Can infinity contain infinity

[u/Loose-Balance3225]

If pi has no end it has to have every combination of numbers but could it hold an infinite combination? Like 1 2 3 4... To infinity

Please help this is missing my brain up

[u/2ndcountable]

If pi is a normal number, as we suspect, it 'must' contain any finite sequence of digits. That is, given any finite sequence of digits, like '3333' or '123456789', it must exist within pi. However, even if pi is normal, it does not have to contain a given infinite sequence of digits. For example, we know for sure that pi does not contain the infinite sequence '333333333...', since that would mean its decimal expansion is eventually periodic and hence that pi is rational, which we know to be false.

[u/mousicle → u/2ndcountable]

As far as normal numbers, right now we have no test as to whether a number is normal or not. The only numbers we know for sure are normal are ones specifically constructed to be normal like 0.123456789101112 ... We do know that almost all numbers are normal though.

[u/Bills_afterMATH → u/mousicle]

What do you mean by test? For example, it’s trivial to show that x is normal in base b iff the sequence (x*b^n) is uniformly distributed mod 1. The set of numbers normal in base b was proven to be Pi_3^0 complete by Ki and Linton in 1994. So any condition that can characterize normality has to be a Pi_3^0 condition. Are you talking about some condition that checks (but can sometimes not determine) if a number is normal based on a finite amount of information?

[u/KattyTheEnby → u/2ndcountable]

For example, we know for sure that pi does not contain the infinite sequence '333333333...', since that would mean its decimal expansion is eventually periodic

How come pi containing an infinite series ov 3s, or any number, at some point implicate that it is periodic?

Can an entropic number, like pi, not contain a series ov random numbers and then have, sandwiched within it, an infinite series ov numbers?*

(* I am necessarily implying that there is some order ov magnitude to the infinities, since there would be an infinitely long sequence ov random numbers on both sides ov the infinitely long sequence ov 3s.)

[u/Head_Evening_5697 → u/KattyTheEnby]

How can you sandwich an infinite sequence inbetween two other sequences?

[u/2ndcountable → u/KattyTheEnby]

The fractional part of the decimal expansion of a number is indexed by N. That is, to the natural number 1 corresponds the first digit of the fractional part, to 2 corresponds the second digit, and so on. Let ai be the i-th digit of the fractional part, i.e. the digit that the natural number i corresponds to. Suppose pi contains the infinite sequence 3333..., starting at the index t. Then, we must have that a_t = 3, a(t+1) = 3, a(t+2) = 3, and so on. Then we have a_i = 3 for all i >= t; Indeed, when i >= t, a_i = a(t+d) = 3, where d = i-t >= 0. It is, however, possible to consider "an infinite sequence with another infinite sequence sandwiched within it", just not within a decimal expansion; For this, you can consider the concept of ordinal numbers. For example, a sequence indexed by w+w could indeed have an infinite sequence of 3s, followed by a different infinite sequence.

[u/KattyTheEnby → u/2ndcountable]

For example, a sequence indexed by w+w

w+w?

I'm guessing you mean in a similar to how, for example, complex numbers work (i.e. 2i + 5), rather than literally multiplying by two. Is that right?

[u/jsundqui → u/KattyTheEnby]

Decimal expansion can only have one infinite sequence, there cannot be different infinite sequences within one decimal expansion. So if, starting at some point, the decimal expansion contains infinite 3's, that's it, there is not room for anything else.

[u/trutheality → u/KattyTheEnby]

(* I am necessarily implying that there is some order ov magnitude to the infinities, since there would be an infinitely long sequence ov random numbers on both sides ov the infinitely long sequence ov 3s.)

That's not going to work here. A contiguous infinite subsequence of an infinite sequence is necessarily one of the latter's tail sequences.