Do all infinite countable sets have the same cardinality?
As I understand uncountable sets have different cardinalities (some infinites are bigger than other), but does that also apply to countable sets?
Asked
Active
Viewed 47 times
0
-
4By definition, countably infinite means that the set is bijective with the integers. – lulu Oct 26 '24 at 11:50
-
Countably infinite sets all have the same cardinality, equal to the cardinality of the natural numbers, and denoted $\aleph_0$. – Deepak Oct 26 '24 at 11:57
-
Fun fact: $\mathbb{R}$ and $\mathbb{R}^3$ has the same cardinality. Check: https://math.stackexchange.com/questions/183361/examples-of-bijective-map-from-mathbbr3-rightarrow-mathbbr – Sujit Bhattacharyya Oct 26 '24 at 11:58
-
@lulu By definition, a set is countable when it has a surjection from the natural numbers or is empty (equivalently, there is an injection into the natural numbers), and it is infinite when it is not finite. So something needs to be proven here. Still it's a duplicate for sure and should be closed. I will try to find duplicates. EDIT: found the duplicate using site search for 1min. – Martin Brandenburg Oct 26 '24 at 12:05
-
@MartinBrandenburg lulu was quite careful to say "countably infinite", not "countable". The OP was less careful in the second sentence, but did also specify the sets are infinite in the first sentence. – Theo Bendit Oct 26 '24 at 12:07
-
@MartinBrandenburg I agree that there is an ambiguity between "countable" and "countably infinite", which is why I specified the latter. The OP also clearly specifies the latter, at least initially, though they lapse into merely "countable" on the second pass. – lulu Oct 26 '24 at 12:12