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I am reading Micheal Searcoid's book: "Elements of Abstract Analysis" ... ...
I am currently focused on understanding Chapter 1: Sets ... and in particular Section 1.4 Ordinals ...
I need some help in fully understanding Theorem 1.4.6 ...
Theorem 1.4.6 reads as follows:
My question regarding the above proof by Micheal Searcoid is as follows:
How do we know that ##\alpha## and ##\beta## are not disjoint? ... indeed ... can they be disjoint?
What happens to the proof if ##\alpha \cap \beta = \emptyset##?
Help will be appreciated ...
Peter
==========================================================================
It may help Physics Forums readers of the above post to have access to the start of Searcoid's section on the ordinals ... so I am providing the same ... as follows:
Hope that helps ...
Peter
I am currently focused on understanding Chapter 1: Sets ... and in particular Section 1.4 Ordinals ...
I need some help in fully understanding Theorem 1.4.6 ...
Theorem 1.4.6 reads as follows:
My question regarding the above proof by Micheal Searcoid is as follows:
How do we know that ##\alpha## and ##\beta## are not disjoint? ... indeed ... can they be disjoint?
What happens to the proof if ##\alpha \cap \beta = \emptyset##?
Help will be appreciated ...
Peter
==========================================================================
It may help Physics Forums readers of the above post to have access to the start of Searcoid's section on the ordinals ... so I am providing the same ... as follows:
Hope that helps ...
Peter
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