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 \(\displaystyle x \subset \alpha\), 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]

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 ?

 

[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