(June 14, 2020 at 12:28 pm)polymath257 Wrote: There is an even deeper level. Alan Turing defined the concept of a 'computable real number'. In essence, it is one for which some computer program can compute the digits in the decimal expansion of that number.
So, for example, pi *is* a computable number: we have programs that will give as many digits of pi as you want. SImilarly for e and sqrt(2), and pretty much any number you have ever heard of.
But what Turing discovered (and it is not difficult to prove) is that there are real numbers (in the usual axiom system) that are not computable: no computer program can compute their decimal expansions. And, in fact, 'most' real numbers are not computable.
In what sense do these uncomputable real numbers exist in some Platonic realm?
This brings me to a somewhat related question I've had on my mind for a while. Why is it that such special numbers "found in nature" such as pi and e are irrational (and seemingly "arbitrary") as opposed to "clean" rational numbers? Is it to do with the particular number system being used (the decimal)?