Ordinals .... Searcoid, Corollary 1.4.5 .... ....

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In summary, the conversation discusses the book "Elements of Abstract Analysis" by Micheal Searcoid, specifically focusing on Chapter 1: Sets and Section 1.4 Ordinals. The reader is seeking help in understanding the Corollary to Theorem 1.4.4, which states that for all ordinals \alpha and \beta, \beta \in \alpha \Longleftrightarrow \beta \subset \alpha. The reader has questions regarding the proof of this corollary and the definition of an ordinal according to Searcoid.
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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 the Corollary to Theorem 1.4.4 ...

Theorem 1.4.4 and its corollary read as follows:
View attachment 8465
Searcoid gives no proof of Corollary 1.4.5 ...Question 1

To prove Corollary 1.4.5 we need to show \(\displaystyle \beta \in \alpha \Longleftrightarrow \beta \subset \alpha\) ... ...
Assume that \(\displaystyle \beta \in \alpha\) ...

Then by Searcoid's definition of an ordinal (Definition 1.4.1 ... see scanned text below) we have \(\displaystyle \beta \subseteq \alpha\) ...

But it is supposed to follow that \(\displaystyle \beta\) is a proper subset of \(\displaystyle \alpha\) ... !

Is Searcoid assuming that \(\displaystyle \beta \neq \alpha\) ... if not how would it follow that \(\displaystyle \beta \subset \alpha\) ... ?
Question 2The Corollary goes on to state that, in particular, if \(\displaystyle \alpha \neq 0\) then \(\displaystyle 0 \in \alpha\) ... can someone please show me how to demonstrate that this is true ... ?
Hope someone can help ...

Peter

============================================================================It may help readers of the above post if the start of the section on ordinals was accessible ... so I am providing that text as follows:
View attachment 8466
 

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    Searcoid - Theorem 1.4.4 ... ....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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Definition 1.4.1. (Ordinal) An ordinal is an ordered set \alpha such that:(i) \alpha is nonempty and transitive: for every x \in \alpha, if y \in x then y \in \alpha;(ii) If x, y \in \alpha and x \neq y, then either x \subseteq y or y \subseteq x. Theorem 1.4.4. Let \alpha be an ordinal. Then \alpha is well-ordered.Corollary 1.4.5. For all ordinals \alpha and \beta, \beta \in \alpha \Longleftrightarrow \beta \subset \alpha.
 

FAQ: Ordinals .... Searcoid, Corollary 1.4.5 .... ....

What are ordinals in the context of mathematics?

Ordinals are a type of number used in mathematics to describe the position or rank of an object in a sequence. They are typically represented by a number and a suffix, such as "1st" or "2nd".

How are ordinals different from other types of numbers?

Unlike other numbers, ordinals have a specific order and cannot be added, subtracted, or multiplied. They are used to describe the relative position of objects in a sequence, rather than their numerical value.

What is the significance of Searcoid's Corollary 1.4.5 in relation to ordinals?

Searcoid's Corollary 1.4.5 is a mathematical principle that states that for any two ordinals, one is either equal to, less than, or greater than the other. This helps establish a clear hierarchy among ordinals and allows for comparisons to be made between them.

Can ordinals be used in real-life situations?

Yes, ordinals can be used to describe the order of events, such as first, second, third, etc. They can also be used in sports rankings, academic standings, and other situations where there is a clear order or ranking of objects.

How are ordinals used in set theory?

In set theory, ordinals are used to describe the order or hierarchy of sets. They are also used to establish the cardinality (size) of a set, as the smallest ordinal that cannot be put into a one-to-one correspondence with the elements of the set.

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