Ordinals .... Searcoid, Theorem 1.4.6 ....

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In summary, ordinals are a type of number that represent the order or position of objects or numbers in a sequence. They are different from cardinals, which represent quantity or size. Searcoid's Theorem 1.4.6 is a mathematical theorem that states that if a set of ordinals is well-ordered, then every subset of that set is also well-ordered. In set theory, ordinals are used to define different levels of infinity and compare the sizes of infinite sets. Ordinals cannot be negative as they start at 0 and continue on to infinity.
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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 1.4.6 ...

Theorem 1.4.6 reads as follows:

View attachment 8467
View attachment 8468
My question regarding the above proof by Micheal Searcoid is as follows:

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

What happens to the proof if \(\displaystyle \alpha \cap \beta = \emptyset\)?
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 8469
View attachment 8470Hope that helps ...

Peter
 

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  • Searcoid - 2 -  Theorem 1.4.6 ... ... 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 - 2 - Start of section on Ordinals ... ... PART 2 ... ......png
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Dear Peter,

Thank you for reaching out for help with understanding Theorem 1.4.6 in Michael Searcoid's book. Theorem 1.4.6 states: "If $\alpha$ and $\beta$ are ordinals, then $\alpha \cup \beta$ is an ordinal." This theorem is important because it shows that the union of two ordinals is itself an ordinal, which is a fundamental concept in abstract analysis.

To answer your question, we do not know for certain that $\alpha$ and $\beta$ are not disjoint. In fact, they could potentially be disjoint. However, the proof of Theorem 1.4.6 does not rely on the assumption that $\alpha$ and $\beta$ are not disjoint. The proof simply shows that if $\alpha$ and $\beta$ are ordinals, then their union $\alpha \cup \beta$ is also an ordinal.

If $\alpha \cap \beta = \emptyset$, the proof still holds. This is because the definition of an ordinal does not require that all of its elements be distinct. In other words, the elements of $\alpha$ and $\beta$ can overlap, but their union will still be an ordinal.

I hope this clarifies your question and helps you better understand Theorem 1.4.6. If you have any further questions, please don't hesitate to ask.
 

FAQ: Ordinals .... Searcoid, Theorem 1.4.6 ....

1. What are ordinals?

Ordinals are a mathematical concept that represents the order of objects or numbers in a sequence. They are often used to describe the position of an object in a list or the rank of a number in a set.

2. How are ordinals different from cardinals?

Ordinals and cardinals are both types of numbers, but they have different uses. Ordinals are used to describe position or order, while cardinals are used to represent quantity or size.

3. What is Searcoid's Theorem 1.4.6?

Searcoid's Theorem 1.4.6 is a mathematical theorem that states that if a set of ordinals is well-ordered, then every subset of that set is also well-ordered. In simpler terms, this means that if a list of objects is in a specific order, then any smaller list within that larger list will also be in a specific order.

4. How are ordinals used in set theory?

In set theory, ordinals are used to define the different levels of infinity. Each ordinal number represents a different level of infinity, and they are used to compare the sizes of different infinite sets.

5. Can ordinals be negative?

No, ordinals cannot be negative. They start at 0 and continue on to infinity. Negative numbers are not considered ordinals because they do not represent a position or order in a sequence.

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