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- Jun 22, 2012

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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