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Proper Subsets of Ordinals ... ... Searcoid, Theorem 1.4.4 ... ...

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,891
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 1.4.4 ...

Theorem 1.4.4 reads as follows:



Searcoid - Theorem 1.4.4 ... ....png



In the above proof by Searcoid we read the following:

"... ... Now, for each \(\displaystyle \gamma \in \beta\) , we have \(\displaystyle \gamma \in \alpha\) by 1.4.2, and the minimality with respect to \(\displaystyle \in\) of \(\displaystyle \beta\) in \(\displaystyle \alpha \text{\\} x\) ensures that \(\displaystyle \gamma \in x\). ... ...


Ca someone please show formally and rigorously that the minimality with respect to \(\displaystyle \in\) of \(\displaystyle \beta\) in \(\displaystyle \alpha \text{\\} x\) ensures that \(\displaystyle \gamma \in x\). ... ...


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 (including Theorem 1.4.2 ... ) ... so I am providing the same ... as follows:



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





It may also help MHB readers to have access to Searcoid's definition of a well order ... so I am providing the text of Searcoid's Definition 1.3.10 ... as follows:




Searcoid - Definition 1.3.10 ... .....png
Searcoid - 2 - Definition 1.3.10 ... .....PART 2 ... ....png



Hope that helps ...

Peter