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Then there are alternatives on a different basis, notably NFU a very impredicative material set theory with a set of all sets and ETCS a structural set theory.
The source for this history, especially the dates, is mostly the English Wikipedia. Everything in standard ZFC ZFC is a pure set , which we will call simply a set; but there are also variations with urelement s and classes.
Urelements may be distinguished from sets and classes since they have no elements although the empty set also has no elements ; sets are usually those classes that are themselves elements members of sets.
If two sets have the same members, then they are equal and themselves members of the same sets. See axiom of extensionality for variations, such as whether this is taken as a definition or an axiomatisation of equality of sets, and how the condition might be strengthened if 10 is left out.
There is an empty set: By 1 , it follows that this set is unique; by even the weakest version of 5 , it is enough to state the existence of some set.
Analogous remarks apply to most of the other axioms. There are many variations, from Bounded Separation to Full Comprehension, which we should probably describe at axiom of separation.
Note that 5 follows from 4 and 6 using classical logic , so it is often left out, except in weak or intuitionistic versions. Again there are many variations, from Weak Replacement to Strong Collection, which we should probably describe at axiom of replacement.
When using intuitionistic logic , it is possible to accept only a weak version of this, such as Subset Collection or even weaker Exponentiation.
Using any but the weakest version of 6 , it is enough to state that there is a set satisfying Peano's axioms of natural numbers , or even any Dedekind-infinite set.
It seems to be uncommon to incorporate 2 into 8 , but in principle 8 implies 2. Note that this set is not unique, nor can we construct a canonical version which is, so we do not give it any name or notation.
This version is the simplest to state in the language of ZFC ZFC ; see axiom of choice for further discussion and weak versions.
It is possible to incorporate 9 into 5 or 6 , but this seems to be rare. For variations including the axiom of anti-foundation , see axiom of foundation.
This scheme can be made into a single axiom even in ZFC ZFC itself although not in versions with intuitionistic logic; in that case it can be made a single axiom only in a class theory.
Zermelo's original version consists of axioms 1—5 and 7—9 , in a somewhat imprecise form which affects the interpretation of 5 of higher-order classical logic.
The modern ZF ZF consists of 1—8 and 10 , using first-order classical logic , the strongest form of 6 that is, Strong Collection, although the standard Replacement is sufficient with classical logic , and the strongest form of 5 possible using only sets and not classes Full Separation.
Since Full Separation follows from Replacement with classical logic , it is often omitted from the list of axioms. The version originally formulated by Fraenkel and Skolem did not include 10 , although the three founders all eventually accepted it.
See also constructive set theory. CZF CZF uses axioms 1—8 and 10 , usually weak forms, in intuitionistic logic ; specifically, it uses Bounded Separation for 5 , Strong Collection for 6 , and an intermediate Subset Collection form of 7.
Shulman gives systematic notation for other versions, which includes those constructive and classical listed above.
The axiom of regularity prevents this from happening. By definition a set z is a subset of a set x if and only if every element of z is also an element of x:.
The Axiom of Power Set states that for any set x , there is a set y that contains every subset of x:. The axiom schema of specification is then used to define the power set P x as the subset of such a y containing the subsets of x exactly:.
Axioms 1—8 define ZF. Alternative forms of these axioms are often encountered, some of which are listed in Jech 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.
For any set X , there is a binary relation 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 provably 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. 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.
One motivation for the ZFC axioms is the cumulative hierarchy of sets introduced by John von Neumann. 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.
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.
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.
As noted earlier, proper classes collections of mathematical objects defined by a property shared by their members which are too big to be sets can only be treated indirectly in ZF and thus ZFC.
Quine's approach built on the earlier approach of Bernays The axiom schemata of replacement and separation each contain infinitely many instances.
Montague included a result first proved in his Ph. 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. Thus, to the extent that ZFC is identified with ordinary mathematics, the consistency of ZFC cannot be demonstrated in ordinary mathematics.
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.
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 there are non-well-founded models that satisfy each axiom of ZFC except the axiom of regularity, that axiom is independent of the other ZFC axioms.
If consistent, ZFC cannot prove the existence of the inaccessible cardinals that category theory requires.
Huge sets of this nature are possible if ZF is augmented with Tarski's axiom. The independence is usually proved by forcing , 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 models , such as in the constructible universe. However, some statements that are true about constructible sets are not consistent with hypothesized large cardinal axioms.
A variation on the method of forcing can also be used to demonstrate the consistency and unprovability of the axiom of choice , i. The consistency of choice can be relatively easily verified by proving that the inner model L satisfies choice.
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.
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.
The project to unify set theorists behind additional axioms to resolve the Continuum Hypothesis or other meta-mathematical ambiguities is sometimes known as "Gödel's program".
One school of thought leans on expanding the "iterative" concept of a set to produce a set-theoretic universe with an interesting and complex but reasonably tractable structure by adopting forcing axioms; another school advocates for a tidier, less cluttered universe, perhaps focused on a "core" inner model.
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. Hence the universe of sets under ZFC is not closed under the elementary operations of the algebra of sets.
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.
The Mizar system and Metamath have adopted Tarski—Grothendieck set theory , an extension of ZFC, so that proofs involving Grothendieck universes encountered in category theory and algebraic geometry can be formalized.