Properties of the Ordinals ....

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In summary, Peter is reading Michael Searcoid's book "Elements of Abstract Analysis" and is currently focused on understanding Chapter 1: Sets, particularly Section 1.4 on Ordinals. Peter is seeking help in understanding Theorem 1.4.3, which states that if a subset of an ordinal is well ordered by membership, then the ordinal itself is also well ordered by membership. Peter also shares some additional information, including the start of Searcoid's section on ordinals and Searcoid's definition of a well order. He also reflects on the original question and wonders if his chain of thinking is going in the right direction.
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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 \(\displaystyle \beta \subseteq \alpha\) 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:
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
 

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  • Searcoid - 1 - Theorem 1.4.3 ... ... PART 1 ... .....png
    Searcoid - 1 - Theorem 1.4.3 ... ... PART 1 ... .....png
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  • Searcoid - 2 - Theorem 1.4.3 ... ... PART 2 ... ......png
    Searcoid - 2 - Theorem 1.4.3 ... ... PART 2 ... ......png
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  • Searcoid - 1 - Start of section on Ordinals ... ... PART 1 ... .....png
    Searcoid - 1 - Start of section on Ordinals ... ... PART 1 ... .....png
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  • Searcoid - Definition 1.3.10 ... .....png
    Searcoid - Definition 1.3.10 ... .....png
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  • Searcoid - 2 - Definition 1.3.10 ... .....PART 2 ... ....png
    Searcoid - 2 - Definition 1.3.10 ... .....PART 2 ... ....png
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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 \(\displaystyle \beta \subseteq \alpha\) 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 have been reflecting on the above post on the ordinals ...Maybe to show that that \(\displaystyle \beta\) is also well ordered by membership, we have to demonstrate that since every subset of \(\displaystyle \alpha\) has a minimum element then every subset of \(\displaystyle \beta\) has a minimum element ... but then that would only be true if every subset of \(\displaystyle \beta\) was also a subset of \(\displaystyle \alpha\) ...

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

Still not sure regarding the original question ...

Peter
 
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Related to Properties of the Ordinals ....

1. What are ordinals?

Ordinals are a type of mathematical number that represents a position or rank in a sequence. They are often used to describe the order of objects or events, and are denoted by the letters "o" or "ω".

2. How are ordinals different from cardinals?

While both ordinals and cardinals are types of numbers, they differ in the way they are used. Ordinals represent the position or rank in a sequence, while cardinals represent the quantity or number of objects in a set. For example, the ordinal "2nd" represents the second position in a sequence, while the cardinal "2" represents a set of two objects.

3. What is the significance of the order type of ordinals?

The order type of an ordinal is a way of categorizing and comparing different ordinals based on their structure and properties. It helps to determine the relationship between different ordinals and their corresponding positions in a sequence.

4. How are ordinals used in set theory?

Ordinals play a crucial role in set theory as they are used to define the size or cardinality of a set. They also help to establish the hierarchy of different sets and their subsets, known as the well-ordering principle.

5. What are some real-life applications of ordinals?

Ordinals are used in various fields, such as computer science, finance, and linguistics. They are also used in everyday life, such as describing the ranking of sports teams or the ordering of items on a menu. In computer science, ordinals are used in programming languages to organize and access data. In finance, they are used to rank investments based on their performance. In linguistics, ordinals are used to describe the order of words in a sentence.

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