Well-Ordering Theorem & Countably Finite Sets: Analysis

In summary, the Well-Ordering Theorem states that every non-empty set of positive integers has a least element, and this principle can be applied to prove the existence of countably infinite sets. Countably finite sets are those that can be mapped one-to-one with the set of natural numbers, and they play a crucial role in analysis as they allow for the study of infinite processes and sequences. This theorem has applications in various fields of mathematics, including set theory, topology, and number theory.
  • #1
ehrenfest
2,020
1

Homework Statement


Isn't the solution at the site http://www.kalva.demon.co.uk/putnam/psoln/psol849.html incomplete because the author assumes he can map the set to Z and we were not given that the set was countably finite? The well-ordering theorem (that states any set can be well-ordered) does not allow you to add indices like that to the set, right?


Homework Equations





The Attempt at a Solution

 
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  • #2
ehrenfest said:

Homework Statement


Isn't the solution at the site http://www.kalva.demon.co.uk/putnam/psoln/psol849.html incomplete because the author assumes he can map the set to Z and we were not given that the set was countably finite? The well-ordering theorem (that states any set can be well-ordered) does not allow you to add indices like that to the set, right?

Where does he map the set to Z? He's given a set with n elements, hence a finite set, so what he does is perfectly well justified.
 
  • #3
d_leet said:
Where does he map the set to Z? He's given a set with n elements, hence a finite set, so what he does is perfectly well justified.

You're right. However, if you were not given that the set were finite or even countable infinite, would you still be allowed to use indices like that? Using the indices i is basically an injection from your set to Z, right?
 
  • #4
ehrenfest said:
You're right. However, if you were not given that the set were finite or even countable infinite, would you still be allowed to use indices like that? Using the indices i is basically an injection from your set to Z, right?
If you well-order the set, then I believe you can pull this off using transfinite recursion.
 

FAQ: Well-Ordering Theorem & Countably Finite Sets: Analysis

What is the Well-Ordering Theorem?

The Well-Ordering Theorem is a fundamental result in mathematics that states that every non-empty set of positive integers has a least element. In other words, any set of positive integers can be arranged in a sequence such that each element is greater than the previous one.

How is the Well-Ordering Theorem used in mathematical proofs?

The Well-Ordering Theorem is often used to prove the existence of certain mathematical objects, such as the natural numbers or prime factorizations. It can also be used to prove the existence of solutions to certain equations or inequalities.

What is a countably finite set?

A countably finite set is a set that contains a finite number of elements. However, unlike a typical finite set, the elements in a countably finite set can be put in a one-to-one correspondence with the natural numbers. This means that the elements can be counted, hence the term "countably finite."

How is the Well-Ordering Theorem related to countably finite sets?

The Well-Ordering Theorem is often used to show that countably finite sets are well-ordered, meaning that they can be arranged in a sequence such that each element is greater than the previous one. This is because any non-empty subset of a countably finite set must have a least element, as guaranteed by the Well-Ordering Theorem.

Can the Well-Ordering Theorem be generalized to sets other than the positive integers?

Yes, the Well-Ordering Theorem can be generalized to other sets, such as the real numbers. However, this requires the use of more advanced mathematical concepts, such as transfinite ordinals. The basic idea of the theorem remains the same, but the proof becomes more complex when dealing with larger sets.

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