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Ordinals ... Searcoid, Corollary 1.4.5 ... ...

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 the Corollary to Theorem 1.4.4 ...

Theorem 1.4.4 and its corollary read as follows:



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



Searcoid gives no proof of Corollary 1.4.5 ...


Question 1

To prove Corollary 1.4.5 we need to show \(\displaystyle \beta \in \alpha \Longleftrightarrow \beta \subset \alpha\) ... ...



Assume that \(\displaystyle \beta \in \alpha\) ...

Then by Searcoid's definition of an ordinal (Definition 1.4.1 ... see scanned text below) we have \(\displaystyle \beta \subseteq \alpha\) ...

But it is supposed to follow that \(\displaystyle \beta\) is a proper subset of \(\displaystyle \alpha\) ... !

Is Searcoid assuming that \(\displaystyle \beta \neq \alpha\) ... if not how would it follow that \(\displaystyle \beta \subset \alpha\) ... ?



Question 2


The Corollary goes on to state that, in particular, if \(\displaystyle \alpha \neq 0\) then \(\displaystyle 0 \in \alpha\) ... can someone please show me how to demonstrate that this is true ... ?



Hope someone can help ...

Peter




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It may help readers of the above post if the start of the section on ordinals was accessible ... so I am providing that text as follows:



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