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 , where and are both ordinals.
We can define in a few rules. First, , or when is 0, the answer will always be the other input, . Eventually, we reach absolute infinity (in this case I will be using the character for absolute infinity, but this is unofficial), or . This will still be , but it is also equal to . For us to get to , whatever we did to get to , we do it again onto , getting an 'absolute absolute' infinity. We can then continue to , which is applying to , is applying to , is applying to , etc.
We then get to , which is the limit of , and we can continue on like this, eventually getting to , which is equal to . To get to , we use whatever we did to get to on , and you are probably getting to pattern now. To get to , we apply to , and we can continue, eventually getting to , which is equal to , ,, and in general, . We can also have all ordinals in the second entry, so .
The function reaches its limit when we have the number . You would need to extend the function or make a new function to create larger numbers.
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 .
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 . 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 , let’s first go from to . No matter how many times we use limit ordinals or successors on , we will never reach .
You may start to see a pattern here.
1 is the first number you cannot reach without increasing the value,
is the first number you cannot reach with successors,
is the first number you cannot reach with limit ordinals or successors,
But what is then? Well, the definition is basically that 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 , which is a contradiction (although there are cardinals and ordinals which contradict themselves).
To get to 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 . This can easily be extended to all numbers beginning with a 1 in LAIF, by just replacing with and with .