In
axiomatic set theory
Set theory is the branch of mathematical logic that studies 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 mathematics, is mostly concer ...
, a function ''f'' :
Ord
Ord or ORD may refer to:
Places
* Ord of Caithness, landform in north-east Scotland
* Ord, Nebraska, USA
* Ord, Northumberland, England
* Muir of Ord, village in Highland, Scotland
* Ord, Skye, a place near Tarskavaig
* Ord River, Western Austr ...
→ Ord is called normal (or a normal function) if and only if it is
continuous (with respect to the
order topology
In mathematics, an order topology is a certain 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, t ...
) and
strictly monotonically increasing. This is equivalent to the following two conditions:
# For every
limit ordinal ''γ'' (i.e. ''γ'' is neither zero nor a successor), it is the case that ''f''(''γ'') =
sup .
# For all ordinals ''α'' < ''β'', it is the case that ''f''(''α'') < ''f''(''β'').
Examples
A simple normal function is given by (see
ordinal arithmetic). But is ''not'' normal because it is not continuous at any limit ordinal; that is, the inverse image of the one-point open set is the set , which is not open when ''λ'' is a limit ordinal. If ''β'' 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 that can be well-ordered. They were introduced by the mathematician Georg Cantor and are named a ...
s
, which connect ordinal and
cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. Th ...
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 second ...
s
.
Properties
If ''f'' is normal, then for any ordinal ''α'',
:''f''(''α'') ≥ ''α''.
Proof: If not, choose ''γ'' minimal such that ''f''(''γ'') < ''γ''. Since ''f'' is strictly monotonically increasing, ''f''(''f''(''γ'')) < ''f''(''γ''), contradicting minimality of ''γ''.
Furthermore, for any non-empty set ''S'' of ordinals, we have
:''f''(sup ''S'') = sup ''f''(''S'').
Proof: "≥" follows from the monotonicity of ''f'' and the definition of the
supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest lo ...
. For "≤", set ''δ'' = sup ''S'' and consider three cases:
* if ''δ'' = 0, then ''S'' = and sup ''f''(''S'') = ''f''(0);
* if ''δ'' = ''ν'' + 1 is a
successor
Successor may refer to:
* An entity that comes after another (see Succession (disambiguation))
Film and TV
* ''The Successor'' (film), a 1996 film including Laura Girling
* ''The Successor'' (TV program), a 2007 Israeli television program Musi ...
, then there exists ''s'' in ''S'' with ν < ''s'', so that ''δ'' ≤ ''s''. Therefore, ''f''(''δ'') ≤ ''f''(''s''), which implies ''f''(δ) ≤ sup ''f''(''S'');
* if ''δ'' is a nonzero limit, pick any ''ν'' < ''δ'', and an ''s'' in ''S'' such that ν < ''s'' (possible since ''δ'' = sup ''S''). Therefore, ''f''(''ν'') < ''f''(''s'') so that ''f''(''ν'') < sup ''f''(''S''), yielding ''f''(''δ'') = sup ≤ sup ''f''(''S''), as desired.
Every normal function ''f'' has arbitrarily large fixed points; see the
fixed-point lemma for normal functions
The fixed-point lemma for normal functions is a basic result in axiomatic set theory stating that any normal function has arbitrarily large fixed points (Levy 1979: p. 117). It was first proved by Oswald Veblen in 1908.
Background and form ...
for a proof. One can create a normal function ''f' '' : Ord → Ord, called the derivative of ''f'', such that ''f' ''(''α'') is the ''α''-th fixed point of ''f''.
For a hierarchy of normal functions, see
Veblen functions.
Notes
References
*{{citation
, first=Peter
, last=Johnstone
, authorlink=Peter Johnstone (mathematician)
, year=1987
, title=Notes on Logic and Set Theory
, publisher=
Cambridge University Press
, isbn=978-0-521-33692-5
, url-access=registration
, url=https://archive.org/details/notesonlogicsett0000john
.
Set theory
Ordinal numbers