Editing Cardinality (section)

=== Aleph numbers ===
{{Main|Aleph number}}
[[File:Aleph0.svg|right|thumb|169x169px|[[Aleph-nought]], aleph-zero, or aleph-null: the smallest infinite cardinal number, and the cardinal number of the set of natural numbers. ]]
The [[aleph numbers]] are a sequence of cardinal numbers that denote the size of [[infinite sets]], denoted with an [[aleph]] <math>\aleph,</math> the first letter of the [[Hebrew alphabet]]. The first aleph number is <math>\aleph_0,</math> called "aleph-nought", "aleph-zero", or "aleph-null", which represents the cardinality of the set of all [[natural numbers]]: <math>\aleph_0 = |\N| = |\{0,1,2,3,\cdots\}| .</math>  Then, <math>\aleph_1</math> represents the next largest cardinality. The most common way this is formalized in set theory is through [[Von Neumann ordinal]]s, known as [[Von Neumann cardinal assignment]].

[[Ordinal number]]s generalize the notion of ''order'' to infinite sets. For example, 2 comes after 1, denoted <math>1 < 2,</math> and 3 comes after both, denoted <math>1 < 2 < 3.</math> Then, one defines a new number, <math>\omega,</math> which comes after every natural number, denoted <math>1 < 2 < 3 < \cdots < \omega.</math> Further <math>\omega < \omega+1 ,</math> and so on. More formally, these ordinal numbers can be defined as follows:

<math>0 := \{\},</math> the [[empty set]], <math>1 := \{0\} ,</math> <math>2 := \{0,1\},</math> <math>3 := \{0,1,2\},</math> and so on. Then one can define <math>m < n \text{, if } \, m \in n,</math> for examlpe, <math>2 \in \{0,1,2\} = 3,</math> therefore <math>2 < 3.</math>  Defining <math>\omega := \{0,1,2,3,\cdots\}</math> (a [[limit ordinal]]) gives <math>\omega</math> the desired property of being the smallest ordinal greater than all finite ordinal numbers. Further, <math>\omega+1 := \{1,2,\cdots,\omega\}</math>, and so on.

Since <math>\omega \sim \N</math> by the natural correspondence, one may define <math>\aleph_0</math> as the set of all finite ordinals. That is, <math>\aleph_0 := \omega.</math> Then, <math>\aleph_1</math> is the set of all countable ordinals (all ordinals <math>\kappa</math> with cardinality <math>|\kappa| \leq \aleph_0</math>), the [[first uncountable ordinal]]. Since a set cannot contain itself, <math>\aleph_1</math> must have a strictly larger cardinality: <math>\aleph_0 < \aleph_1.</math> Furthermore, <math>\aleph_2</math> is the set of all ordinals with cardinality <math>\aleph_1,</math> and so on. By the [[well-ordering theorem]], there cannot exist any set with cardinality between <math>\aleph_0</math> and <math>\aleph_1,</math> and every infinite set has some cardinality corresponding to some aleph <math>\aleph_\alpha,</math> for some ordinal <math>\alpha.</math>