LAIF

LAIF, standing for Large Absolute Infinity Function, is a function made for defining or comparing cardinals that are absolutely infinite or larger. LAIF is found in the form of $$\infin(\alpha,\beta)$$, where $$\alpha$$and $$\beta$$are both ordinals.

Definition
We can define $$\infin(\alpha,\beta)$$in a few rules. First, $$\infin(\alpha,0)=\alpha$$, or when $$\beta$$is 0, the answer will always be the other input, $$\alpha$$. Eventually, we reach absolute infinity (in this case I will be using the character $$\Psi$$for absolute infinity, but this is unofficial), or $$\infin(\Psi,0)$$. This will still be $$\Psi$$, but it is also equal to $$\infin(0,1)$$. For us to get to $$\infin(1,1)$$, whatever we did to get to $$\Psi$$, we do it again onto $$\Psi$$, getting an 'absolute absolute' infinity. We can then continue to $$\infin(2,1)$$, which is applying $$\Psi$$to $$\infin(1,1)$$, $$\infin(3,1)$$is applying $$\Psi$$to $$\infin(2,1)$$, $$\infin(4,1)$$is applying $$\Psi$$to $$\infin(3,1)$$, etc.

We then get to $$\infin(\omega,1)$$, which is the limit of $$\infin(0,1),\infin(1,1),\infin(2,1),\infin(3,1),...$$, and we can continue on like this, eventually getting to $\kappa=\infin(\kappa,0)$, which is equal to $$\infin(0,2)$$. To get to $$\infin(1,2)$$, we use whatever we did to get to $$\infin(0,2)$$on $$\infin(1,2)$$, and you are probably getting to pattern now. To get to $$\infin(2,2)$$, we apply $$\infin(0,2)$$to $$\infin(1,2)$$, and we can continue, eventually getting to $$\kappa=\infin(\kappa,2)$$, which is equal to $$\infin(0,3)$$, $$\kappa=\infin(\kappa,3)=\infin(0,4)$$,$$\kappa=\infin(\kappa,4)=\infin(0,5)$$, and in general, $$\kappa=\infin(\kappa,\beta)=\infin(0,\beta+1)$$. We can also have all ordinals in the second entry, so $$\infin(0,\omega)=\{\infin(0,0),\infin(0,1),\infin(0,2),\infin(0,3)...\}$$.

The function reaches its limit when we have the number $$\kappa=\infin(0,\kappa)$$. You would need to extend the function or make a new function to create larger numbers.

Details on $$\Psi$$
When absolute infinity was previously defined, the actual way of obtaining it was left very vague. To help with this, here is a more well-defined description of $$\Psi$$.

Before we imagine absolute infinity, let’s instead imagine the jump from 0 to 1. Pretty simple, right? We’re just going to the next whole number (the technical name for this is the successor). Next, let’s take the jump from 1 to $$\omega$$. No matter how many times we take the successor of 1, we never reach omega. This is called a limit ordinal, although this is a pretty oversimplified definition. Before we take the jump to $$\Psi$$, let’s first go from $$\omega$$ to $$\Omega$$. No matter how many times we use limit ordinals or successors on $$\omega$$, we will never reach $$\Omega$$.

You may start to see a pattern here.

1 is the first number you cannot reach without increasing the value,

$$\omega$$ is the first number you cannot reach with successors,

$$\Omega$$ is the first number you cannot reach with limit ordinals or successors,

But what is $$\Psi$$ then? Well, the definition is basically that $$\Psi$$is the first number you cannot reach with successors, limit ordinals, or non-self-referential ‘the first number you cannot reach’ definitions.

That non-self-referential portion is important, or else you would be able to say that $$\Psi>\Psi$$, which is a contradiction (although there are cardinals and ordinals which contradict themselves).

Reaching $$\infty(1,1)$$
To get to $$\infty(1,1)$$ from absolute infinity, we simply just say that this number is the first number that cannot be reached with successors, limit ordinals, and non-self-referential ‘the first number you cannot reach’ definitions on $$\infty(0,1)$$. This can easily be extended to all numbers beginning with a 1 in LAIF, by just replacing $$0,1$$with $$\beta,1$$ and $$1,1$$ with $$\beta+1,1$$.