In
axiomatic set theory, 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 Austral ...
→ Ord is called normal (or a normal function) if and only if it is
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** 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, th ...
) 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 ...
''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
In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation. Each can be defined in essentially two different ways: either by constructing an expl ...
). 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. T ...
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 H ...
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. 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 function
In mathematics, the Veblen functions are a hierarchy of normal functions ( continuous strictly increasing functions from ordinals to ordinals), introduced by Oswald Veblen in . If φ0 is any normal function, then for any non-zero ordinal α, φ ...
s.
Notes
References
*{{citation
, first=Peter
, last=Johnstone
, authorlink=Peter Johnstone (mathematician)
, year=1987
, title=Notes on Logic and Set Theory
, publisher=
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer.
Cambridge University Pre ...
, isbn=978-0-521-33692-5
, url-access=registration
, url=https://archive.org/details/notesonlogicsett0000john
.
Set theory
Ordinal numbers