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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
Attachments

Searcoid  1  Theorem 1.4.6 ... ... PART 1 ... ......png12.2 KB · Views: 419

Searcoid  2  Theorem 1.4.6 ... ... PART 2 ... .......png21.5 KB · Views: 410

Searcoid  1  Start of section on Ordinals ... ... PART 1 ... .....png31.4 KB · Views: 402

?temp_hash=8f9cff7393f9c0d811f87ad59134570a.png12.2 KB · Views: 388

?temp_hash=8f9cff7393f9c0d811f87ad59134570a.png21.5 KB · Views: 429

?temp_hash=8f9cff7393f9c0d811f87ad59134570a.png31.4 KB · Views: 435

Searcoid  2  Start of section on Ordinals ... ... PART 2 ... ......png39.7 KB · Views: 378

?temp_hash=c13d2e58ed130749d22a8e1847b2f2a4.png39.7 KB · Views: 401
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