Ordinals .... Searcoid, Theorem 1.4.6 .

[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 need some help in fully understanding Theorem 1.4.6 ...

Theorem 1.4.6 reads as follows:

My question regarding the above proof by Micheal Searcoid is as follows:

How do we know that $\alpha$ and $\beta$ are not disjoint? ... indeed ... can they be disjoint?

What happens to the proof if $\alpha \cap \beta = \emptyset$?
Help will be appreciated ...

Peter
==========================================================================It may help Physics Forums readers of the above post to have access to the start of Searcoid's section on the ordinals ... so I am providing the same ... as follows:

Hope that helps ...

Peter

 

[andrewkirk]

They can be disjoint. If they are disjoint then $\alpha\cap\beta=\emptyset$ which is the ordinal 0. The proof still works.

To get a feel for this, consider the case where say $\alpha=0=\emptyset$. If you follow the proof through with this, you'll see that it still works and that either $\beta=\alpha=0$ or $0=\alpha\subset\beta$. The empty set is a subset of every later ordinal, because it is a subset of every set. More generally, every ordinal is a subset of every greater ordinal.

 

[Math Amateur]

Thanks for the help Andrew ...

Peter