In
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
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 ...
is a strongly inaccessible cardinal if it is
uncountable,
regular, and a
strong limit cardinal.
A cardinal is a weakly inaccessible cardinal if it is uncountable, regular, and a
weak limit cardinal.
Since about 1950, "inaccessible cardinal" has typically meant "strongly inaccessible cardinal" whereas before it has meant "weakly inaccessible cardinal". Weakly inaccessible cardinals were introduced by . Strongly inaccessible cardinals were introduced by and ; in the latter they were referred to along with
as ''Grenzzahlen'' (
English "limit numbers").
Every strongly inaccessible cardinal is a weakly inaccessible cardinal. The
generalized continuum hypothesis implies that all weakly inaccessible cardinals are strongly inaccessible as well.
The two notions of an inaccessible cardinal
describe a cardinality
which can not be obtained as the cardinality of a result of typical set-theoretic operations involving only sets of cardinality less than
. Hence the word "inaccessible". By mandating that inaccessible cardinals are uncountable, they turn out to be very large.
In particular, inaccessible cardinals need not exist at all. That is, it is believed that there are models of
Zermelo-Fraenkel set theory, even with the
axiom of choice
In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from e ...
(ZFC), for which no inaccessible cardinals exist. On the other hand, it also believed that there are models of ZFC for which even strongly inaccessible cardinals
do exist. That ZFC can accommodate these large sets, but does not necessitate them, provides an introduction to the
large cardinal axioms. See also
Models and consistency.
The existence of a strongly inaccessible cardinal is equivalent to the existence of a
Grothendieck universe.
If
is a strongly inaccessible cardinal then the
von Neumann stage is a Grothendieck universe.
Conversely, if
is a Grothendieck universe then there is a strongly inaccessible cardinal
such that
. As expected from their correspondence with strongly inaccessible cardinals, Grothendieck universes are very well-closed under set-theoretic operations.
An
ordinal is a weakly inaccessible cardinal if and only if it is a regular ordinal and it is a limit of regular ordinals. (Zero, one, and are regular ordinals, but not limits of regular ordinals.)
From some perspectives, the requirement that a weakly or strongly inaccessible cardinal be uncountable is unnatural or unnecessary. Even though is countable, it is regular and is a strong limit cardinal. is also the smallest weak limit regular cardinal. Assuming the axiom of choice, every other infinite cardinal number is either regular or a weak limit cardinal. However, only a rather large cardinal number can be both. Since a cardinal larger than is necessarily uncountable, if is also regular and a weak limit cardinal then must be a weakly inaccessible cardinal.
Models and consistency
Suppose that
is a cardinal number.
Zermelo–Fraenkel set theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes suc ...
with Choice (ZFC) implies that the
th level of the
Von Neumann universe is a
model
A model is an informative representation of an object, person, or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin , .
Models can be divided in ...
of ZFC whenever
is strongly inaccessible. Furthermore, ZF implies that the
Gödel universe is a model of ZFC whenever
is weakly inaccessible. Thus, ZF together with "there exists a weakly inaccessible cardinal" implies that ZFC is consistent. Therefore, inaccessible cardinals are a type of
large cardinal.
If
is a standard model of ZFC and
is an inaccessible in
, then
#
is one of the intended models of
Zermelo–Fraenkel set theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes suc ...
;
#
is one of the intended models of Mendelson's version of
Von Neumann–Bernays–Gödel set theory which excludes global choice, replacing limitation of size by replacement and ordinary choice;
# and
is one of the intended models of
Morse–Kelley set theory.
Here,
is the set of Δ
0-definable subsets of ''X'' (see
constructible universe). It is worth pointing out that the first claim can be weakened:
does not need to be inaccessible, or even a cardinal number, in order for to be a standard model of ZF (see
below).
Suppose
is a model of ZFC. Either
contains no strong inaccessible or, taking
to be the smallest strong inaccessible in
,
is a standard model of ZFC which contains no strong inaccessibles. Thus, the consistency of ZFC implies consistency of ZFC+"there are no strong inaccessibles". Similarly, either contains no weak inaccessible or, taking
to be the smallest ordinal which is weakly inaccessible relative to any standard sub-model of
, then
is a standard model of ZFC which contains no weak inaccessibles. So consistency of ZFC implies consistency of ZFC+"there are no weak inaccessibles". This shows that ZFC cannot prove the existence of an inaccessible cardinal, so ZFC is consistent with the non-existence of any inaccessible cardinals.
The issue whether ZFC is consistent with the existence of an inaccessible cardinal is more subtle. The proof sketched in the previous paragraph that the consistency of ZFC implies the consistency of ZFC + "there is not an inaccessible cardinal" can be formalized in ZFC. However, assuming that ZFC is consistent, no proof that the consistency of ZFC implies the consistency of ZFC + "there is an inaccessible cardinal" can be formalized in ZFC. This follows from
Gödel's second incompleteness theorem, which shows that if ZFC + "there is an inaccessible cardinal" is consistent, then it cannot prove its own consistency. Because ZFC + "there is an inaccessible cardinal" does prove the consistency of ZFC, if ZFC proved that its own consistency implies the consistency of ZFC + "there is an inaccessible cardinal" then this latter theory would be able to prove its own consistency, which is impossible if it is consistent.
There are arguments for the existence of inaccessible cardinals that cannot be formalized in ZFC. One such argument, presented by , is that the class of all ordinals of a particular model ''M'' of set theory would itself be an inaccessible cardinal if there was a larger model of set theory extending ''M'' and preserving powerset of elements of ''M''.
Existence of a proper class of inaccessibles
There are many important axioms in set theory which assert the existence of a proper class of cardinals which satisfy a predicate of interest. In the case of inaccessibility, the corresponding axiom is the assertion that for every cardinal ''μ'', there is an inaccessible cardinal which is strictly larger, . Thus, this axiom guarantees the existence of an infinite tower of inaccessible cardinals (and may occasionally be referred to as the inaccessible cardinal axiom). As is the case for the existence of any inaccessible cardinal, the inaccessible cardinal axiom is unprovable from the axioms of ZFC. Assuming ZFC, the inaccessible cardinal axiom is equivalent to the universe axiom of
Grothendieck and
Verdier: every set is contained in a
Grothendieck universe. The axioms of ZFC along with the universe axiom (or equivalently the inaccessible cardinal axiom) are denoted ZFCU (not to be confused with ZFC with
urelements). This axiomatic system is useful to prove for example that every
category has an appropriate
Yoneda embedding.
This is a relatively weak large cardinal axiom since it amounts to saying that ∞ is 1-inaccessible in the language of the next section, where ∞ denotes the least ordinal not in V, i.e. the class of all ordinals in your model.
''α''-inaccessible cardinals and hyper-inaccessible cardinals
The term "''α''-inaccessible cardinal" is ambiguous and different authors use inequivalent definitions. One definition is that
a cardinal is called ''α''-inaccessible, for any ordinal ''α'', if is inaccessible and for every ordinal ''β'' < ''α'', the set of ''β''-inaccessibles less than is unbounded in (and thus of cardinality , since is regular). In this case the 0-inaccessible cardinals are the same as strongly inaccessible cardinals. Another possible definition is that a cardinal is called ''α''-weakly inaccessible if is regular and for every ordinal ''β'' < ''α'', the set of ''β''-weakly inaccessibles less than is unbounded in κ. In this case the 0-weakly inaccessible cardinals are the regular cardinals and the 1-weakly inaccessible cardinals are the weakly inaccessible cardinals.
The ''α''-inaccessible cardinals can also be described as fixed points of functions which count the lower inaccessibles. For example, denote by ''ψ''
0(''λ'') the ''λ''
th inaccessible cardinal, then the fixed points of ''ψ''
0 are the 1-inaccessible cardinals. Then letting ''ψ''
''β''(''λ'') be the ''λ''
th ''β''-inaccessible cardinal, the fixed points of ''ψ''
''β'' are the (''β''+1)-inaccessible cardinals (the values ''ψ''
''β''+1(''λ'')). If ''α'' is a limit ordinal, an ''α''-inaccessible is a fixed point of every ''ψ''
''β'' for ''β'' < ''α'' (the value ''ψ''
''α''(''λ'') is the ''λ''
th such cardinal). This process of taking fixed points of functions generating successively larger cardinals is commonly encountered in the study of
large cardinal numbers.
The term hyper-inaccessible is ambiguous and has at least three incompatible meanings. Many authors use it to mean a regular limit of strongly inaccessible cardinals (1-inaccessible). Other authors use it to mean that is -inaccessible. (It can never be -inaccessible.) It is occasionally used to mean
Mahlo cardinal.
The term ''α''-hyper-inaccessible is also ambiguous. Some authors use it to mean ''α''-inaccessible. Other authors use the definition that
for any ordinal ''α'', a cardinal is ''α''-hyper-inaccessible if and only if is hyper-inaccessible and for every ordinal ''β'' < ''α'', the set of ''β''-hyper-inaccessibles less than is unbounded in .
Hyper-hyper-inaccessible cardinals and so on can be defined in similar ways, and as usual this term is ambiguous.
Using "weakly inaccessible" instead of "inaccessible", similar definitions can be made for "weakly ''α''-inaccessible", "weakly hyper-inaccessible", and "weakly ''α''-hyper-inaccessible".
Mahlo cardinals are inaccessible, hyper-inaccessible, hyper-hyper-inaccessible, ... and so on.
Two model-theoretic characterisations of inaccessibility
Firstly, a cardinal is inaccessible if and only if has the following
reflection property: for all subsets
, there exists
such that
is an
elementary substructure of
. (In fact, the set of such ''α'' is
closed unbounded in .) Therefore,
is
-
indescribable for all ''n'' ≥ 0. On the other hand, there is not necessarily an ordinal
such that
, and if this holds, then
must be the
th inaccessible cardinal.
It is provable in ZF that
has a somewhat weaker reflection property, where the substructure
is only required to be 'elementary' with respect to a finite set of formulas. Ultimately, the reason for this weakening is that whereas the model-theoretic satisfaction relation can be defined, semantic truth itself (i.e.
) cannot, due to
Tarski's theorem.
Secondly, under ZFC
Zermelo's categoricity theorem can be shown, which states that
is inaccessible if and only if
is a model of
second order ZFC.
In this case, by the reflection property above, there exists
such that
is a standard model of (
first order) ZFC. Hence, the existence of an inaccessible cardinal is a stronger hypothesis than the existence of a transitive model of ZFC.
Inaccessibility of
is a
property over
, while a cardinal
being inaccessible (in some given model of
containing
) is
.
[K. J. Devlin, "Indescribability Properties and Small Large Cardinals" (1974). In ''ISILC Logic Conference: Proceedings of the International Summer Institute and Logic Colloquium, Kiel 1974'', Lecture Notes in Mathematics, vol. 499 (1974)]
See also
*
Worldly cardinal, a weaker notion
*
Mahlo cardinal, a stronger notion
*
Club set
*
Inner model
*
Von Neumann universe
*
Constructible universe
Works cited
*
*
*
*
*
*. English translation: .
References
{{Reflist
Large cardinals