I came across a statement that "every countable set is the range of a 1-1 function defined on J" under Definition 2.7 on Page 26 of the book The principles of Mathematic Analysis. So must a countable set be infinite? If no, what does that statement mean? If yes, is a finite set a countable set? Thanks.
Asked
Active
Viewed 82 times
0
-
11It depends on how exactly it has been defined. Some authors allow "countable" to refer to any set which can be put into bijection with any subset (finite or otherwise) of the natural numbers in which case yes finite sets may be countable. If a distinction needs to be made, we call the sets "finite" and "countably infinite" sets respectively. Other authors reserve the word "countable" to refer only to those infinite sets which are in bijection with the entirety of $\Bbb N$. – JMoravitz Feb 10 '19 at 03:27
-
Yes, it varies by the author whether finite sets are considered "countable". But the sources I'm familiar with generally say that finite sets are countable. It does seem a bit strange to consider ${1,2,3}$ to not be "countable"... – Jair Taylor Feb 10 '19 at 03:37
-
Related questions: (1), (2), (3), and (4). None of these are precise duplicates, but the answers provided therein seem to address the question here. I am going to flag this as a duplicate of (3), as this answer to that question seems to do the job. – Xander Henderson Feb 10 '19 at 04:18