- #1

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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 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