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- Jun 22, 2012

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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 \(\displaystyle \beta \subseteq \alpha\) so that \(\displaystyle \beta\) is also well ordered by membership. ... ...

To conclude that \(\displaystyle \beta\) 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 \(\displaystyle \beta\) 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 follows:

Hope that helps,

Peter

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 \(\displaystyle \beta \subseteq \alpha\) so that \(\displaystyle \beta\) is also well ordered by membership. ... ...

To conclude that \(\displaystyle \beta\) 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 \(\displaystyle \beta\) 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 follows:

Hope that helps,

Peter

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