# Thread: |all finite sentences| (countable alphabet) = countable, right?

1. ## |all finite sentences| (countable alphabet) = countable, right?

That is, the cardinality of the set of all finite sentences, when the alphabet is countable infinite -- I have the impression that this is countable, but when attempting to prove it, I end up making my collection look the same as an uncountable collection. Any pointers?

2. Originally Posted by nomadreid
That is, the cardinality of the set of all finite sentences, when the alphabet is countable infinite -- I have the impression that this is countable, but when attempting to prove it, I end up making my collection look the same as an uncountable collection. Any pointers?

The set of all finite sentences is the (at most) infinite countable union of the set of all sentences of one letter and

the set of all sentences of two letters and...etc.

As a countable union of countable sets is countable we're done.

Tonio

3. That makes sense. Thanks, Tonio.