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

[Math Amateur]

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:
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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:

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Hope that helps ...

Peter

 

[mmwave]

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.