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Main Question or Discussion Point
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 gives no proof of Corollary 1.4.5 ...
To prove Corollary 1.4.5 we need to show ##\beta \in \alpha \Longleftrightarrow \beta \subset \alpha## ... ...
Assume that ##\beta \in \alpha## ...
Then by Searcoid's definition of an ordinal (Definition 1.4.1 ... see scanned text below) we have ##\beta \subseteq \alpha## ...
But it is supposed to follow that ##\beta## is a proper subset of ##\alpha## ... !
Is Searcoid assuming that ##\beta \neq \alpha##? ... ... if not how would it follow that \beta \subset \alpha ... ?
Hope someone can help ...
Peter
============================================================================
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:
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 the Corollary to Theorem 1.4.4 ...
Theorem 1.4.4 and its corollary read as follows:
Searcoid gives no proof of Corollary 1.4.5 ...
To prove Corollary 1.4.5 we need to show ##\beta \in \alpha \Longleftrightarrow \beta \subset \alpha## ... ...
Assume that ##\beta \in \alpha## ...
Then by Searcoid's definition of an ordinal (Definition 1.4.1 ... see scanned text below) we have ##\beta \subseteq \alpha## ...
But it is supposed to follow that ##\beta## is a proper subset of ##\alpha## ... !
Is Searcoid assuming that ##\beta \neq \alpha##? ... ... if not how would it follow that \beta \subset \alpha ... ?
Hope someone can help ...
Peter
============================================================================
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:
Hope that helps ...
Peter
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