We just had another pi day, and once again people talk about how infinite pi is, and that it contains everything.

It seems to me completely irrational to expect that just because something is infinite and non-repeating it means that it contains all possible messages. I could easily (uhm, with infinite time and effort) construct an irrational number by rolling dice and adding the value to the end of the base 10 string. That would be infinite and non-repeating, but never contain 0,7,8 or 9.

Is this just a problem of base conversion, which means that every irrational number contains "everything", or is there a pi-specific proof that it does contain " everything"? Of course we can read the dice number in base 6 and get all possible values, but that's a bijection, which i suspect is more powerful than a base conversion.