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/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/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.