In
axiomatic set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
, a function is called normal (or a normal function) if it is
continuous (with respect to the
order topology
In mathematics, an order topology is a specific topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets.
If ''X'' is a totally ordered set, ...
) and
strictly monotonically increasing. This is equivalent to the following two conditions:
# For every
limit ordinal
In set theory, a limit ordinal is an ordinal number that is neither zero nor a successor ordinal. Alternatively, an ordinal λ is a limit ordinal if there is an ordinal less than λ, and whenever β is an ordinal less than λ, then there exists a ...
(i.e. is neither zero nor a
successor
Successor may refer to:
* An entity that comes after another (see Succession (disambiguation))
Film and TV
* ''The Successor'' (1996 film), a film including Laura Girling
* The Successor (2023 film), a French drama film
* ''The Successor'' ( ...
), it is the case that .
# For all ordinals , it is the case that .
Examples
A simple normal function is given by (see
ordinal arithmetic). But is ''not'' normal because it is not continuous at any limit ordinal (for example,
). If is a fixed ordinal, then the functions , (for ), and (for ) are all normal.
More important examples of normal functions are given by the
aleph number
In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets. They were introduced by the mathematician Georg Cantor and are named after the symbol he used t ...
s
, which connect ordinal and
cardinal number
In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number. For dealing with the cas ...
s, and by the
beth number
In mathematics, particularly in set theory, the beth numbers are a certain sequence of infinite cardinal numbers (also known as transfinite numbers), conventionally written \beth_0, \beth_1, \beth_2, \beth_3, \dots, where \beth is the Hebrew lett ...
s
.
Properties
If is normal, then for any ordinal ,
:.
Proof: If not, choose minimal such that . Since is strictly monotonically increasing, , contradicting minimality of .
Furthermore, for any
non-empty set of ordinals, we have
:.
Proof: "≥" follows from the monotonicity of and the definition of the
supremum
In mathematics, the infimum (abbreviated inf; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique, ...
. For "", set and consider three cases:
* if , then and ;
* if is a successor, then there exists in with , so that . Therefore, , which implies ;
* if is a nonzero limit, pick any , and an in such that (possible since ). Therefore, so that , yielding , as desired.
Every normal function has arbitrarily large fixed points; see the
fixed-point lemma for normal functions for a proof. One can create a normal function , called the derivative of , such that is the -th fixed point of .
For a hierarchy of normal functions, see
Veblen functions.
Notes
References
*
{{refend
Set theory
Ordinal numbers