system
Given the following post, respond as if you're commenting on Reddit:
human
If something is infinite, is it also necessarily exhaustive? Is the "infinite monkeys on typewriters will write Shakespeare" trope true?

Not sure if I used the precise terminology ("exhaustive"), but the "an infinite number of monkeys typing on typewriters will eventually write Shakespeare" adage is a misrepresentation of infinity, correct?  Like for instance, I could have an infinite set of numbers that never included the number 1234, right?  It could just have 1233 and then expand into infinite numbers that start with 1233 without ever including 1234, and still meet the definition of "infinite", right?


I guess my question really is: does something have to include all possible outcomes to truly be "infinite"?  Or can something have infinite outcomes but not all possible outcomes?
gpt
In the monkey example it is assumed the output of the typing is one long string of alphanumeric symbols for which each symbol is chosen uniformly from a fixed and finite alphabet. So for any postion, the chance of it being one symbol is the same as for any other symbol. 

Such a string of uniformly chosen symbols has the property of being normal. This means that not only do all individual symbols occur with the same probability, but all possible strings of length 2 have the same chance of appearing. The same goes for strings of any finite length. (See Wikipedia's article on normal numbers for a more precise definition.) 

The entire works of Shakespeare is just a finite string of symbols. So it will eventually appear, and it will appear infinitely often, and just as often as any other string of the same length.

Your intuition is correct though. Not all infinite strings are normal. There is no reason an infinite string of digits has to contain the string 1234 or, indeed, any given finite string.