# ZFC and the Axiom of Power Sets ...

#### Peter

##### Well-known member
MHB Site Helper
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

#### Andrei

##### Member
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
Thanks for the help AndreI ...

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

Thanks again ...

Peter