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Ordinals ... Searcoid, Theorem 1.4.6 ...

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,918
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:




Searcoid - 1 -  Theorem 1.4.6 ... ... PART 1 ... ......png
Searcoid - 2 -  Theorem 1.4.6 ... ... PART 2 ... .......png






My question regarding the above proof by Micheal Searcoid is as follows:

How do we know that \(\displaystyle \alpha\) and \(\displaystyle \beta\) are not disjoint? ... indeed ... can they be disjoint?

What happens to the proof if \(\displaystyle \alpha \cap \beta = \emptyset\)?



Help will be appreciated ...

Peter



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It may help MHB
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:



Searcoid - 1 -  Start of section on Ordinals  ... ... PART 1 ... .....png
Searcoid - 2 -  Start of section on Ordinals  ... ... PART 2 ... ......png





Hope that helps ...

Peter
 
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