Properties of the Ordinals ....

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SUMMARY

The discussion centers on understanding Theorem 1.4.3 from Micheal Searcoid's book "Elements of Abstract Analysis," specifically regarding the well-ordering of ordinals. The theorem states that if $$\beta \subseteq \alpha$$, then $$\beta$$ is well-ordered by membership. Participants seek clarification on the formal demonstration required to prove that a subset of an ordinal is also well-ordered, emphasizing the necessity of showing that every subset of $$\beta$$ has a minimum element. The conversation highlights the importance of rigorous proof in set theory and ordinal analysis.

PREREQUISITES
  • Understanding of set theory concepts, particularly ordinals.
  • Familiarity with well-ordering principles as defined in mathematical literature.
  • Knowledge of formal proof techniques in mathematics.
  • Basic comprehension of the structure and content of "Elements of Abstract Analysis" by Micheal Searcoid.
NEXT STEPS
  • Study the concept of well-ordering in detail, focusing on Searcoid's Definition 1.3.10.
  • Review the proof techniques used in set theory, particularly those related to ordinals.
  • Examine additional examples of well-ordered sets to solidify understanding.
  • Explore the implications of Theorem 1.4.3 in broader mathematical contexts.
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Mathematics students, educators, and researchers interested in set theory, particularly those studying ordinals and well-ordering principles in abstract analysis.

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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:
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
 

Attachments

  • Searcoid - 1 - Theorem 1.4.3 ... ... PART 1 ... .....png
    Searcoid - 1 - Theorem 1.4.3 ... ... PART 1 ... .....png
    1.8 KB · Views: 132
  • Searcoid - 2 - Theorem 1.4.3 ... ... PART 2 ... ......png
    Searcoid - 2 - Theorem 1.4.3 ... ... PART 2 ... ......png
    13.6 KB · Views: 119
  • Searcoid - 1 - Start of section on Ordinals ... ... PART 1 ... .....png
    Searcoid - 1 - Start of section on Ordinals ... ... PART 1 ... .....png
    32.5 KB · Views: 125
  • Searcoid - Definition 1.3.10 ... .....png
    Searcoid - Definition 1.3.10 ... .....png
    9 KB · Views: 120
  • Searcoid - 2 - Definition 1.3.10 ... .....PART 2 ... ....png
    Searcoid - 2 - Definition 1.3.10 ... .....PART 2 ... ....png
    9.1 KB · Views: 116
Last edited:
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Peter said:
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 follows:Hope 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
 
Last edited:

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