Proving Dedekind Infiniteness of Countable Sets | Solution Attempt

[cragar]

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.

 

[Dick]

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?