How big of an infinity can we approach? At first I naively thought about: $$ \lim_{i \to \aleph_0}\aleph_i $$ However, this approaches $\aleph_{\aleph_0}$ which is $\aleph_{\omega}$
Let's define: $$ a_i := \aleph_{\aleph_{..._{\aleph_i}}} $$
Where the non-negative integer $i$ is one less than the total number of alephs.
For example, $a_0 = \aleph_0$, $a_1 = \aleph_{\aleph_1}$, ...
Or maybe it would make more sense to think about $a_i$ as a function $f: ? \to ?$ (can someone help out with the domain and codomain - is this possible?)
How close can we get to Absolute Infinity?
Perhaps it doesn't make sense to talk about approaching the Absolute Infinity, so let's ask instead, how close can we get to something we'll call $\infty^2$? Where $\infty^2$ is the largest type of infinity that's not the Absolute Infinity (though there isn't a largest infinity I don't think).
I apologize for how poorly this is defined. Maybe you could help me out.
Maybe a better way of saying this is: how can we come up with a construction (a limit? I'm not sure what to call it) that approaches the largest not-absolute infinity?
... I keep rereading this question and I'm killing myself over the phrasing. This is fairly new to me and the topic is so abstract, that I'm not sure how to even ask questions that make sense let alone provide constructions that do.
With the inherent complications that infinity brings with it, I'm not even sure what questions are legal when talking about infinity.
Any insight into this is appreciated.
I'm afraid that Absolute Infinity doesn't exist...
According to Cantor's theorem, for any set $A$: $$\vert \mathcal P(A) \vert > \vert A \vert$$ so you can build a stricly increasing sequence of cardinals defined by $\beth_0=\aleph_0$ and $\beth_{n+1}=2^{\beth_n}$.
So if the Absolute Infinite would exist $$2^{\text{Absolute Infinite}}$$ would be bigger than it. Not so good for an Absolute Infinite!