Ordinals .... Searcoid, Theorem 1.4.6 .

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In summary, Theorem 1.4.6 in Michael Searcoid's book "Elements of Abstract Analysis" discusses the concept of ordinals and how they relate to sets. The proof of the theorem shows that if ##\alpha## and ##\beta## are not disjoint, then ##\alpha\cap\beta## is the ordinal 0. However, the proof still holds if ##\alpha## and ##\beta## are disjoint. This is because every ordinal is a subset of every greater ordinal, including the empty set.
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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:
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?temp_hash=8f9cff7393f9c0d811f87ad59134570a.png


My question regarding the above proof by Micheal Searcoid is as follows:

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

What happens to the proof if ##\alpha \cap \beta = \emptyset##?
Help will be appreciated ...

Peter
==========================================================================It may help Physics Forums 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:
?temp_hash=8f9cff7393f9c0d811f87ad59134570a.png

?temp_hash=c13d2e58ed130749d22a8e1847b2f2a4.png


Hope that helps ...

Peter
 

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  • #2
They can be disjoint. If they are disjoint then ##\alpha\cap\beta=\emptyset## which is the ordinal 0. The proof still works.

To get a feel for this, consider the case where say ##\alpha=0=\emptyset##. If you follow the proof through with this, you'll see that it still works and that either ##\beta=\alpha=0## or ##0=\alpha\subset\beta##. The empty set is a subset of every later ordinal, because it is a subset of every set. More generally, every ordinal is a subset of every greater ordinal.
 
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Thanks for the help Andrew ...

Peter
 

1. What are ordinals in mathematics?

Ordinals are a concept in mathematics that extends the idea of natural numbers to include infinite numbers and numbers that represent order or position in a series. They are often used in set theory and other branches of mathematics to describe the order or progression of elements.

2. How are ordinals represented in mathematics?

Ordinals are usually represented using lowercase Greek letters, such as α, β, γ, etc. They can also be represented using other symbols, such as the "ω" symbol for the first infinite ordinal.

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 it has a least element. This means that in a well-ordered set of ordinals, there is always a smallest or first element.

4. How are ordinals different from cardinal numbers?

Ordinals and cardinal numbers are both types of numbers used in mathematics, but they represent different concepts. Ordinals represent order or position in a series, while cardinal numbers represent quantity or size. For example, the ordinal "3rd" represents the third element in a series, while the cardinal number "3" represents a set with three elements.

5. What is the significance of ordinals in mathematics?

Ordinals play an important role in mathematics, particularly in set theory and logic. They help to describe the order and structure of mathematical objects and are used to prove theorems and make mathematical constructions. They also have applications in computer science, where they are used to represent and manipulate data structures.

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