 #1
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MHB
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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:
View attachment 8467
View attachment 8468
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
==========================================================================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:
View attachment 8469
View attachment 8470Hope 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:
View attachment 8467
View attachment 8468
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
==========================================================================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:
View attachment 8469
View attachment 8470Hope that helps ...
Peter
Attachments

Searcoid  1  Theorem 1.4.6 ... ... PART 1 ... ......png4.8 KB · Views: 76

Searcoid  2  Theorem 1.4.6 ... ... PART 2 ... .......png7.6 KB · Views: 66

Searcoid  1  Start of section on Ordinals ... ... PART 1 ... .....png32.5 KB · Views: 71

Searcoid  2  Start of section on Ordinals ... ... PART 2 ... ......png36.9 KB · Views: 81
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