Unless otherwise specified, we will assume ZF axioms. Recall the following ZF axiom:

**Axiom of Infinity**:

We will call the set of **natural numbers**. That is,

We can then define and iterate as before, whence by applying the axiom of infinity again we can obtain and so on. All such sets generated by this process are called **ordinals**. This in turn gives us the canonical linear ordering of the ordinals where iff

**Definition 1**. An ordinal is a **successor ordinal** iff A **limit ordinal** is an ordinal which is not a successor ordinal.

**Proposition 2. (Transfinite Induction)** Let denote the class of all ordinals (in accordance to VBG notion of class) and be a class. If

- and
- if is a limit ordinal and for all then

then

*Proof.* Since is a linear ordering on the ordinals, let be the least ordinal such that But then which is a contradiction. Hence

We can *index* ordinals with ordinals to generate the notion of a sequence of ordinals. We define an **nondecreasing (nonincreasing) sequence** of ordinals an ordered set of ordinals where iff

**Definition 3.** Let be an nondecreasing sequence of ordinals and be a limit ordinal and Then we define the **limit** of the sequence as

A dual definition can be defined for nonincreasing sequences, in which case the limits can be respectively distinguished as *left* and *right* limits. A sequence is **continuous** if for every limit ordinal in the indexing subclass we have

An example of a sequence which is not continuous may be one of the form where are both limit ordinals. So in this case

(since the sup is actually a max in this case).

**Definition 4. (Ordinal Arithmetic)** We define

**Addition:**

**Multiplication:**

**Exponentiation:**

It follows that addition and multiplication are both associative, but not commutative. In particular one can see that

and

**Definition 5.** For a set we define its **cardinality**, denoted as the unique ordinal with which the set has a bijection. The corresponding subclass of ordinals is called the class of **cardinals**.

**Proposition 6.** If then

*Proof.* For every define

Hence the mapping is a bijection between and

Hence in this context, Cantor’s theorem immediately follows:

**Proposition 7.** Let and Then

*Proof.* We have

Also if and are bijections, then we can define by and this is easily seen to be a bijection.

And if then we can define which is also seen to be a bijection.

It thus follows that addition and multiplication of cardinals are commutative (that is, ordinal operations are commutative on this subclass).

Above we said and But

and

for any finite ordinal with Knuth notation

having raisings of Consider the following convention for defining countable infinite ordinals:

They are all called countable infinite ordinals since for any one of them To clarify when we are talking about cardinal numbers versus ordinal numbers, we will use aleph notation: From Cantor’s theorem above, we know that if then That is, the ordinal corresponding to must be greater than all of the countable ordinals above, otherwise its cardinality would be This necessitates the notion of **uncountable ordinals** and corresponding **uncountable cardinals**. We use subscripts to characterize these: etc.

**Definition 8.** An infinite cardinal is a **successor cardinal** iff is a successor ordinal, and it is a **limit cardinal** iff is a limit ordinal.

**Definition 9.** Let be a limit ordinal. An increasing –sequence with a limit ordinal is **cofinal** in if And if is an ordinal, then we define its **cofinality** as

It is easy to verify that iff is a successor ordinal. Also and for any finite ordinal

**Proposition 10.**

*Proof.* Let Then (in particular it is the smallest such ). Now if then certainly Now since is cofinal in a subsequence of indices is cofinal in (where cofinality can be chosen to be ). So

But then is cofinal in That is,

whence

**Definition 11.** An ordinal is **regular** if It is **singular** if it is not regular.

**Corollary 12.** If is a limit ordinal, then is a regular cardinal.

**Theorem 13.** If is an infinite cardinal, then

[1] Jech, Thomas. * Set Theory*. 3rd Edition. Springer Monographs in Mathematics. Springer-Verlag. 2000.