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?
There are two questions here that need to be distinguished: (1) Does the decimal expansion of pi contain a large but finite string of consecutive 7's--say, 1000 consecutive 7's? (2) Does it contain an infinite string of consecutive 7's? The second question is the easier one. The only way that pi could contain a string of infinitely many consecutive 7's is if all the digits are 7's from some point on. And, as William has pointed out, that can't happen because pi is irrational. But the first question is harder. The digits of pi "look random." Imagine a number whose digits are generated by some random process--for example, we might roll a 10-sided die repeatedly to generate the digits. One could compute the probability that a string of 1000 consecutive 7's appears in the first n digits of this number. For n =1000, this number would be extremely small--you'd have to roll 1000 consecutive 7's on your first 1000 rolls, and that's very unlikely. But as n increases, the probability increases, and...