> Is it not outputting all the real numbers between 0 and 10? > By tracing a suitable path down the tree we can find any > real number we care to. (Yes, no, maybe?) > > On the other hand, there is a hypothesis that the digits > of pi are "normal" -- containing all possible 2-digit > sequences (not only, but they all occur 1/100 of the time.) > Likewise all possible 3-digit sequences, etc. If true, > then pi contains somewhere in its decimal expansion, > sequentially, an encoding of the complete works of > Shakespeare (any edition), complete encodings of the > bible (any edition, any translation.) The sum total > of human output all in one number, not just past knowledge, > but all books yet to be written as well! [Imagine the > enormity of the knowledge contained in the whole tree! > Perhaps we should call it the "god tree (tm)".]
The tree "paradox" is a standard technique/idea in set theory and logic, and a good starting point for those interested would be to google "infinite binary tree" and "binary tree" AND "real numbers".
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Incidentally, for pi to have this property, a much weaker hypothesis than "pi is a normal number" suffices.
Almost all real numbers, in the sense of Lebesgue measure, are normal (i.e. the set of non-normal numbers has Lebesgue measure zero), but the opposite is true in the case of Baire category (almost all real numbers, in the sense of Baire category, are NOT normal). Thus, while the set of normal numbers is really big in the sense of Lebesgue measure, it's also really small in the sense of Baire category.
On the other hand, the property of having all finite strings of decimals appearing in a real number's decimal expansion holds for real numbers that form a much larger set of real numbers. Indeed, this set of real numbers is really big in the sense of Lebesgue measure AND really big in the sense of Baire category. In fact, the distinction is even more extreme than this, as I indicate in the post below.