Understanding Uncountable Well-Ordered Sets: An Intuitive Explanation

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In summary, Hurkyl defines an uncountable well ordered set which is isomorphic to another set with the same properties if both sets have an ordinal as their smallest element. He responds to a question about the set by saying that either you understand the proof or you don't and that the set exists because ordinals are well ordered.
  • #1
zefram_c
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I can't get a good intuitive grasp on this set. Folland defines it as follows:

Folland said:
There is an uncountable well ordered set S such that Ix={y in S: y<x} is countable for each x in S. If S' is another set with the same properties, then S and S' are order isomorphic.

Proof: Uncountable well-ordered sets exist by the WEP, let X be one. Either X has the desired property or there is a minimal element x0 s.t. Ix0 is uncountable, in which case let S be Ix0

My questions / problems with it are somewhat as follows:

1) I don't see how the set can be uncountable if any initial segment is countable.

2) How do we know that the element x0 used in the proof exists?
 
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  • #2
If you want an analogy, consider the first infinite ordinal, [itex]\omega[/itex]. While it is an infinite ordinal, but every initial segment is finite... if you can understand this, you should be able to understand the first uncountable ordinal.


x0 exists because ordinals are well ordered. Any collection of ordinals has a smallest element.
 
  • #3
Did you understand Hurkyl's response to your question?

I.e. either you meant to ask: why does it follow logically that the statements given are true? (which is an immediate consequence of their definitions and of logic)

or you meant to ask: "I understand the proof, but how can this be possible?"

Hurkyl answered the second version of your question.

re-reading, it seems your question 1) was the psychological one,
and your 2) was a (tauto)logical one.

ah yes, you said you wanted an intuitive grasp of the set, so Hurkyl understood you correctly.
 
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  • #4
Thanks Hurkyl - after reading your response, I went back and checked. Turns out my understanding of well-ordering was incorrect; I now get the logic. The analogy gave me food for thought, so I will think on it for now and see if I can persuade myself that it works.
 

FAQ: Understanding Uncountable Well-Ordered Sets: An Intuitive Explanation

What is a set of countable ordinals?

A set of countable ordinals is a collection of numbers that can be counted and arranged in a specific order. Each ordinal represents a unique position in the counting sequence.

How are countable ordinals different from cardinal numbers?

Countable ordinals represent the order or position of a number in a counting sequence, while cardinal numbers represent the quantity or size of a set. For example, the ordinal "first" represents the first position in a counting sequence, while the cardinal number "one" represents a set with one element.

Can a set of countable ordinals be infinite?

Yes, a set of countable ordinals can be infinite. For example, the set of natural numbers (1, 2, 3, ...) forms an infinite set of countable ordinals.

What is the relationship between set theory and countable ordinals?

Countable ordinals are an important concept in set theory, as they help to define and classify different types of sets. In particular, they are used to classify sets that can be well-ordered, meaning that the elements can be arranged in a specific order with no infinite decreasing sequences.

What are some real-world applications of countable ordinals?

Countable ordinals have applications in computer science, specifically in the study of algorithms and data structures. They can also be used to analyze and compare the complexity of different problems and algorithms. Additionally, they are used in mathematics to study the properties of infinite sets and to prove theorems about them.

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