# Proper Subsets of Ordinals ... ... Another Question ... ... Searcoid, Theorem 1.4.4 ... ...

#### 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 have another question regarding the proof of Theorem 1.4.4 ... In the above proof by Searcoid we read the following:

"... ... Moreover, since $$\displaystyle x \subset \alpha$$, we have $$\displaystyle \delta \in \alpha$$. But $$\displaystyle \beta \in \alpha$$ and $$\displaystyle \alpha$$ is totally ordered, so we must have $$\displaystyle \delta \in \beta$$ or $$\displaystyle \delta = \beta$$ or $$\displaystyle \beta \in \delta$$ ... ... "

My question is regarding the three alternatives $$\displaystyle \delta \in \beta$$ or $$\displaystyle \delta = \beta$$ or $$\displaystyle \beta \in \delta$$ ... ...

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

So if we follow the definition exactly in the quote above there are only two alternatives .... $$\displaystyle \delta \in \beta$$ or $$\displaystyle \beta \in \delta$$ ... ...

My question is ... where does the $$\displaystyle =$$ alternative come from ... ?

How does the $$\displaystyle =$$ alternative follow from the definition of totally ordered ... ?

Help will be appreciated ...

Peter

#### steenis

##### Well-known member
MHB Math Helper
Now ... where $$\displaystyle (S, <)$$ is a partially ordered set ... $$\displaystyle S$$ is said to be totally ordered by $$\displaystyle <$$ if and only if for every pair of distinct members $$\displaystyle x, y \in S$$, either $$\displaystyle x < y$$ or $$\displaystyle y < x$$ ... ..
Peter
Please, read this definition very very carefully, and ask yourself: what if the pair of members is/are not distinct ?

#### Peter

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