Back when I studied chemistry I was introduced to Avogadros number which is numerically around 6.022x10**23 and represents the number of particles in a mole of a substance. (Google around for various explanations etc.). As it turns out, a mole of a substance weighs the same as it's atomic weight in grammes. A mole of water for example weighs 18 grammes which has a volume of 18 ml. Now start to think about the number of atoms of water in the world's oceans...
The point of this long winded rambling is that once you get into the area of counting molecules you need to use some really big numbers. (and that's without going sub-atomic). These numbers are big but they are still finite.
Infinity is not a number but the concept of a quantity that has no limits. In limiting the form of the JAPH in terms of the total number of characters that it can contain, you have defined limits. In order to say that there are an infinite number of JAPHs you need to expand at least one parameter so that it can always be increased. You could for example represent the JAPH using any number of character sets where each character could be expressed by an arbitrary number of bytes...
I could go on and on and on (ad infinitum)
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