Proper Subsets of Ordinals .... .... Searcoid, Theorem 1.4.4 .... ....

In summary, proper subsets of ordinals are sets that contain only some, but not all, of the elements of another set of ordinals. To determine if a set is a proper subset of ordinals, one must check if all of its elements are also elements of the original set and if it has at least one element that is not in the original set. These subsets are significant in set theory and mathematical logic, as they help establish the hierarchy and relationships of sets. Theorem 1.4.4 states that for any set of ordinals, there is a least ordinal that is not an element of that set. Proper subsets of ordinals are also used in mathematical research to study properties and relationships of sets and in various areas of
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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 1.4.4 ...

Theorem 1.4.4 reads as follows:
View attachment 8457
In the above proof by Searcoid we read the following:

"... ... Now, for each \(\displaystyle \gamma \in \beta\) , we have \(\displaystyle \gamma \in \alpha\) by 1.4.2, and the minimality with respect to \(\displaystyle \in\) of \(\displaystyle \beta\) in \(\displaystyle \alpha \text{\\} x\) ensures that \(\displaystyle \gamma \in x\). ... ... Ca someone please show formally and rigorously that the minimality with respect to \(\displaystyle \in\) of \(\displaystyle \beta\) in \(\displaystyle \alpha \text{\\} x\) ensures that \(\displaystyle \gamma \in x\). ... ... 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 (including Theorem 1.4.2 ... ) ... so I am providing the same ... as follows:
View attachment 8458It 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 8459
View attachment 8460
Hope that helps ...

Peter
 

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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
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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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Dear Peter,

Thank you for sharing your questions and providing more context from Searcoid's book. I am happy to help you understand Theorem 1.4.4 in more detail.

First, let's review Definition 1.3.10, which states that a well-order on a set X is a relation that is reflexive, transitive, and linear on X. This means that every element in X is related to itself, the relation is transitive (if a is related to b and b is related to c, then a is related to c), and the relation is linear (for any two elements a and b in X, either a is related to b or b is related to a).

Now, let's look at Theorem 1.4.4. The proof starts by stating that for each ordinal \gamma in the set \beta, we know that \gamma is also in the set \alpha (from Theorem 1.4.2). This is because \beta is a subset of \alpha, and since \gamma is in \beta, it must also be in \alpha.

Next, the proof states that the minimality with respect to \in of \beta in \alpha \text{\\} x ensures that \gamma \in x. This means that since \gamma is in \alpha and \beta is the smallest element of \alpha that is not in x, \gamma must be in x. This is because if \gamma was not in x, then \beta would not be the smallest element of \alpha \text{\\} x.

To show this rigorously, we can prove by contradiction. Assume that \gamma is not in x. This means that \gamma is in \alpha \text{\\} x, which is a subset of \alpha. But this contradicts the fact that \beta is the smallest element of \alpha \text{\\} x, since \gamma is in \alpha \text{\\} x and is smaller than \beta. Therefore, our assumption was wrong and \gamma must be in x.

I hope this explanation helps you better understand Theorem 1.4.4. Let me know if you have any further questions.
 

FAQ: Proper Subsets of Ordinals .... .... Searcoid, Theorem 1.4.4 .... ....

What is a proper subset of ordinals?

A proper subset of ordinals is a set that contains some of the elements of another set of ordinals, but not all of them. In other words, it is a subset that is not equal to the original set.

How do you determine if a set is a proper subset of ordinals?

To determine if a set is a proper subset of ordinals, you need to check if all of its elements are also elements of the original set of ordinals, and if it has at least one element that is not in the original set.

What is the significance of proper subsets of ordinals?

Proper subsets of ordinals are important in set theory and mathematical logic, as they help to establish the hierarchy of sets and their properties. They also allow us to compare the sizes of different sets and understand their relationships.

What is Theorem 1.4.4 in the context of proper subsets of ordinals?

Theorem 1.4.4 is a theorem in set theory that states that for any set of ordinals, there is a least ordinal that is not an element of that set. In other words, for any set of ordinals, there is always a larger ordinal that is not included in that set.

How are proper subsets of ordinals used in mathematical research?

Proper subsets of ordinals are used in mathematical research to study the properties and relationships of sets, and to establish the foundations of mathematics. They are also used in various areas of mathematics, such as in topology and measure theory.

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