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/Bills_afterMATH]

Whether it’s “larger” depends on the notion. They have the same cardinality, but the set of even numbers has density 1/2 compared to the set of natural numbers which has density 1.

[u/shwilliams4 → u/Bills_afterMATH]

My math degree says that’s wrong but I’m old. Can you send a paper on this?

[u/Bills_afterMATH → u/shwilliams4]

My math degree says there's a lot more than cardinality that's frequently used to compare infinite sets (measure, Hausdorff dimension, Hausdorff measure (when Hausdorff dimensions are equal), many variants of asymptotic densities (when comparing subsets of the natural numbers), Packing dimension, winning for Schmidt's game, meagre vs comeagre, etc.). All of these notions are fairly old and I'm surprised that you only considered cardinality.

Specifically, for the vanilla version of density, you consider any A \subset \mathbb{N}$. You can define the density of $A$ to be

$d(A) := \lim_{n \to \infty} \frac{A \cap [1,n]} {n},$

assuming it exists (you can easily construct examples of subsets of the natural numbers that don't have a density). For the example being considered, you can check that $d(2\mathbb{N})=1/2$. You can also show that the density of the set of square free numbers is $6/\pi^2$. There are also variants of density taking a liminf or limsup (to make sure every subset has a density) and there are also uniform versions (for example, upper Banach density shows up a lot in ergodic Ramsey theory).

Edit: forgot to mention that density is NOT $\sigma$-additive. For example, every singleton has density 0 and the set of natural numbers has density 1. So, you don't want to use this as a measure.

[u/shwilliams4 → u/Bills_afterMATH]

Yep I’m old and out of the loop