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

#### Peter

##### Well-known member
MHB Site Helper
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 ...

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:

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