Beyond absolute infinity

Beyond the absolute infinite, there are still mathematical objects for which it might be argued are 'larger' in size than the collection of all sets. The concept of size becomes a matter of mapping with objects of this nature.

## Classes

Classes are a second type of collection in extensions of ZFC, analogous to sets but able to contain *both* sets and classes, whereas sets can *only* contain other sets. With classes, one can define operations analogous to power setting, and through reasoning isomorphic to Cantor's diagonal argument, one can create classes even 'larger' than the class of all sets, or "absolute infinity," like the class B within the following set theory sentence which uses overline-epsilon as a class membership predicate.

Beyond classes one can define higher groupings, each with their own predicate, even creating second-order predicates which range over membership-like first-order predicates and create a hierarchy of objects each bigger than all possible objects of the last type. This in turn can be extended by finding a way to continuously create new orders of logic and diagonalize over them in a process akin to the Rayo function.

## Past mathematics

A few things are needed to assure higher types of objects "larger" than infinity:

* The objects must follow a well-ordered hierarchy in which any two members can be compared.
* The objects must be defined unambiguously in a language with binary truth predicates.
* The objects must be definable through sentences that preserve consistency of the given theory.

In the same way certain large cardinal axioms force the behavior of the set-theoretic universe to change as a whole, there may come a point where a large object axiom is simply "too strong" to ever be formulated without decreasing its own size. At this point entirely new logical concepts would have to be invented, or a more philosophcial approach to objects larger than absolute infinity might be taken. If one were feeling especially precocious, one could say the totality of *all things* counts as something past all prior mentioned schema of objects, languages, and so on. This would be ill-defined but hold at least a modicum of conceptual merit as being an unbeatable (albeit trivial) conceptualization of a large object.

Admitting intuition of size, one could argue that an infinite set of cells in 3D space wherein every cell has one grain of sand is somehow "less full" than an infinite set of cells in 3D space wherein every cell is packed with sand. According to set theory, these two spaces have the same amount of sand, but intuitively one feels more full, so one might consider the sand packed infinite cell space to be "larger" than the single-grain infinite cell space. On account of this, one might imagine the number of the totality of all things as comprising an infinite set of rooms, and how dense the rooms are relate to the perceived "size" of this totality. With this slightly cheaty way, we can visualize what *feels* to be the densest and most full thing.

It is likely there is a special neural network within the brain responsible for perception of size and feeling of grandeur. If this could be activated directly and put to full throttle, it would technically produce the feeling of having felt the largest thing in existence, even beyond the wonder and awe produced from contemplating proper classes in set theory.