Is this for philosophers, mathematicians, or logicians? But here goes:
Given that the decimal places of pi continue to infinity, does this imply that somewhere in the sequence of numbers of pi there must be, for instance, a huge (and possibly infinite) number of the same number repeated? 77777777777777777777777777... , say?
If Pi goes on forever, you might think it must be. After all, if you checked pi to the first googol decimal places you obviously would't find an infinite number of anything. Try a googlplex! Still nothing.
But we haven't scratched the surface, even though the universe would have fizzled out by now. If pi's decimal places go on forever, there may be, (not just 77777777777777... or 1515151515151) but all of them, in all combinations, forever. After all, you only have to say "You've only checked a googolplex. There's still an infinite number to to check. The universe is long gone, but pi goes on and on."
Philosophers, mathematicians, logicians, any ideas?