Proving Dedekind Infiniteness of Countable Sets | Solution Attempt

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SUMMARY

The discussion centers on proving that every countable set is Dedekind infinite, defined as having a one-to-one mapping onto its proper subset. The proposed solution involves well-ordering the countable set and selecting an element 'a' to ensure that all other elements are mapped to values greater than 'a'. This approach requires a more rigorous definition of the mapping process to validate the proof. The necessity for clarity in defining countable sets is also emphasized.

PREREQUISITES
  • Understanding of Dedekind infinite sets
  • Familiarity with countable sets and their properties
  • Knowledge of well-ordering principles
  • Basic concepts of one-to-one mappings in set theory
NEXT STEPS
  • Study the formal definition of countable sets in set theory
  • Explore the principles of well-ordering and its implications
  • Learn about one-to-one mappings and their applications in proofs
  • Investigate rigorous proof techniques in mathematical logic
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Mathematics students, educators, and anyone interested in set theory and the foundations of mathematics will benefit from this discussion.

cragar
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Homework Statement


Call a set X Dedekind infinite if there is a 1-to-1 mapping of X onto
its proper subset.
Prove that every countable set is Dedekind infinite.

The Attempt at a Solution


I want to say that every countable set can be well ordered.
I guess I could just pick some element from our set X and call it a.
And then make sure everything from our set gets mapped to something
larger than a. So we have a 1-to-1 mapping to our proper subset.
I probably need to be more rigorous about how this mapping takes place.
 
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cragar said:

Homework Statement


Call a set X Dedekind infinite if there is a 1-to-1 mapping of X onto
its proper subset.
Prove that every countable set is Dedekind infinite.

The Attempt at a Solution


I want to say that every countable set can be well ordered.
I guess I could just pick some element from our set X and call it a.
And then make sure everything from our set gets mapped to something
larger than a. So we have a 1-to-1 mapping to our proper subset.
I probably need to be more rigorous about how this mapping takes place.

Yes, you should be more explicit about what the mapping is. What's the definition of countable set?
 

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