MHB Properties of the Ordinals ....

[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 1.4.3 ...

Theorem 1.4.3 reads as follows:
View attachment 8451
View attachment 8452In the above proof by Searcoid we read the following:

"... ... Then $$\beta \subseteq \alpha$$ so that $$\beta$$ is also well ordered by membership. ... ... To conclude that $$\beta$$ is also well ordered by membership, don't we have to show that a subset of an ordinal is well ordered?

Indeed, how would we demonstrate formally and rigorously that $$\beta$$ is also well ordered by membership. ... ... ?
Help will be appreciated ...

Peter
==========================================================================It may help MHB 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:
View attachment 8453

It may also help MHB readers to have access to Searcoid's definition of a well order ... so I am providing the text of Searcoid's Definition 1.3.10 ... as follows:

View attachment 8454
View attachment 8455Hope that helps,

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 need some help in fully understanding Theorem 1.4.3 ...

Theorem 1.4.3 reads as follows:

In the above proof by Searcoid we read the following:

"... ... Then $$\beta \subseteq \alpha$$ so that $$\beta$$ is also well ordered by membership. ... ... To conclude that $$\beta$$ is also well ordered by membership, don't we have to show that a subset of an ordinal is well ordered?

Indeed, how would we demonstrate formally and rigorously that $$\beta$$ is also well ordered by membership. ... ... ?
Help will be appreciated ...

Peter
==========================================================================It may help MHB 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:It may also help MHB readers to have access to Searcoid's definition of a well order ... so I am providing the text of Searcoid's Definition 1.3.10 ... as followsope that helps,

Peter

I have been reflecting on the above post on the ordinals ...Maybe to show that that $$\beta$$ is also well ordered by membership, we have to demonstrate that since every subset of $$\alpha$$ has a minimum element then every subset of $$\beta$$ has a minimum element ... but then that would only be true if every subset of $$\beta$$ was also a subset of $$\alpha$$ ...

Is the above chain of thinking going in the right direction ...?

Still not sure regarding the original question ...

Peter