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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:
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
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 $$b = \{ s, t, r \}$$ 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