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
- if is a limit ordinal and for all 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
It follows that addition and multiplication are both associative, but not commutative. In particular one can see that
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
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
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,
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
 Jech, Thomas. Set Theory. 3rd Edition. Springer Monographs in Mathematics. Springer-Verlag. 2000.