Zermelo–Fraenkel set theory
Encyclopedia
In mathematics, Zermelo–Fraenkel set theory with the axiom of choice, named after mathematicians Ernst Zermelo
and Abraham Fraenkel and commonly abbreviated ZFC, is one of several axiomatic system
s that were proposed in the early twentieth century to formulate a theory of sets without the paradoxes of naive set theory
such as Russell's paradox
. Specifically, ZFC does not allow unrestricted comprehension. Today ZFC is the standard form of axiomatic set theory and as such is the most common foundation of mathematics
.
ZFC has a single primitive ontological
notion, that of a hereditary
well-founded set, and a single ontological assumption, namely that all individual
s in the universe of discourse are such sets. Thus, ZFC is a set theory without urelements (elements of sets which are not themselves sets). ZFC does not formalize the notion of class
es (collections of mathematical objects defined by a property shared by their members) and specifically does not include proper classes (objects that have members but cannot be members themselves).
ZFC is a one-sorted theory in first-order logic
. The signature has equality and a single primitive binary relation
, set membership, which is usually denoted ∈. The formula a ∈ b means that the set a is a member of the set b (which is also read, "a is an element of b" or "a is in b").
There are many equivalent formulations of the ZFC axiom
s. Most of the ZFC axioms state the existence of particular sets. For example, the axiom of pairing
says that given any two sets a and b there is a new set {a, b} containing exactly a and b. Other axioms describe properties of set membership. A goal of the ZFC axioms is that each axiom should be true if interpreted as a statement about the collection of all sets in the von Neumann universe
(also known as the cumulative hierarchy).
The metamathematics
of ZFC has been extensively studied. Landmark results in this area established the independence of the continuum hypothesis
from ZFC, and of the axiom of choice from the remaining ZFC axioms.
proposed the first axiomatic set theory, Zermelo set theory
. This axiomatic theory did not allow the construction of the ordinal number
s. While most of "ordinary mathematics" can be developed without ever using ordinals, ordinals are an essential tool in most set-theoretic investigations. Moreover, one of Zermelo's axioms invoked a concept, that of a "definite" property, whose operational meaning was not clear. In 1922, Abraham Fraenkel and Thoralf Skolem
independently proposed operationalizing a "definite" property as one that could be formulated as a first order theory
whose atomic formula
s were limited to set membership and identity. They also independently proposed replacing the axiom schema of specification
with the axiom schema of replacement
. Appending this schema, as well as the axiom of regularity
(first proposed by Dimitry Mirimanoff in 1917), to Zermelo set theory yields the theory denoted by ZF. Adding to ZF either the axiom of choice (AC) or a statement that is equivalent to it yields ZFC.
All formulations of ZFC imply that at least one set exists. Kunen includes an axiom that directly asserts the existence of a set, in addition to the axioms given below. Its omission here can be justified in two ways. Many authors require a nonempty domain of discourse
as part of the semantics of the first-order logic in which ZFC is formalized. Also the axiom of infinity (below) also implies that at least one set exists, as it begins with an existential quantifier, so its presence makes superfluous an axiom (merely) asserting the existence of a set.
The converse of this axiom follows from the substitution property of equality. If the background logic does not include equality "=", x=y may be defined as an abbreviation for the following formula (Hatcher 1982, p. 138, def. 1):
In this case, the axiom of extensionality can be reformulated as
which says that if x and y have the same elements, then they belong to the same sets (Fraenkel et al. 1973).
is any property which may characterize the elements x of z, then there is a subset y of z containing those x in z which satisfy the property. The "restriction" to z is necessary to avoid Russell's paradox
and its variants. More formally, let
be any formula in the language of ZFC with free variables among 
. So y is not free in 
. Then:
This axiom is part of Z, but can be redundant in ZF, in that it may follow from the axiom schema of replacement
, with (as here) or without the axiom of the empty set.
The axiom of specification can be used to prove the existence of the empty set
, denoted
, once the existence of at least one set is established (see above). A common way to do this is to use an instance of specification for a property which all sets do not have. For example, if w is a set which already exists, the empty set can be constructed as
.
If the background logic includes equality, it is also possible to define the empty set as
.
Thus the axiom of the empty set is implied by the nine axioms presented here. The axiom of extensionality implies the empty set is unique, if it exists. It is common to make a definitional extension that adds the symbol
to the language of ZFC.
This axiom is part of Z, but is redundant in ZF because it follows from the axiom schema of replacement
applied to any two-member set. The existence of such a set is assured by either the axiom of infinity
, or by the axiom of the power set applied twice to the empty set.
there is a set A containing every set that is a member of some member of 
be any formula in the language of ZFC whose free variables are among 
, so that in particular B is not free in 
. Then:
Less formally, this axiom states that if the domain
of a definable function f is a set, and f(x) is a set for any x in that domain, then the range
of f is a subclass of a set, subject to a restriction needed to avoid paradoxes. The form stated here, in which B may be larger than strictly necessary, is sometimes called the axiom schema of collection.
abbreviate 
, where 
is some set. Then there exists a set X such that the empty set 
is a member of X and, whenever a set y is a member of X, then 
is also a member of X.
More colloquially, there exists a set X having infinitely many members. The minimal set X satisfying the axiom of infinity is the von Neumann ordinal ω, which can also be thought of as the set of natural numbers
.
abbreviate 
For any set x, there is a set y which is a superset
of the power set of x. The power set of x is the class whose members are all of the subsets of x.
Alternative forms of axioms 1–8 are often encountered, some of which are listed in Jech (2003). Some ZF axiomatizations include an axiom asserting that the empty set exists
. The axioms of pairing, union, replacement, and power set are often stated so that the members of the set x whose existence is being asserted are just those sets which the axiom asserts x must contain.
R which well-orders X. This means R is a linear order on X such that every nonempty subset
of X has a member which is minimal under R.
Given axioms 1-8, there are many statements equivalent to axiom 9, the best known of which is the axiom of choice (AC), which goes as follows. Let X be a set whose members are all non-empty. Then there exists a function f from X to the union of the members of X, called a "choice function", such that for all one has . Since the existence of a choice function when X is a finite set is easily proved from axioms 1-8, AC only matters for certain infinite sets. AC is characterized as nonconstructive because it asserts the existence of a choice set but says nothing about how the choice set is to be "constructed." Much research has sought to characterize the definability (or lack thereof) of certain sets whose existence AC asserts.
(Shoenfield 1977, sec. 2). In this viewpoint, the universe of set theory is built up in stages, with one stage for each ordinal number
. At stage 0 there are no sets yet. At each following stage, a set is added to the universe if all of its elements have been added at previous stages. Thus the empty set is added at stage 1, and the set containing the empty set is added at stage 2; see Hinman (2005, p. 467). The collection of all sets that are obtained in this way, over all the stages, is known as V. The sets in V can be arranged into a hierarchy by assigning to each set the first stage at which that set was added to V.
It is provable that a set is in V if and only if the set is pure and well-founded; and provable that V satisfies all the axioms of ZFC, if the class of ordinals has appropriate reflection properties. For example, suppose that a set x is added at stage α, which means that every element of x was added at a stage earlier than α. Then every subset of x is also added at stage α, because all elements of any subset of x were also added before stage α. This means that any subset of x which the axiom of separation can construct is added at stage α, and that the powerset of x will be added at the next stage after α. For a complete argument that V satisfies ZFC see Shoenfield (1977).
The picture of the universe of sets stratified into the cumulative hierarchy is characteristic of ZFC and related axiomatic set theories such as Von Neumann–Bernays–Gödel set theory
(often called NBG) and Morse–Kelley set theory
. The cumulative hierarchy is not compatible with other set theories such as New Foundations
.
It is possible to change the definition of V so that at each stage, instead of adding all the subsets of the union of the previous stages, subsets are only added if they are definable in a certain sense. This results in a more "narrow" hierarchy which gives the constructible universe
L, which also satisfies all the axioms of ZFC, including the axiom of choice. It is independent from the ZFC axioms whether V = L. Although the structure of L is more regular and well behaved than that of V, few mathematicians argue that V = L should be added to ZFC as an additional axiom.
(1961) included a result first proved in his 1957 Ph.D. thesis: if ZFC is consistent, it is impossible to axiomatize ZFC using only finitely many axioms. On the other hand, Von Neumann–Bernays–Gödel set theory
(NBG) can be finitely axiomatized. The ontology of NBG includes proper classes
as well as sets; a set is any class that can be a member of another class. NBG and ZFC are equivalent set theories in the sense that any theorem
not mentioning classes and provable in one theory can be proved in the other.
Gödel's second incompleteness theorem says that a recursively axiomatizable system that can interpret Robinson arithmetic
can prove its own consistency only if it is inconsistent. Moreover, Robinson arithmetic can be interpreted in general set theory
, a small fragment of ZFC. Hence the consistency
of ZFC cannot be proved within ZFC itself (unless it is actually inconsistent). Thus, to the extent that ZFC is identified with ordinary mathematics, the consistency of ZFC cannot be demonstrated in ordinary mathematics. The consistency of ZFC does follow from the existence of a weakly inaccessible cardinal
, which is unprovable in ZFC if ZFC is consistent. Nevertheless, it is deemed unlikely that ZFC harbors an unsuspected contradiction; it is widely believed that if ZFC were inconsistent, that fact would have been uncovered by now. This much is certain — ZFC is immune to the classic paradoxes of naive set theory
: Russell's paradox
, the Burali-Forti paradox
, and Cantor's paradox
.
Abian and LaMacchia (1978) studied a subtheory of ZFC consisting of the axioms of extensionality, union, powerset, replacement, and choice. Using models
, they proved this subtheory consistent, and proved that each of the axioms of extensionality, replacement, and power set is independent of the four remaining axioms of this subtheory. If this subtheory is augmented with the axiom of infinity, each of the axioms of union, choice, and infinity is independent of the five remaining axioms. Because non-well-founded set theory
is a model
of ZFC without the axiom of regularity, that axiom is independent of the other ZFC axioms.
If consistent, ZFC cannot prove the existence of the inaccessible cardinal
s that category theory
requires. Huge sets of this nature are possible if ZF is augmented with Tarski's axiom
(Tarski 1939). Assuming that axiom turns the axioms of infinity
, power set
, and choice (7 − 9 above) into theorems.
, whereby it is shown that every countable transitive model
of ZFC (sometimes augmented with large cardinal axioms) can be expanded to satisfy the statement in question. A different expansion is then shown to satisfy the negation of the statement. An independence proof by forcing automatically proves independence from arithmetical statements, other concrete statements, and large cardinal axioms. Some statements independent of ZFC can be proven to hold in particular inner model
s, such as in the constructible universe
. However, some statements that are true about constructible sets are not consistent with hypothesized large cardinal axioms.
Forcing proves that the following statements are independent of ZFC:
Remarks:
A variation on the method of forcing
can also be used to demonstrate the consistency and unprovability of the axiom of choice, i.e., that the axiom of choice is independent of ZF. The consistency of choice can be (relatively) easily verified by proving that the inner model L satisfies choice. (Thus every model of ZF contains a submodel of ZFC, so that Con(ZF) implies Con(ZFC).) Since forcing preserves choice, we cannot directly produce a model contradicting choice from a model satisfying choice. However, we can use forcing to create a model which contains a suitable submodel, namely one satisfying ZF but not C.
Another method of proving independence results, one owing nothing to forcing, is based on Gödel's second incompleteness theorem. This approach employs the statement whose independence is being examined, to prove the existence of a set model of ZFC, in which case Con(ZFC) is true. Since ZFC satisfies the conditions of Gödel's second theorem, the consistency of ZFC is unprovable in ZFC (provided that ZFC is, in fact, consistent). Hence no statement allowing such a proof can be proved in ZFC. This method can prove that the existence of large cardinals is not provable in ZFC, but cannot prove that assuming such cardinals, given ZFC, is free of contradiction.
ZFC has been criticized both for being excessively strong and for being excessively weak, as well as for its failure to capture objects such as proper classes and the universal set.
Many mathematical theorems can be proven in much weaker systems than ZFC, such as Peano arithmetic and second order arithmetic (as explored by the program of reverse mathematics
). Saunders Mac Lane
and Solomon Feferman
have both made this point. Some of "mainstream mathematics" (mathematics not directly connected with axiomatic set theory) is beyond Peano arithmetic and second order arithmetic, but still, all such mathematics can be carried out in ZC (Zermelo set theory
with choice), another theory weaker than ZFC. Much of the power of ZFC, including the axiom of regularity and the axiom schema of replacement, is included primarily to facilitate the study of the set theory itself.
On the other hand, among axiomatic set theories, ZFC is comparatively weak. Unlike New Foundations
, ZFC does not admit the existence of a universal set
. Hence the universe
of sets under ZFC is not closed under the elementary operations of the algebra of sets
. Unlike von Neumann–Bernays–Gödel set theory
and Morse–Kelley set theory
(MK), ZFC does not admit the existence of proper classes. These ontological
restrictions are required for ZFC to avoid Russell's paradox
, but critics argue these restrictions make the ZFC axioms fail to capture the informal concept of set. A further comparative weakness of ZFC is that the axiom of choice included in ZFC is weaker than the axiom of global choice
included in MK.
There are numerous mathematical statements undecidable in ZFC. These include the continuum hypothesis
, the Whitehead problem
, and the Normal Moore space conjecture
. Some of these conjectures are provable with the addition of axioms such as Martin's axiom
, large cardinal axioms to ZFC. Some others are decided in ZF+AD where AD is the axiom of determinacy
, a strong supposition incompatible with choice. One attraction of large cardinal axioms is that they enable many results from ZF+AD to be established in ZFC adjoined by some large cardinal axiom (see projective determinacy). The Mizar system
has adopted Tarski–Grothendieck set theory instead of ZFC so that proofs involving Grothendieck universe
s (encountered in category theory and algebraic geometry) can be formalized.
Related axiomatic set theories:
Ernst Zermelo
Ernst Friedrich Ferdinand Zermelo was a German mathematician, whose work has major implications for the foundations of mathematics and hence on philosophy. He is known for his role in developing Zermelo–Fraenkel axiomatic set theory and his proof of the well-ordering theorem.-Life:He graduated...
and Abraham Fraenkel and commonly abbreviated ZFC, is one of several axiomatic system
Axiomatic system
In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A mathematical theory consists of an axiomatic system and all its derived theorems...
s that were proposed in the early twentieth century to formulate a theory of sets without the paradoxes of naive set theory
Naive set theory
Naive set theory is one of several theories of sets used in the discussion of the foundations of mathematics. The informal content of this naive set theory supports both the aspects of mathematical sets familiar in discrete mathematics , and the everyday usage of set theory concepts in most...
such as Russell's paradox
Russell's paradox
In the foundations of mathematics, Russell's paradox , discovered by Bertrand Russell in 1901, showed that the naive set theory created by Georg Cantor leads to a contradiction...
. Specifically, ZFC does not allow unrestricted comprehension. Today ZFC is the standard form of axiomatic set theory and as such is the most common foundation of mathematics
Foundations of mathematics
Foundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, type theory and recursion theory...
.
ZFC has a single primitive ontological
Ontology
Ontology is the philosophical study of the nature of being, existence or reality as such, as well as the basic categories of being and their relations...
notion, that of a hereditary
Hereditary set
In set theory, a hereditary set is a set all of whose elements are hereditary sets. That is, all elements of the set are themselves sets, as are all elements of the elements, and so on...
well-founded set, and a single ontological assumption, namely that all individual
Individual
An individual is a person or any specific object or thing in a collection. Individuality is the state or quality of being an individual; a person separate from other persons and possessing his or her own needs, goals, and desires. Being self expressive...
s in the universe of discourse are such sets. Thus, ZFC is a set theory without urelements (elements of sets which are not themselves sets). ZFC does not formalize the notion of class
Class (set theory)
In set theory and its applications throughout mathematics, a class is a collection of sets which can be unambiguously defined by a property that all its members share. The precise definition of "class" depends on foundational context...
es (collections of mathematical objects defined by a property shared by their members) and specifically does not include proper classes (objects that have members but cannot be members themselves).
ZFC is a one-sorted theory in first-order logic
First-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...
. The signature has equality and a single primitive binary relation
Binary relation
In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = . More generally, a binary relation between two sets A and B is a subset of...
, set membership, which is usually denoted ∈. The formula a ∈ b means that the set a is a member of the set b (which is also read, "a is an element of b" or "a is in b").
There are many equivalent formulations of the ZFC axiom
Axiom
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...
s. Most of the ZFC axioms state the existence of particular sets. For example, the axiom of pairing
Axiom of pairing
In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of pairing is one of the axioms of Zermelo–Fraenkel set theory.- Formal statement :...
says that given any two sets a and b there is a new set {a, b} containing exactly a and b. Other axioms describe properties of set membership. A goal of the ZFC axioms is that each axiom should be true if interpreted as a statement about the collection of all sets in the von Neumann universe
Von Neumann universe
In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted V, is the class of hereditary well-founded sets...
(also known as the cumulative hierarchy).
The metamathematics
Metamathematics
Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories...
of ZFC has been extensively studied. Landmark results in this area established the independence of the continuum hypothesis
Continuum hypothesis
In mathematics, the continuum hypothesis is a hypothesis, advanced by Georg Cantor in 1874, about the possible sizes of infinite sets. It states:Establishing the truth or falsehood of the continuum hypothesis is the first of Hilbert's 23 problems presented in the year 1900...
from ZFC, and of the axiom of choice from the remaining ZFC axioms.
History
In 1908, Ernst ZermeloErnst Zermelo
Ernst Friedrich Ferdinand Zermelo was a German mathematician, whose work has major implications for the foundations of mathematics and hence on philosophy. He is known for his role in developing Zermelo–Fraenkel axiomatic set theory and his proof of the well-ordering theorem.-Life:He graduated...
proposed the first axiomatic set theory, Zermelo set theory
Zermelo set theory
Zermelo set theory, as set out in an important paper in 1908 by Ernst Zermelo, is the ancestor of modern set theory. It bears certain differences from its descendants, which are not always understood, and are frequently misquoted...
. This axiomatic theory did not allow the construction of the ordinal number
Ordinal number
In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
s. While most of "ordinary mathematics" can be developed without ever using ordinals, ordinals are an essential tool in most set-theoretic investigations. Moreover, one of Zermelo's axioms invoked a concept, that of a "definite" property, whose operational meaning was not clear. In 1922, Abraham Fraenkel and Thoralf Skolem
Thoralf Skolem
Thoralf Albert Skolem was a Norwegian mathematician known mainly for his work on mathematical logic and set theory.-Life:...
independently proposed operationalizing a "definite" property as one that could be formulated as a first order theory
First-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...
whose atomic formula
Atomic formula
In mathematical logic, an atomic formula is a formula with no deeper propositional structure, that is, a formula that contains no logical connectives or equivalently a formula that has no strict subformulas. Atoms are thus the simplest well-formed formulas of the logic...
s were limited to set membership and identity. They also independently proposed replacing the axiom schema of specification
Axiom schema of specification
In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom schema of specification, axiom schema of separation, subset axiom scheme or axiom schema of restricted comprehension, is a schema of axioms in Zermelo-Fraenkel set theory...
with the axiom schema of replacement
Axiom schema of replacement
In set theory, the axiom schema of replacement is a schema of axioms in Zermelo–Fraenkel set theory that asserts that the image of any set under any definable mapping is also a set...
. Appending this schema, as well as the axiom of regularity
Axiom of regularity
In mathematics, the axiom of regularity is one of the axioms of Zermelo-Fraenkel set theory and was introduced by...
(first proposed by Dimitry Mirimanoff in 1917), to Zermelo set theory yields the theory denoted by ZF. Adding to ZF either the axiom of choice (AC) or a statement that is equivalent to it yields ZFC.
The axioms
There are many equivalent formulations of the ZFC axioms; for a rich but somewhat dated discussion of this fact, see Fraenkel et al. (1973). The following particular axiom set is from Kunen (1980). The axioms per se are expressed in the symbolism of first order logic. The associated English prose is only intended to aid the intuition.All formulations of ZFC imply that at least one set exists. Kunen includes an axiom that directly asserts the existence of a set, in addition to the axioms given below. Its omission here can be justified in two ways. Many authors require a nonempty domain of discourse
Domain of discourse
In the formal sciences, the domain of discourse, also called the universe of discourse , is the set of entities over which certain variables of interest in some formal treatment may range...
as part of the semantics of the first-order logic in which ZFC is formalized. Also the axiom of infinity (below) also implies that at least one set exists, as it begins with an existential quantifier, so its presence makes superfluous an axiom (merely) asserting the existence of a set.
1. Axiom of extensionality
Two sets are equal (are the same set) if they have the same elements.The converse of this axiom follows from the substitution property of equality. If the background logic does not include equality "=", x=y may be defined as an abbreviation for the following formula (Hatcher 1982, p. 138, def. 1):
In this case, the axiom of extensionality can be reformulated as
which says that if x and y have the same elements, then they belong to the same sets (Fraenkel et al. 1973).
2. Axiom of regularity (also called the Axiom of foundation)
Every non-empty set x contains a member y such that x and y are disjoint sets.3. Axiom schema of specification (also called the axiom schema of separation or of restricted comprehension)
If z is a set, and is any property which may characterize the elements x of z, then there is a subset y of z containing those x in z which satisfy the property. The "restriction" to z is necessary to avoid Russell's paradoxRussell's paradox
In the foundations of mathematics, Russell's paradox , discovered by Bertrand Russell in 1901, showed that the naive set theory created by Georg Cantor leads to a contradiction...
and its variants. More formally, let
be any formula in the language of ZFC with free variables among . So y is not free in . Then:This axiom is part of Z, but can be redundant in ZF, in that it may follow from the axiom schema of replacement
Axiom schema of replacement
In set theory, the axiom schema of replacement is a schema of axioms in Zermelo–Fraenkel set theory that asserts that the image of any set under any definable mapping is also a set...
, with (as here) or without the axiom of the empty set.
The axiom of specification can be used to prove the existence of the empty set
Empty set
In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
, denoted
, once the existence of at least one set is established (see above). A common way to do this is to use an instance of specification for a property which all sets do not have. For example, if w is a set which already exists, the empty set can be constructed as.If the background logic includes equality, it is also possible to define the empty set as
.Thus the axiom of the empty set is implied by the nine axioms presented here. The axiom of extensionality implies the empty set is unique, if it exists. It is common to make a definitional extension that adds the symbol
to the language of ZFC.4. Axiom of pairing
If x and y are sets, then there exists a set which contains x and y as elements.This axiom is part of Z, but is redundant in ZF because it follows from the axiom schema of replacement
Axiom schema of replacement
In set theory, the axiom schema of replacement is a schema of axioms in Zermelo–Fraenkel set theory that asserts that the image of any set under any definable mapping is also a set...
applied to any two-member set. The existence of such a set is assured by either the axiom of infinity
Axiom of infinity
In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of infinity is one of the axioms of Zermelo-Fraenkel set theory...
, or by the axiom of the power set applied twice to the empty set.
5. Axiom of union
For any set there is a set A containing every set that is a member of some member of 6. Axiom schema of replacement
Let be any formula in the language of ZFC whose free variables are among , so that in particular B is not free in . Then:Less formally, this axiom states that if the domain
Domain (mathematics)
In mathematics, the domain of definition or simply the domain of a function is the set of "input" or argument values for which the function is defined...
of a definable function f is a set, and f(x) is a set for any x in that domain, then the range
Range (mathematics)
In mathematics, the range of a function refers to either the codomain or the image of the function, depending upon usage. This ambiguity is illustrated by the function f that maps real numbers to real numbers with f = x^2. Some books say that range of this function is its codomain, the set of all...
of f is a subclass of a set, subject to a restriction needed to avoid paradoxes. The form stated here, in which B may be larger than strictly necessary, is sometimes called the axiom schema of collection.
7. Axiom of infinity
Let abbreviate , where is some set. Then there exists a set X such that the empty set is a member of X and, whenever a set y is a member of X, then is also a member of X.More colloquially, there exists a set X having infinitely many members. The minimal set X satisfying the axiom of infinity is the von Neumann ordinal ω, which can also be thought of as the set of natural numbers
.8. Axiom of power set
Let abbreviate For any set x, there is a set y which is a supersetSuperSet
SuperSet Software was a group founded by friends and former Eyring Research Institute co-workers Drew Major, Dale Neibaur, Kyle Powell and later joined by Mark Hurst...
of the power set of x. The power set of x is the class whose members are all of the subsets of x.
Alternative forms of axioms 1–8 are often encountered, some of which are listed in Jech (2003). Some ZF axiomatizations include an axiom asserting that the empty set exists
Axiom of empty set
In axiomatic set theory, the axiom of empty set is an axiom of Zermelo–Fraenkel set theory, the fragment thereof Burgess calls ST, and Kripke–Platek set theory.- Formal statement :...
. The axioms of pairing, union, replacement, and power set are often stated so that the members of the set x whose existence is being asserted are just those sets which the axiom asserts x must contain.
9. Well-ordering theorem
For any set X, there is a binary relationBinary relation
In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = . More generally, a binary relation between two sets A and B is a subset of...
R which well-orders X. This means R is a linear order on X such that every nonempty subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
of X has a member which is minimal under R.
Given axioms 1-8, there are many statements equivalent to axiom 9, the best known of which is the axiom of choice (AC), which goes as follows. Let X be a set whose members are all non-empty. Then there exists a function f from X to the union of the members of X, called a "choice function", such that for all one has . Since the existence of a choice function when X is a finite set is easily proved from axioms 1-8, AC only matters for certain infinite sets. AC is characterized as nonconstructive because it asserts the existence of a choice set but says nothing about how the choice set is to be "constructed." Much research has sought to characterize the definability (or lack thereof) of certain sets whose existence AC asserts.
Motivation via the cumulative hierarchy
One motivation for the ZFC axioms is the cumulative hierarchy of sets introduced by John von NeumannJohn von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...
(Shoenfield 1977, sec. 2). In this viewpoint, the universe of set theory is built up in stages, with one stage for each ordinal number
Ordinal number
In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
. At stage 0 there are no sets yet. At each following stage, a set is added to the universe if all of its elements have been added at previous stages. Thus the empty set is added at stage 1, and the set containing the empty set is added at stage 2; see Hinman (2005, p. 467). The collection of all sets that are obtained in this way, over all the stages, is known as V. The sets in V can be arranged into a hierarchy by assigning to each set the first stage at which that set was added to V.
It is provable that a set is in V if and only if the set is pure and well-founded; and provable that V satisfies all the axioms of ZFC, if the class of ordinals has appropriate reflection properties. For example, suppose that a set x is added at stage α, which means that every element of x was added at a stage earlier than α. Then every subset of x is also added at stage α, because all elements of any subset of x were also added before stage α. This means that any subset of x which the axiom of separation can construct is added at stage α, and that the powerset of x will be added at the next stage after α. For a complete argument that V satisfies ZFC see Shoenfield (1977).
The picture of the universe of sets stratified into the cumulative hierarchy is characteristic of ZFC and related axiomatic set theories such as Von Neumann–Bernays–Gödel set theory
Von Neumann–Bernays–Gödel set theory
In the foundations of mathematics, von Neumann–Bernays–Gödel set theory is an axiomatic set theory that is a conservative extension of the canonical axiomatic set theory ZFC. A statement in the language of ZFC is provable in NBG if and only if it is provable in ZFC. The ontology of NBG includes...
(often called NBG) and Morse–Kelley set theory
Morse–Kelley set theory
In the foundation of mathematics, Morse–Kelley set theory or Kelley–Morse set theory is a first order axiomatic set theory that is closely related to von Neumann–Bernays–Gödel set theory...
. The cumulative hierarchy is not compatible with other set theories such as New Foundations
New Foundations
In mathematical logic, New Foundations is an axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of the theory of types of Principia Mathematica. Quine first proposed NF in a 1937 article titled "New Foundations for Mathematical Logic"; hence the name...
.
It is possible to change the definition of V so that at each stage, instead of adding all the subsets of the union of the previous stages, subsets are only added if they are definable in a certain sense. This results in a more "narrow" hierarchy which gives the constructible universe
Constructible universe
In mathematics, the constructible universe , denoted L, is a particular class of sets which can be described entirely in terms of simpler sets. It was introduced by Kurt Gödel in his 1938 paper "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis"...
L, which also satisfies all the axioms of ZFC, including the axiom of choice. It is independent from the ZFC axioms whether V = L. Although the structure of L is more regular and well behaved than that of V, few mathematicians argue that V = L should be added to ZFC as an additional axiom.
Metamathematics
The axiom schemata of replacement and separation each contain infinitely many instances. MontagueRichard Montague
Richard Merett Montague was an American mathematician and philosopher.-Career:At the University of California, Berkeley, Montague earned an B.A. in Philosophy in 1950, an M.A. in Mathematics in 1953, and a Ph.D. in Philosophy 1957, the latter under the direction of the mathematician and logician...
(1961) included a result first proved in his 1957 Ph.D. thesis: if ZFC is consistent, it is impossible to axiomatize ZFC using only finitely many axioms. On the other hand, Von Neumann–Bernays–Gödel set theory
Von Neumann–Bernays–Gödel set theory
In the foundations of mathematics, von Neumann–Bernays–Gödel set theory is an axiomatic set theory that is a conservative extension of the canonical axiomatic set theory ZFC. A statement in the language of ZFC is provable in NBG if and only if it is provable in ZFC. The ontology of NBG includes...
(NBG) can be finitely axiomatized. The ontology of NBG includes proper classes
Class (set theory)
In set theory and its applications throughout mathematics, a class is a collection of sets which can be unambiguously defined by a property that all its members share. The precise definition of "class" depends on foundational context...
as well as sets; a set is any class that can be a member of another class. NBG and ZFC are equivalent set theories in the sense that any theorem
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
not mentioning classes and provable in one theory can be proved in the other.
Gödel's second incompleteness theorem says that a recursively axiomatizable system that can interpret Robinson arithmetic
Robinson arithmetic
In mathematics, Robinson arithmetic, or Q, is a finitely axiomatized fragment of Peano arithmetic , first set out in R. M. Robinson . Q is essentially PA without the axiom schema of induction. Since Q is weaker than PA, it is incomplete...
can prove its own consistency only if it is inconsistent. Moreover, Robinson arithmetic can be interpreted in general set theory
General set theory
General set theory is George Boolos's name for a fragment of the axiomatic set theory Z. GST is sufficient for all mathematics not requiring infinite sets, and is the weakest known set theory whose theorems include the Peano axioms.-Ontology:...
, a small fragment of ZFC. Hence the consistency
Consistency proof
In logic, a consistent theory is one that does not contain a contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if and only if it has a model, i.e. there exists an interpretation under which all...
of ZFC cannot be proved within ZFC itself (unless it is actually inconsistent). Thus, to the extent that ZFC is identified with ordinary mathematics, the consistency of ZFC cannot be demonstrated in ordinary mathematics. The consistency of ZFC does follow from the existence of a weakly inaccessible cardinal
Inaccessible cardinal
In set theory, an uncountable regular cardinal number is called weakly inaccessible if it is a weak limit cardinal, and strongly inaccessible, or just inaccessible, if it is a strong limit cardinal. Some authors do not require weakly and strongly inaccessible cardinals to be uncountable...
, which is unprovable in ZFC if ZFC is consistent. Nevertheless, it is deemed unlikely that ZFC harbors an unsuspected contradiction; it is widely believed that if ZFC were inconsistent, that fact would have been uncovered by now. This much is certain — ZFC is immune to the classic paradoxes of naive set theory
Naive set theory
Naive set theory is one of several theories of sets used in the discussion of the foundations of mathematics. The informal content of this naive set theory supports both the aspects of mathematical sets familiar in discrete mathematics , and the everyday usage of set theory concepts in most...
: Russell's paradox
Russell's paradox
In the foundations of mathematics, Russell's paradox , discovered by Bertrand Russell in 1901, showed that the naive set theory created by Georg Cantor leads to a contradiction...
, the Burali-Forti paradox
Burali-Forti paradox
In set theory, a field of mathematics, the Burali-Forti paradox demonstrates that naively constructing "the set of all ordinal numbers" leads to a contradiction and therefore shows an antinomy in a system that allows its construction...
, and Cantor's paradox
Cantor's paradox
In set theory, Cantor's paradox is derivable from the theorem that there is no greatest cardinal number, so that the collection of "infinite sizes" is itself infinite...
.
Abian and LaMacchia (1978) studied a subtheory of ZFC consisting of the axioms of extensionality, union, powerset, replacement, and choice. Using models
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
, they proved this subtheory consistent, and proved that each of the axioms of extensionality, replacement, and power set is independent of the four remaining axioms of this subtheory. If this subtheory is augmented with the axiom of infinity, each of the axioms of union, choice, and infinity is independent of the five remaining axioms. Because non-well-founded set theory
Non-well-founded set theory
Non-well-founded set theories are variants of axiomatic set theory which allow sets to contain themselves and otherwise violate the rule of well-foundedness...
is a model
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
of ZFC without the axiom of regularity, that axiom is independent of the other ZFC axioms.
If consistent, ZFC cannot prove the existence of the inaccessible cardinal
Inaccessible cardinal
In set theory, an uncountable regular cardinal number is called weakly inaccessible if it is a weak limit cardinal, and strongly inaccessible, or just inaccessible, if it is a strong limit cardinal. Some authors do not require weakly and strongly inaccessible cardinals to be uncountable...
s that category theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
requires. Huge sets of this nature are possible if ZF is augmented with Tarski's axiom
Tarski-Grothendieck set theory
Tarski–Grothendieck set theory is an axiomatic set theory that was introduced as part of the Mizar system for formal verification of proofs. Its characteristic axiom is Tarski's axiom . The theory is a non-conservative extension of Zermelo–Fraenkel set theory...
(Tarski 1939). Assuming that axiom turns the axioms of infinity
Axiom of infinity
In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of infinity is one of the axioms of Zermelo-Fraenkel set theory...
, power set
Axiom of power set
In mathematics, the axiom of power set is one of the Zermelo–Fraenkel axioms of axiomatic set theory.In the formal language of the Zermelo–Fraenkel axioms, the axiom reads:...
, and choice (7 − 9 above) into theorems.
Independence in ZFC
Many important statements are independent of ZFC (see list of statements undecidable in ZFC). The independence is usually proved by forcingForcing (mathematics)
In the mathematical discipline of set theory, forcing is a technique invented by Paul Cohen for proving consistency and independence results. It was first used, in 1963, to prove the independence of the axiom of choice and the continuum hypothesis from Zermelo–Fraenkel set theory...
, whereby it is shown that every countable transitive model
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
of ZFC (sometimes augmented with large cardinal axioms) can be expanded to satisfy the statement in question. A different expansion is then shown to satisfy the negation of the statement. An independence proof by forcing automatically proves independence from arithmetical statements, other concrete statements, and large cardinal axioms. Some statements independent of ZFC can be proven to hold in particular inner model
Inner model
In mathematical logic, suppose T is a theory in the languageL = \langle \in \rangleof set theory.If M is a model of L describing a set theory and N is a class of M such that \langle N, \in_M, \ldots \rangle...
s, such as in the constructible universe
Constructible universe
In mathematics, the constructible universe , denoted L, is a particular class of sets which can be described entirely in terms of simpler sets. It was introduced by Kurt Gödel in his 1938 paper "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis"...
. However, some statements that are true about constructible sets are not consistent with hypothesized large cardinal axioms.
Forcing proves that the following statements are independent of ZFC:
- Continuum hypothesisContinuum hypothesisIn mathematics, the continuum hypothesis is a hypothesis, advanced by Georg Cantor in 1874, about the possible sizes of infinite sets. It states:Establishing the truth or falsehood of the continuum hypothesis is the first of Hilbert's 23 problems presented in the year 1900...
- Diamond principleDiamondsuitIn mathematics, and particularly in axiomatic set theory, the diamond principle ◊ is a combinatorial principle introduced by that holds in the constructible universe and that implies the continuum hypothesis...
- Suslin hypothesisSuslin's problemIn mathematics, Suslin's problem is a question about totally ordered sets posed by Mikhail Yakovlevich Suslin in a work published posthumously in 1920....
- Kurepa hypothesis
- Martin's axiomMartin's axiomIn the mathematical field of set theory, Martin's axiom, introduced by , is a statement which is independent of the usual axioms of ZFC set theory. It is implied by the continuum hypothesis, so certainly consistent with ZFC, but is also known to be consistent with ZF + ¬ CH...
(which is not a ZFC axiom) - Axiom of Constructibility (V=L)Axiom of constructibilityThe axiom of constructibility is a possible axiom for set theory in mathematics that asserts that every set is constructible. The axiom is usually written as V = L, where V and L denote the von Neumann universe and the constructible universe, respectively.- Implications :The axiom of...
(which is also not a ZFC axiom).
Remarks:
- The consistency of V=L is provable by inner modelInner modelIn mathematical logic, suppose T is a theory in the languageL = \langle \in \rangleof set theory.If M is a model of L describing a set theory and N is a class of M such that \langle N, \in_M, \ldots \rangle...
s but not forcing: every model of ZF can be trimmed to become a model of ZFC+V=L. - The Diamond Principle implies the Continuum Hypothesis and the negation of the Suslin Hypothesis.
- Martin's axiom plus the negation of the Continuum Hypothesis implies the Suslin Hypothesis.
- The constructible universeConstructible universeIn mathematics, the constructible universe , denoted L, is a particular class of sets which can be described entirely in terms of simpler sets. It was introduced by Kurt Gödel in his 1938 paper "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis"...
satisfies the Generalized Continuum Hypothesis, the Diamond Principle, Martin's Axiom and the Kurepa Hypothesis.
A variation on the method of forcing
Forcing (mathematics)
In the mathematical discipline of set theory, forcing is a technique invented by Paul Cohen for proving consistency and independence results. It was first used, in 1963, to prove the independence of the axiom of choice and the continuum hypothesis from Zermelo–Fraenkel set theory...
can also be used to demonstrate the consistency and unprovability of the axiom of choice, i.e., that the axiom of choice is independent of ZF. The consistency of choice can be (relatively) easily verified by proving that the inner model L satisfies choice. (Thus every model of ZF contains a submodel of ZFC, so that Con(ZF) implies Con(ZFC).) Since forcing preserves choice, we cannot directly produce a model contradicting choice from a model satisfying choice. However, we can use forcing to create a model which contains a suitable submodel, namely one satisfying ZF but not C.
Another method of proving independence results, one owing nothing to forcing, is based on Gödel's second incompleteness theorem. This approach employs the statement whose independence is being examined, to prove the existence of a set model of ZFC, in which case Con(ZFC) is true. Since ZFC satisfies the conditions of Gödel's second theorem, the consistency of ZFC is unprovable in ZFC (provided that ZFC is, in fact, consistent). Hence no statement allowing such a proof can be proved in ZFC. This method can prove that the existence of large cardinals is not provable in ZFC, but cannot prove that assuming such cardinals, given ZFC, is free of contradiction.
Criticisms
- For criticism of set theory in general, see Objections to set theory
ZFC has been criticized both for being excessively strong and for being excessively weak, as well as for its failure to capture objects such as proper classes and the universal set.
Many mathematical theorems can be proven in much weaker systems than ZFC, such as Peano arithmetic and second order arithmetic (as explored by the program of reverse mathematics
Reverse mathematics
Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Its defining method can briefly be described as "going backwards from the theorems to the axioms", in contrast to the ordinary mathematical practice of...
). Saunders Mac Lane
Saunders Mac Lane
Saunders Mac Lane was an American mathematician who cofounded category theory with Samuel Eilenberg.-Career:...
and Solomon Feferman
Solomon Feferman
Solomon Feferman is an American philosopher and mathematician with major works in mathematical logic.He was born in New York City, New York, and received his Ph.D. in 1957 from the University of California, Berkeley under Alfred Tarski...
have both made this point. Some of "mainstream mathematics" (mathematics not directly connected with axiomatic set theory) is beyond Peano arithmetic and second order arithmetic, but still, all such mathematics can be carried out in ZC (Zermelo set theory
Zermelo set theory
Zermelo set theory, as set out in an important paper in 1908 by Ernst Zermelo, is the ancestor of modern set theory. It bears certain differences from its descendants, which are not always understood, and are frequently misquoted...
with choice), another theory weaker than ZFC. Much of the power of ZFC, including the axiom of regularity and the axiom schema of replacement, is included primarily to facilitate the study of the set theory itself.
On the other hand, among axiomatic set theories, ZFC is comparatively weak. Unlike New Foundations
New Foundations
In mathematical logic, New Foundations is an axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of the theory of types of Principia Mathematica. Quine first proposed NF in a 1937 article titled "New Foundations for Mathematical Logic"; hence the name...
, ZFC does not admit the existence of a universal set
Universal set
In set theory, a universal set is a set which contains all objects, including itself. In set theory as usually formulated, the conception of a set of all sets leads to a paradox...
. Hence the universe
Universe
The Universe is commonly defined as the totality of everything that exists, including all matter and energy, the planets, stars, galaxies, and the contents of intergalactic space. Definitions and usage vary and similar terms include the cosmos, the world and nature...
of sets under ZFC is not closed under the elementary operations of the algebra of sets
Algebra of sets
The algebra of sets develops and describes the basic properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion...
. Unlike von Neumann–Bernays–Gödel set theory
Von Neumann–Bernays–Gödel set theory
In the foundations of mathematics, von Neumann–Bernays–Gödel set theory is an axiomatic set theory that is a conservative extension of the canonical axiomatic set theory ZFC. A statement in the language of ZFC is provable in NBG if and only if it is provable in ZFC. The ontology of NBG includes...
and Morse–Kelley set theory
Morse–Kelley set theory
In the foundation of mathematics, Morse–Kelley set theory or Kelley–Morse set theory is a first order axiomatic set theory that is closely related to von Neumann–Bernays–Gödel set theory...
(MK), ZFC does not admit the existence of proper classes. These ontological
Ontology
Ontology is the philosophical study of the nature of being, existence or reality as such, as well as the basic categories of being and their relations...
restrictions are required for ZFC to avoid Russell's paradox
Russell's paradox
In the foundations of mathematics, Russell's paradox , discovered by Bertrand Russell in 1901, showed that the naive set theory created by Georg Cantor leads to a contradiction...
, but critics argue these restrictions make the ZFC axioms fail to capture the informal concept of set. A further comparative weakness of ZFC is that the axiom of choice included in ZFC is weaker than the axiom of global choice
Axiom of global choice
In class theories, the axiom of global choice is a stronger variant of the axiom of choice which applies to proper classes as well as sets.- Statement :The axiom can be expressed in various ways which are equivalent:...
included in MK.
There are numerous mathematical statements undecidable in ZFC. These include the continuum hypothesis
Continuum hypothesis
In mathematics, the continuum hypothesis is a hypothesis, advanced by Georg Cantor in 1874, about the possible sizes of infinite sets. It states:Establishing the truth or falsehood of the continuum hypothesis is the first of Hilbert's 23 problems presented in the year 1900...
, the Whitehead problem
Whitehead problem
In group theory, a branch of abstract algebra, the Whitehead problem is the following question:Shelah proved that Whitehead's problem is undecidable within standard ZFC set theory.-Refinement:...
, and the Normal Moore space conjecture
Moore space (topology)
In mathematics, more specifically point-set topology, a Moore space is a developable regular Hausdorff space. Equivalently, a topological space X is a Moore space if the following conditions hold:...
. Some of these conjectures are provable with the addition of axioms such as Martin's axiom
Martin's axiom
In the mathematical field of set theory, Martin's axiom, introduced by , is a statement which is independent of the usual axioms of ZFC set theory. It is implied by the continuum hypothesis, so certainly consistent with ZFC, but is also known to be consistent with ZF + ¬ CH...
, large cardinal axioms to ZFC. Some others are decided in ZF+AD where AD is the axiom of determinacy
Axiom of determinacy
The axiom of determinacy is a possible axiom for set theory introduced by Jan Mycielski and Hugo Steinhaus in 1962. It refers to certain two-person games of length ω with perfect information...
, a strong supposition incompatible with choice. One attraction of large cardinal axioms is that they enable many results from ZF+AD to be established in ZFC adjoined by some large cardinal axiom (see projective determinacy). The Mizar system
Mizar system
The Mizar system consists of a language for writing strictly formalized mathematical definitions and proofs, a computer program which is able to check proofs written in this language, and a library of definitions and proved theorems which can be referenced and used in new articles. Mizar has goals...
has adopted Tarski–Grothendieck set theory instead of ZFC so that proofs involving Grothendieck universe
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...
s (encountered in category theory and algebraic geometry) can be formalized.
See also
- Foundation of mathematics
- Inner modelInner modelIn mathematical logic, suppose T is a theory in the languageL = \langle \in \rangleof set theory.If M is a model of L describing a set theory and N is a class of M such that \langle N, \in_M, \ldots \rangle...
- Large cardinal axiom
Related axiomatic set theories:
- Morse–Kelley set theoryMorse–Kelley set theoryIn the foundation of mathematics, Morse–Kelley set theory or Kelley–Morse set theory is a first order axiomatic set theory that is closely related to von Neumann–Bernays–Gödel set theory...
- Von Neumann–Bernays–Gödel set theoryVon Neumann–Bernays–Gödel set theoryIn the foundations of mathematics, von Neumann–Bernays–Gödel set theory is an axiomatic set theory that is a conservative extension of the canonical axiomatic set theory ZFC. A statement in the language of ZFC is provable in NBG if and only if it is provable in ZFC. The ontology of NBG includes...
- Tarski–Grothendieck set theory
- Constructive set theoryConstructive set theoryConstructive set theory is an approach to mathematical constructivism following the program of axiomatic set theory. That is, it uses the usual first-order language of classical set theory, and although of course the logic is constructive, there is no explicit use of constructive types...
- Internal set theoryInternal set theoryInternal set theory is a mathematical theory of sets developed by Edward Nelson that provides an axiomatic basis for a portion of the non-standard analysis introduced by Abraham Robinson. Instead of adding new elements to the real numbers, the axioms introduce a new term, "standard", which can be...
External links
- Stanford Encyclopedia of PhilosophyStanford Encyclopedia of PhilosophyThe Stanford Encyclopedia of Philosophy is a freely-accessible online encyclopedia of philosophy maintained by Stanford University. Each entry is written and maintained by an expert in the field, including professors from over 65 academic institutions worldwide...
articles by Thomas JechThomas JechThomas J. Jech is a mathematician specializing in set theory who was at Penn State for more than 25 years. He was educated at Charles University and is now at the of the Academy of Sciences of the Czech Republic....
: - Metamath version of the ZFC axioms — A concise and nonredundant axiomatization. The background first order logic is defined especially to facilitate machine verification of proofs.
- A derivation in Metamath of a version of the separation schema from a version of the replacement schema.