If pi is truly infinite wouldn’t it eventually express a sequence of information which would be self aware if expressed in binary in a programmatic system?
My understanding (which might be wrong) is that just because PI is infinite and non-repeating, doesn't necessarily mean that every conceivable pattern of digits is present.
As a contrived example, consider the pattern:
01 001 0001 00001 etc.
This pattern is infinite and never repeats but we will never see two consecutive "1"s next to each other.
Yes, it doesn't necessarily follow, but it is indeed conjectured that pi is a normal number, meaning all digits appear with the same frequency, but it is not known yet. https://en.wikipedia.org/wiki/Normal_number
The same frequency does not imply every subsequence appears. Consider the modification which rewrites every sequence of 123 to 132. All digits will have the same frequency but 123 will never appear.
You haven't read the link you posted though. Every digit appearing with the same frequency means a number is simply normal and it is not enough to get you what you want in this case (as pointed out by sibling comment). Normal number is a number where every possible string of length n has the same frequency of 10^(-n)
No, you haven't read the link he posted. https://en.wikipedia.org/wiki/Normal_number#Definitions: "A disjunctive sequence is a sequence in which every finite string appears. A normal sequence is disjunctive". If Pi is normal, then it is also disjunctive.
When I was at university, one of the senior number theory professors allegedly said during a tutorial that he accepts the normality of pi on the basis of "proof by why the hell wouldn't it be". With tongue in cheek, of course.
If your argument is "These algorithms have differing degrees of computational complexity" then that doesn't actually demonstrate that one can't be algorithmically determined
Describe the n-th digit of an irrational number without calculating all previous positions of the number.
If pi were a sequence of digits, there is no algorithm to calculate it other than by calculating pi but there is one for op's number. The very fact that he could show the algorithm for creating the sequence of numbers in his post is indicative of that.
For pi such an algorithm doesn't exist (other than calculating pi itself).
I wanted to emphasize this by talking about the "sequence of digits" in my original reply but apparently I failed at explaining this well.
We know for a fact that pi is truly infinite, there's no "if" there. But we are not sure whether it contains every sequence of (e.g.) decimal digits.
Either way, your proposition works for "the list (or concatenation) of all positive integers in ascending order" as well. There is no deep insight in it, even if it were also true for pi.
if you accept the premise behind this question (which I wouldn't dispute) then theoretically any information at all would be self aware given the right computer
What you want is a disjunctive number, also called rich number or universe number.
It is an infinite number where every possible sequence of digits is present, and therefore, such a number contains the code of a self aware program, as well as the complete description of our own universe (hence the name "universe number") and even the simulation that runs it, if such things exist.
We don't know if pi is a disjunctive number, for what we know, though unlikely, the decimal representation of pi may only have a finite number of zeroes. It means we don't have the answer to your question.
As a contrived example, consider the pattern:
01 001 0001 00001 etc.
This pattern is infinite and never repeats but we will never see two consecutive "1"s next to each other.