Is every countable set infinite?

[Ronn]

Question is the same as the title.

What I think is that since every countable set C ~ Z+( all positive integers) and Z+ is infinite, then C is also infinite. Sounds straightfoward but I need to check it.

Thanks,

Ronn

 

[HallsofIvy]

Yes, using that definition- a set having a one-to-one correspondence with the positive real numbers- is necessarily infinite. You can use the phrase "at most countable" to include finite sets if you like.

Be aware, however, that some books use the term "countable" to include finite sets- defining countable to mean there is a one-to-one function from the set into (not necessarily "onto") the positive integers - and then say "countably infinite" for the situation above.

 

[Ronn]

Thanks for the really fast reply. I've got another related question.

1. If A is a subset of B and B is countable, then A is at most countable?

If 1 is correct, then A is either finite or countable by definition.

2. What if A is known to be infinite,then is it safe to say A is countable? i.e., Since A is either finite or countable, if A is infinite it is the other case where A is countable.

Thanks again!

Ronn

 

[Dick]

Yes, 1 is correct. For 2, sure, if A is contained in the countable set B, then it's at most countable. So if it's infinite, then it's countable.