1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Is every countable set infinite?

  1. Sep 24, 2008 #1
    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
     
  2. jcsd
  3. Sep 24, 2008 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    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.
     
  4. Sep 24, 2008 #3
    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
     
  5. Sep 24, 2008 #4

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?