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It is provable that plain ZF ZF , if consistent, cannot be finitely axiomatized in its own first-order language; NBG NBG escapes this conclusion by extending the language with the notion of classes.
One can also rework all of the weak versions of set theory above into a class theory like NBG NBG , which is conservative over the original set theory.
One can also use a class theory like MK MK , although this destroys any attempt to use a weak version of 5. These additional axioms are most commonly studied in the context of a material set theory, but they work just as well in a structural set theory.
Of course, one can add a large cardinal to MK MK to get something even stronger. It is often convenient to assume that one always has more large cardinals when necessary.
You cannot say this in an absolute sense, but you can adopt the axiom that every set belongs to some Grothendieck universe.
This is not the last word, however; you can make it stronger by adding classes in the style of MK MK , or even adding a cardinal which is inaccessible from TG TG.
In fact, we have barely begun the large cardinals known to modern set theory! However, it is incompatible with the sufficiently large cardinals: AD AD is inconsistent with the full axiom of choice, although it is consistent with dependent choice.
The equiconsistency of projective determinacy with a large cardinal assertion can be regarded as a step in this direction.
This uses 1—4 , Bounded Separation for 5 , and 7—10 , with Weak Replacement following from 5 and 7. English versions of the early key texts on set theory by Zermelo, Fraenkel, Skolem, von Neumann et al.
There are many texts which discuss ZFC and the cumulative hierarchy from a traditional material set-theoretic perspective.
A good example is. Last revised on May 2, at Set Theory, 2nd ed. Introduction to Mathematical Logic, 4th ed. Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more.
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Visit our list of: High Yield Dividend Stocks. News Market Warp for November 9: The consistency of a theory such as ZFC cannot be proved within the theory itself.
Formally, ZFC is a one-sorted theory in first-order logic. The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the s.
However, the discovery of paradoxes in naive set theory , such as Russell's paradox , led to the desire for a more rigorous form of set theory that was free of these paradoxes.
In , Ernst Zermelo proposed the first axiomatic set theory , Zermelo set theory. In , Fraenkel and Thoralf Skolem independently proposed operationalizing a "definite" property as one that could be formulated as a first order theory whose atomic formulas 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 , to Zermelo set theory yields the theory denoted by ZF.
The following particular axiom set is from Kunen 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.
First, in the standard semantics of first-order logic in which ZFC is typically formalized, the domain of discourse must be nonempty.
Consequently, it is a theorem of every first-order theory that something exists. However, as noted above, because in the intended semantics of ZFC there are only sets, the interpretation of this logical theorem in the context of ZFC is that some set exists.
Hence, there is no need for a separate axiom asserting that a set exists. Second, however, even if ZFC is formulated in so-called free logic , in which it is not provable from logic alone that something exists, the axiom of infinity below asserts that an infinite set exists.
This implies that a set exists and so, once again, it is superfluous to include an axiom asserting as much. The converse of this axiom follows from the substitution property of equality.
Every non-empty set x contains a member y such that x and y are disjoint sets. This implies, for example, that no set is an element of itself and that every set has an ordinal rank.
Subsets are commonly constructed using set builder notation. Note that the axiom schema of specification can only construct subsets, and does not allow the construction of sets of the more general form:.
This restriction is necessary to avoid Russell's paradox and its variants that accompany naive set theory with unrestricted comprehension.
In some other axiomatizations of ZF, this axiom is redundant in that it follows from the axiom schema of replacement and the axiom of the empty set.
For example, if w is any existing set, the empty set can be constructed 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 does not depend on w. If x and y are sets, then there exists a set which contains x and y as elements.
The axiom schema of specification must be used to reduce this to a set with exactly these two elements. The axiom of pairing is part of Z, but is redundant in ZF because it follows from the axiom schema of replacement, if we are given a set with at least two elements.
The existence of a set with at least two elements is assured by either the axiom of infinity , or by the axiom schema of specification and the axiom of the power set applied twice to any set.
The union over the elements of a set exists. The axiom schema of replacement asserts that the image of a set under any definable function will also fall inside a set.
More colloquially, there exists a set X having infinitely many members. It must be established, however, that these members are all different, because if two elements are the same, the sequence will loop around in a finite cycle of sets.
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. 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.
This makes it conservative over ZFC ZFC and also allows for a finite axiomatisation; we replace the formulas in 5 and 6 with classes, and add some special cases of 5 for subclasses, one for each logical connective.