Grammatical and Mathematical Tip of the Day

The local Humane Society is running an ad on the radio asking for help taking care of all the pets that have been abandoned by people who have lost their jobs and/or homes and can no longer afford to keep them. While this is a very important message, their phrasing really irritates me every time I hear the ad.

They say "a countless number of pets" have been abandoned. This is wrong on many levels:

1. You cannot say "a countless number." It doesn't sound right. Go with either "countless pets" or "a number of pets."
2. If you're going to say "a countless number" and ignore the grammar, then I'm going to get pedantic on you and say that the number of pets cannot be countless. You see, countless means "too many to be counted," and it is impossible to have a number of pets that is too large to be counted. Too large to count easily or quickly? Sure! But it can't be too large to be counted. Here's why:

There are only non-negative integer numbers of pets. That is to say, you can't have 3.7 pets or √3 pets or -28 pets. Therefore, the number of pets must be countable.

In other words, if you took all the pets and lined them all up, you could map them one-to-one with the set of natural numbers (non-negative integers), kind of like this: {(Fido, 0), (Goldie, 1), (Muffin, 2), (Fifi, 3), ...}. Therefore, we could take the size of the set of one-to-one mappings of pets and natural numbers, and determine the number of pets.

I understand that what they really mean is an unknown and probably very large number of pets that need help. But they should express it in a more mathematically accurate way, especially if they're counting on the support of mathematicians.