Axiom of Power Sets ....

[Math Amateur]

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:

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 \(\displaystyle b = \{ s, t, r \}\) then \(\displaystyle a = \{ s, t \}\) is a proper subset of \(\displaystyle b\) ... and the existence of a is guaranteed by the Axiom of Power Sets ...

Help will be much appreciated ...

Peter

 

[Andrei1]

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.

 

[Math Amateur]

Thanks for the help AndreI ...

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

Thanks again ...

Peter