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ZFC and the Axiom of Power Sets ...

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,916
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 \(\displaystyle b = \{ s, t, r \}\) 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
 

Andrei

Member
Jan 18, 2013
36
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.
 

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,916
Thanks for the help AndreI ...

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

Thanks again ...

Peter