MHB Understanding Proper Subsets of Ordinals in Searcoid's Theorem 1.4.4 - Peter

[Math Amateur]

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 have another question regarding the proof of Theorem 1.4.4 ...

Theorem 1.4.4 reads as follows:
View attachment 8463
In the above proof by Searcoid we read the following:

"... ... Moreover, since $$x \subset \alpha$$, we have $$\delta \in \alpha$$. But $$\beta \in \alpha$$ and $$\alpha$$ is totally ordered, so we must have $$\delta \in \beta$$ or $$\delta = \beta$$ or $$\beta \in \delta$$ ... ... "My question is regarding the three alternatives $$\delta \in \beta$$ or $$\delta = \beta$$ or $$\beta \in \delta$$ ... ...Now ... where $$(S, <)$$ is a partially ordered set ... $$S$$ is said to be totally ordered by $$<$$ if and only if for every pair of distinct members $$x, y \in S$$, either $$x < y$$ or $$y < x$$ ... ..So if we follow the definition exactly in the quote above there are only two alternatives ... $$\delta \in \beta$$ or $$\beta \in \delta$$ ... ...My question is ... where does the $$=$$ alternative come from ... ?

How does the $$=$$ alternative follow from the definition of totally ordered ... ?Help will be appreciated ...

Peter

 

[steenis]

Now ... where $$(S, <)$$ is a partially ordered set ... $$S$$ is said to be totally ordered by $$<$$ if and only if for every pair of distinct members $$x, y \in S$$, either $$x < y$$ or $$y < x$$ ... ..
Peter

Please, read this definition very very carefully, and ask yourself: what if the pair of members is/are not distinct ?

 

[Math Amateur]

Please, read this definition very very carefully, and ask yourself: what if the pair of members is/are not distinct ?

Thanks Steenis ...

See that key term is "distinct"... if not distinct then members are equal ... enough said ...

Thanks for your help ...

Peter