Understanding Proper Subsets of Ordinals in Searcoid's Theorem 1.4.4 - Peter

In summary, the conversation is about understanding Chapter 1: Sets and specifically Section 1.4 Ordinals. There is a question about the proof of Theorem 1.4.4, which states that a partially ordered set is totally ordered if for every pair of distinct members, there is a clear order between them. The question is about the three possible alternatives in the proof: \delta \in \beta, \delta = \beta, or \beta \in \delta, and how the = alternative fits into the definition of totally ordered. The answer is that if the pair of members is not distinct, then they are considered equal and the = alternative applies.
  • #1
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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 have another question regarding the proof of Theorem 1.4.4 ...

Theorem 1.4.4 reads as follows:
View attachment 8463
In the above proof by Searcoid we read the following:

"... ... Moreover, since \(\displaystyle x \subset \alpha\), we have \(\displaystyle \delta \in \alpha\). But \(\displaystyle \beta \in \alpha\) and \(\displaystyle \alpha\) is totally ordered, so we must have \(\displaystyle \delta \in \beta\) or \(\displaystyle \delta = \beta\) or \(\displaystyle \beta \in \delta\) ... ... "My question is regarding the three alternatives \(\displaystyle \delta \in \beta\) or \(\displaystyle \delta = \beta\) or \(\displaystyle \beta \in \delta\) ... ...Now ... where \(\displaystyle (S, <)\) is a partially ordered set ... \(\displaystyle S\) is said to be totally ordered by \(\displaystyle <\) if and only if for every pair of distinct members \(\displaystyle x, y \in S\), either \(\displaystyle x < y\) or \(\displaystyle y < x\) ... ..So if we follow the definition exactly in the quote above there are only two alternatives ... \(\displaystyle \delta \in \beta\) or \(\displaystyle \beta \in \delta\) ... ...My question is ... where does the \(\displaystyle =\) alternative come from ... ?

How does the \(\displaystyle =\) alternative follow from the definition of totally ordered ... ?Help will be appreciated ...

Peter
 

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Peter said:
Now ... where \(\displaystyle (S, <)\) is a partially ordered set ... \(\displaystyle S\) is said to be totally ordered by \(\displaystyle <\) if and only if for every pair of distinct members \(\displaystyle x, y \in S\), either \(\displaystyle x < y\) or \(\displaystyle y < x\) ... ..
Peter

Please, read this definition very very carefully, and ask yourself: what if the pair of members is/are not distinct ?
 
  • #3
steenis said:
Please, read this definition very very carefully, and ask yourself: what if the pair of members is/are not distinct ?

Thanks Steenis ...

See that key term is "distinct"... if not distinct then members are equal ... enough said ...

Thanks for your help ...

Peter
 

1. What is the significance of ordinals in Searcoid's Theorem 1.4.4?

Ordinals are used in Searcoid's Theorem 1.4.4 to represent the relative order of elements in a set. This allows for the comparison of different subsets of a set and helps to establish the proper subset relationship.

2. How does Searcoid's Theorem 1.4.4 define proper subsets?

Searcoid's Theorem 1.4.4 defines a proper subset as a subset that contains fewer elements than the original set. In other words, all elements in the proper subset must also be present in the original set, but the proper subset may not contain all the elements of the original set.

3. Can you provide an example of a proper subset in Searcoid's Theorem 1.4.4?

Yes, for example, if we have a set A = {1, 2, 3, 4} and a proper subset B = {1, 2, 3}, then B is a proper subset of A because it contains fewer elements than A, but all elements in B are also present in A.

4. How is the concept of ordinals used in Searcoid's Theorem 1.4.4?

The concept of ordinals is used to establish the relative order of elements in a set. By utilizing ordinals, we can compare different subsets of a set and determine whether one subset is a proper subset of another.

5. What are some practical applications of understanding proper subsets in Searcoid's Theorem 1.4.4?

Understanding proper subsets in Searcoid's Theorem 1.4.4 can be useful in various fields, such as mathematics, computer science, and data analysis. It can help in organizing and comparing data sets, identifying relationships between sets, and proving theorems in mathematics and computer science. It also has practical applications in set theory, combinatorics, and other areas of mathematics.

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