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I found the following definition:

Definition. A set is countable iff its cardinality is either finite or equal to $\aleph_0$. A set is denumerable iff its cardinality is exactly $\aleph_0$. A set is uncountable iff its cardinality is greater than $\aleph_0$.

The null set is countable. The finite set, {A, B, C}, is countable. The infinite set, $\mathbb{N}$, is countable and denumerable. Sets with a larger cardinality than $\mathbb{N}$ are uncountable.

I have trouble with seeing the difference between countable and denumerable, apart from the part that the cardinality is finite. Isn't "A set is countable iff its cardinality equal to $\aleph_0$" and "A set is denumerable iff its cardinality is exactly $\aleph_0$" the same?

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  • $\begingroup$ $\{1\}$ is countable but not denumerable. $\endgroup$ – Santiago Canez Sep 11 '15 at 18:45
  • $\begingroup$ But countable set may be finite also! $\endgroup$ – Arpit Kansal Sep 11 '15 at 18:46
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    $\begingroup$ Maybe this will help: If a countable set is not finite, then it is denumerable. In fact if you rule out "finite" the two definitions are the same, countable and denumerable. $\endgroup$ – user2566092 Sep 11 '15 at 18:46
  • $\begingroup$ The answer to your final question is yes. However, ‘A set is countable iff its cardinality is equal to $\aleph_0$’ is different from the definition that you quote. The difference between the terms as defined here is precisely that countable is denumerable or finite. (You should note that this is a somewhat non-standard usage of denumerable. In my experience it usually means finite or countably infinite, just as countable does.) $\endgroup$ – Brian M. Scott Sep 11 '15 at 18:47
  • $\begingroup$ @SantiagoCanez Yes, that's what's in the definition, I tried to rule that one out because I did get that one. $\endgroup$ – Garth Marenghi Sep 11 '15 at 18:49
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Every square is a rectangle, but not every rectangle is a square. Similarly, every denumerable set is countable, but not every countable set is denumerable. If you want, think of "denumerable" as an abbreviation for "countable and infinite" (or think of "countable" as an abbreviation for "denumerable or finite").

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  • $\begingroup$ Got it! But could you elborate as to why the author chose to not only say denumerable, but countable as well for the infinite set $\mathbb{N}$? $\endgroup$ – Garth Marenghi Sep 12 '15 at 8:33
  • $\begingroup$ @GarthMarenghi They did that just to clarify the relationship between the two terms. $\endgroup$ – Noah Schweber Sep 12 '15 at 18:44
  • $\begingroup$ I managed to track down the author's email address and asked him the question I asked here. His repsonse was "I was introducing the terms to readers who might not have known them before. I thought that some redundancy was a price worth paying to show that that "countable" and "denumerable" were not mutually exclusive." So you were completely right. $\endgroup$ – Garth Marenghi Sep 12 '15 at 21:03
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Basically, the issue is a terminological one, and thus subject to change.

There are book were "countable" is not used at all.

See :

page 100 : Definition 5. [A set] $A$ is finite if and only if ...

page 150 : Definition 23. A set is infinite if and only if it is not finite.

page 151 : Theorem 41. The set $\omega$ of natural numbers is infinite.

Definition 24. A set is denumerable if and only if it is equipollent to the set $\omega$ of all natural numbers.

Theorem 43. Every denumerable set is infinite.

page 191 : Theorem 59. The set of real numbers is not denumerable.


Other authors "suppress" denumerable; see

Definition I.11.14 $A$ is countable iff $A \preccurlyeq \omega$. $A$ is finite iff $A \preccurlyeq n$ for some $n \in \omega$. "infinite" means "not finite". "uncountable" means "not countable". $A$ is countably infinite iff $A$ is countable and infinite.

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  • $\begingroup$ Interesting to see different uses or choices etc. being used to denote the same thing. Sometimes confusing though :\ $\endgroup$ – Garth Marenghi Sep 12 '15 at 21:05

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