GG5: ～な countable; enumerable; denumerable.
可算	21305
可算な	597
可算の	642

That "～な" is not strongly supported.
I think the key point is how the Japanese is glossed; not the precise definitions of the English words in the glossing.

The previous commenter, Robin Scott, claimed that the Wiktionary entry and Wikipedia entry supported the idea that a countable set could be finite. But Wiktionary gives two different definitions of the word in mathematics, and Wikipedia goes into some detail about the two different definitions. The previously-mentioned text by Walter Rudin gives the notion that countable means infinite only, but the book I used at university (G.F. Simmons "Introduction to Topology and Modern Analysis") defines countable to mean either finite or countably infinite.

https://en.wiktionary.org/wiki/countable#Adjective
(mathematics, of a set) finite or countably infinite; having a one-to-one correspondence (bijection) with a subset of the natural numbers.
https://www.collinsdictionary.com/dictionary/english/countable
able to be counted using the natural numbers; finite or denumerable
https://en.wikipedia.org/wiki/Countable_set
The terms enumerable and denumerable may also be used, e.g. referring to countable and countably infinite respectively, but as definitions vary the reader is once again advised to check the definition in use

A countable set can be finite.
The kokugos and 世界大百科事典 treat 可算集合 and 可付番集合 as synonyms.
Several sources say that "countable" and "enumerable" are equivalent whereas "denumerable" only means "countably infinite".

@@ -13,2 +13,3 @@
-<field>&ling;</field>
-<gloss>countable</gloss>
+<xref type="see" seq="2077860">可算名詞</xref>
+<field>&gramm;</field>
+<gloss>countable (of a noun)</gloss>
@@ -17,0 +19 @@
+<xref type="see" seq="2443750">可算集合</xref>
@@ -19 +21 @@
-<gloss>countable</gloss>
+<gloss>countable (of a set)</gloss>

The original entry wasn't great.

According to the translation of the Rudin book, 可付番 is denumerable or enumerable in mathematics. I'm not entirely sure whether this is the same as "countable", since "countable" in mathematics means that it is infinite. The Rudin translation gives "たかだか可算" for "at most countable" which means either finite or countable (infinite but denumerable). I'll try to look up in other texts later on. Unfortunately it seems that fools like myself rush in to add dictionary entries without the necessary expertise.