In
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 conce ...
, an
uncountable
In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal numb ...
cardinal
Cardinal or The Cardinal may refer to:
Animals
* Cardinal (bird) or Cardinalidae, a family of North and South American birds
**''Cardinalis'', genus of cardinal in the family Cardinalidae
**''Cardinalis cardinalis'', or northern cardinal, the ...
is inaccessible if it cannot be obtained from smaller cardinals by the usual operations of
cardinal arithmetic
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. The ...
. More precisely, a cardinal is strongly inaccessible if it is uncountable, it is not a sum of fewer than cardinals smaller than , and
implies
.
The term "inaccessible cardinal" is ambiguous. Until about 1950, it meant "weakly inaccessible cardinal", but since then it usually means "strongly inaccessible cardinal". An uncountable cardinal is weakly inaccessible if it is a
regular weak limit cardinal
In mathematics, limit cardinals are certain cardinal numbers. A cardinal number ''λ'' is a weak limit cardinal if ''λ'' is neither a successor cardinal nor zero. This means that one cannot "reach" ''λ'' from another cardinal by repeated succe ...
. It is strongly inaccessible, or just inaccessible, if it is a regular strong limit cardinal (this is equivalent to the definition given above). Some authors do not require weakly and strongly inaccessible cardinals to be uncountable (in which case is strongly inaccessible). Weakly inaccessible cardinals were introduced by , and strongly inaccessible ones by and .
Every strongly inaccessible cardinal is also weakly inaccessible, as every strong limit cardinal is also a weak limit cardinal. If the
generalized continuum hypothesis
In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that
or equivalently, that
In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to ...
holds, then a cardinal is strongly inaccessible if and only if it is weakly inaccessible.
(
aleph-null
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 af ...
) is a regular strong limit cardinal. Assuming the
axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collectio ...
, every other infinite cardinal number is regular or a (weak) limit. However, only a rather large cardinal number can be both and thus weakly inaccessible.
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.) A cardinal which is weakly inaccessible and also a strong limit cardinal is strongly inaccessible.
The assumption of the existence of a strongly inaccessible cardinal is sometimes applied in the form of the assumption that one can work inside a
Grothendieck universe In mathematics, a Grothendieck universe is a set ''U'' with the following properties:
# If ''x'' is an element of ''U'' and if ''y'' is an element of ''x'', then ''y'' is also an element of ''U''. (''U'' is a transitive set.)
# If ''x'' and ''y'' a ...
, the two ideas being intimately connected.
Models and consistency
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 such as ...
with Choice (ZFC) implies that the
th level of the
Von Neumann universe
In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted by ''V'', is the class of hereditary well-founded sets. This collection, which is formalized by Zermelo–Fraenkel set theory (Z ...
is a
model
A model is an informative representation of an object, person or system. The term originally denoted the Plan_(drawing), plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a mea ...
of ZFC whenever
is strongly inaccessible. And 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
In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers. Cardinals with such properties are, as the name suggests, generally very "large" (for example, bigger than the least Î ...
.
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 such as ...
; and
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
In the foundations of mathematics, Morse–Kelley set theory (MK), Kelley–Morse set theory (KM), Morse–Tarski set theory (MT), Quine–Morse set theory (QM) or the system of Quine and Morse is a first-order axiomatic set theory that is closely ...
. Here
is the set of Δ
0 definable subsets of ''X'' (see
constructible universe). However,
does not need to be inaccessible, or even a cardinal number, in order for to be a standard model of ZF (see
below
Below may refer to:
*Earth
*Ground (disambiguation)
*Soil
*Floor
*Bottom (disambiguation)
Bottom may refer to:
Anatomy and sex
* Bottom (BDSM), the partner in a BDSM who takes the passive, receiving, or obedient role, to that of the top or ...
).
Suppose
is a model of ZFC. Either V 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 In mathematics, a Grothendieck universe is a set ''U'' with the following properties:
# If ''x'' is an element of ''U'' and if ''y'' is an element of ''x'', then ''y'' is also an element of ''U''. (''U'' is a transitive set.)
# If ''x'' and ''y'' a ...
. 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 ''α'' any ordinal, 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 In mathematics, a Mahlo cardinal is a certain kind of large cardinal number. Mahlo cardinals were first described by . As with all large cardinals, none of these varieties of Mahlo cardinals can be proven to exist by ZFC (assuming ZFC is consist ...
.
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 cardinal In mathematics, a Mahlo cardinal is a certain kind of large cardinal number. Mahlo cardinals were first described by . As with all large cardinals, none of these varieties of Mahlo cardinals can be proven to exist by ZFC (assuming ZFC is consist ...
s 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 In model theory, a branch of mathematical logic, two structures ''M'' and ''N'' of the same signature ''σ'' are called elementarily equivalent if they satisfy the same first-order ''σ''-sentences.
If ''N'' is a substructure of ''M'', one often ...
of
. (In fact, the set of such ''α'' is
closed unbounded in .) Equivalently,
is
-
indescribable for all ''n'' ≥ 0.
It is provable in ZF that ∞ satisfies 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 it can be shown 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
In mathematics and other formal sciences, first-order or first order most often means either:
* "linear" (a polynomial of degree at most one), as in first-order approximation and other calculus uses, where it is contrasted with "polynomials of high ...
) ZFC. Hence, the existence of an inaccessible cardinal is a stronger hypothesis than the existence of a transitive model of ZFC.
See also
*
Worldly cardinal In mathematical set theory, a worldly cardinal is a cardinal κ such that the rank ''V''κ is a model of Zermelo–Fraenkel set theory.
Relationship to inaccessible cardinals
By Zermelo's theorem on inaccessible cardinals, every inaccessible c ...
, a weaker notion
*
Mahlo cardinal In mathematics, a Mahlo cardinal is a certain kind of large cardinal number. Mahlo cardinals were first described by . As with all large cardinals, none of these varieties of Mahlo cardinals can be proven to exist by ZFC (assuming ZFC is consist ...
, a stronger notion
*
Club set
In mathematics, particularly in mathematical logic and set theory, a club set is a subset of a limit ordinal that is closed under the order topology, and is unbounded (see below) relative to the limit ordinal. The name ''club'' is a contraction o ...
*
Inner model
In set theory, a branch of mathematical logic, an inner model for a theory ''T'' is a substructure of a model ''M'' of a set theory that is both a model for ''T'' and contains all the ordinals of ''M''.
Definition
Let L = \langle \in \rangle be ...
*
Von Neumann universe
In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted by ''V'', is the class of hereditary well-founded sets. This collection, which is formalized by Zermelo–Fraenkel set theory (Z ...
*
Constructible universe
Works cited
*
*
*
*
*
*. English translation: .
{{Mathematical logic
Large cardinals