Well-Ordering Theorem & Countably Finite Sets: Analysis

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Homework Help Overview

The discussion revolves around the well-ordering theorem and its application to a set that is not explicitly stated to be countably finite. Participants are analyzing the implications of mapping a set to the integers and questioning the assumptions made regarding the nature of the set.

Discussion Character

  • Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants are questioning whether the mapping of a set to the integers is justified without the set being defined as countably finite. They explore the implications of using indices and the conditions under which such mappings are valid.

Discussion Status

The discussion is ongoing with participants raising valid points about the assumptions made in the original solution. There is a productive exchange regarding the conditions necessary for applying the well-ordering theorem and the implications of using indices for mapping.

Contextual Notes

There is uncertainty regarding the nature of the set in question, specifically whether it is finite or countably infinite, which affects the validity of the arguments presented. Participants are navigating these constraints while discussing the theorem's application.

ehrenfest
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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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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.
 
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?
 
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.
 

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