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/Idksonameiguess]

If pi has no end it has to have every combination of numbers

Nope. Take 0.3333... for example, has no end and yet doesn't have every combination of numbers. We have no proof that pi contains every combination of numbers, but it is hypothesized.

We do, however, know the following:

Let a be some infinite combination of digits. If pi contains a, then

pi = n/10^k + a, with n and k being whole numbers (this is just by definition). Therefore, it can't contain 2 different infinite combinations which do not contain each other. This is true in general for any number, not just pi.

[u/NessaSola → u/Idksonameiguess]

A non-rational example is 0.12122122212222... where the amount of digit '2' following each Nth '1' is N.

To expand on what you said, one open question is whether pi is a normal number. If it's normal, it contains every sequence. If not, perhaps it's like 0.333... or 0.12122...

[u/Idksonameiguess → u/NessaSola]

Just to give an idea of a normal number, we can consider the "Champernowne's constant", designed specifically to be normal, defined as 0.123456789101112...

If I'm not mistaken, the only normal number we have are constructed, and we haven't proven any naturally appearing constant (like e, pi or the likes) to be normal.

[u/Bills_afterMATH → u/Idksonameiguess]

It’s true if you add the condition that the number must be computable. For example, every one of Chaitan’s constants are normal is all bases (as they are Martin-Löf random), but they aren’t computable.