Cardinality

Introduction

In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = {2, 4, 6} contains 3 elements, and therefore A has a cardinality of 3. The cardinality of a set A is usually denoted |A| with a vertical bar on each side; this is the same notation as absolute value, and the meaning depends on context.

The cardinality of the natural numbers is denoted aleph-null (ℵ0) while the cardinality of the real numbers is denoted by "c" and is also referred to as the cardinality of the continuum. Cantor showed, using the diagonal argument, that c >ℵ0.

The continuum hypothesis says that ℵ1 =2^ℵ0 i.e. 2^ℵ0 is the smallest cardinal number bigger than ℵ0, i.e. there is no set whose cardinality is strictly between that of the integers and that of the real numbers.

An infinite set A is called countably infinite (or countable) if it has the same cardinality as N. In other words, there is a bijection A → N.

An infinite set A is called uncountably infinite (or uncountable) if it is not countable. In other words, there exists no bijection A→ N.

These definitions suggest that even among the class of infinite sets, there are different "sizes of infinity." In the sense of cardinality, countably infinite sets are "smaller" than uncountably infinite sets. Of course, finite sets are "smaller" than any infinite sets, but the distinction between countable and uncountable gives a way of comparing sizes of infinite sets as well. Below are some examples of countable and uncountable sets.

Levels

For the purpose and simplicity of the wiki, we divide Cardinality into easy to understand categories.

None: For characters at 11-C as they lack cardinality and energy altogether.

Finite: For characters that range all the way from 11-B to High 3-A, these characters have a finite set of cardinality.

ℵ0: For characters that range from High 3-A to 2-C, these characters affect an infinite sized universe contained within a finite sized volume.

Beyond ℵ0: For characters that range from 2-B to 2-A, these characters affect multiple universes contained within a space-time bubble.

ℵ1: For characters that are in High 2-A, this is where one affects an infinite sized multiverse or an infinite sized universe within a infinite sized volume.

ℵ2 or higher accessible: For characters that are in 1-C, this is where one approaches the larger uncountable infinities or transfinite cardinals within an accessible cardinal.

Inaccessible Cardinal: For characters that are in 1-B, affecting things within an inaccessible cardinal.

Absolute Infinity: For characters that are in 1-A, affecting things that deal with the absolute infinite.

Inapplicable: For characters that are in High 1-A, being transcendent by things such as cardinality.