> I recently came to understand why an infinite > series of coin flips will never result in all > heads or all tails even though either of such a > series is in fact one of the possible series among > all the infinite possibilities.
Umm. No, you still need to work on that, I think.
The reason an infinite series of coin flips wil never result in all heads or all tails is because an infinite series will never "result" [at least not in the "cardinals" kind of infinity].
You can't, at least in a proof, talk about doing something with an infinite series "at the end", or "after it's done". There doesn't seem to _be_ an "end". It is _never_ "done".
You cannot talk about what eventuates "after" an infinite series. Unlike a finite series, an infinite series isn't a product, it's a process.
The best you can do is to describe the situation after any finite number of steps, and then describe a situation the infinite series "approximates more and more closely with more and more steps", if there is any such; or describe a situation "after the next step" if you are, say, doing a proof by induction.
Or so I understand.
Now the theory of "ordinals", which I do not understand _at all_, works under entirely different rules, and in particular, you _can_ stick something onto the far end of an infinity and have that make sense.
