Difference between revisions of "Zermelo-Fraenkel Axioms"
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| − | + | The '''Zermelo-Fraenkel Axioms''' are a set of axioms that compiled by Ernst Zermelo and Abraham Fraenkel that make it very convenient for set theorists to determine whether a given collection of objects with a given property describable by the language of [[set theory]] could be called a [[set]]. As shown by paradoxes such as [[Russell's Paradox]], some restrictions must be put on which collections to call sets. | |
| − | == The Axiom of Extensionability == | + | ==Axioms== |
| + | === The Axiom of Extensionability === | ||
This axiom establishes the most basic property of sets - a set is completely characterized by its elements alone. <br/> | This axiom establishes the most basic property of sets - a set is completely characterized by its elements alone. <br/> | ||
'''Statement:''' If two sets have the same elements, they are identical. | '''Statement:''' If two sets have the same elements, they are identical. | ||
| − | == The Null Set Axiom == | + | === The Null Set Axiom === |
This axiom ensures that there is at least one set. <br/> | This axiom ensures that there is at least one set. <br/> | ||
'''Statement:''' There exists a set called the null set (or [[empty set]]) which contains no elements. | '''Statement:''' There exists a set called the null set (or [[empty set]]) which contains no elements. | ||
| − | == The Axiom of Subset Selection == | + | === The Axiom of Subset Selection === |
This axiom declares subsets of a given set as sets themselves. <br/> | This axiom declares subsets of a given set as sets themselves. <br/> | ||
'''Statement:''' Given a set <math>A</math>, and a formula <math>\phi(a)</math> with one free variable, there exists a set whose elements are precisely those elements of <math>A</math> which satisfy <math>\phi</math>. | '''Statement:''' Given a set <math>A</math>, and a formula <math>\phi(a)</math> with one free variable, there exists a set whose elements are precisely those elements of <math>A</math> which satisfy <math>\phi</math>. | ||
| − | == The Power Set Axiom == | + | === The Power Set Axiom === |
This axiom allows us to construct a bigger set from a given set. <br/> | This axiom allows us to construct a bigger set from a given set. <br/> | ||
'''Statement:''' Given a set <math>A</math>, there is a set containing all the subsets of A and no other element. | '''Statement:''' Given a set <math>A</math>, there is a set containing all the subsets of A and no other element. | ||
| − | == The Axiom of Replacement == | + | === The Axiom of Replacement === |
This axiom allows us, given a set, to construct other sets of the same size. <br/> | This axiom allows us, given a set, to construct other sets of the same size. <br/> | ||
'''Statement:''' Given a set <math>A</math> and a bijective binary relation describable in the language of set theory, there is a set which consists of exactly those elements related to elements in <math>A</math>. | '''Statement:''' Given a set <math>A</math> and a bijective binary relation describable in the language of set theory, there is a set which consists of exactly those elements related to elements in <math>A</math>. | ||
| − | == The Axiom of Union == | + | === The Axiom of Union === |
This axiom allows us to take unions of two or more sets. <br/> | This axiom allows us to take unions of two or more sets. <br/> | ||
'''Statement:''' Given a set <math>A</math>, there exists a set with exactly those elements which belong to some element of <math>A</math>. | '''Statement:''' Given a set <math>A</math>, there exists a set with exactly those elements which belong to some element of <math>A</math>. | ||
| − | == The Axiom of Infinity == | + | === The Axiom of Infinity === |
This gives us at least one infinite set. <br/> | This gives us at least one infinite set. <br/> | ||
'''Statement:''' There exists a set <math>A</math> containing the null set, such that for all <math>a</math> in <math>A</math>, <math>\{a\}</math> is also in <math>A</math>. | '''Statement:''' There exists a set <math>A</math> containing the null set, such that for all <math>a</math> in <math>A</math>, <math>\{a\}</math> is also in <math>A</math>. | ||
| − | == The Axiom of Foundation == | + | === The Axiom of Foundation === |
This makes sure no set contains itself, thus avoiding certain paradoxical situations. <br/> | This makes sure no set contains itself, thus avoiding certain paradoxical situations. <br/> | ||
'''Statement:''' The relation ''belongs to'' is [[well-founded]]. | '''Statement:''' The relation ''belongs to'' is [[well-founded]]. | ||
| − | == The Axiom of Choice == | + | === The Axiom of Choice === |
This allows to find a choice set for any arbitrary collection of sets. <br/> | This allows to find a choice set for any arbitrary collection of sets. <br/> | ||
Revision as of 18:49, 21 December 2007
The Zermelo-Fraenkel Axioms are a set of axioms that compiled by Ernst Zermelo and Abraham Fraenkel that make it very convenient for set theorists to determine whether a given collection of objects with a given property describable by the language of set theory could be called a set. As shown by paradoxes such as Russell's Paradox, some restrictions must be put on which collections to call sets.
Contents
Axioms
The Axiom of Extensionability
This axiom establishes the most basic property of sets - a set is completely characterized by its elements alone.
Statement: If two sets have the same elements, they are identical.
The Null Set Axiom
This axiom ensures that there is at least one set.
Statement: There exists a set called the null set (or empty set) which contains no elements.
The Axiom of Subset Selection
This axiom declares subsets of a given set as sets themselves.
Statement: Given a set 
, and a formula 
with one free variable, there exists a set whose elements are precisely those elements of which satisfy 
.
The Power Set Axiom
This axiom allows us to construct a bigger set from a given set.
Statement: Given a set , there is a set containing all the subsets of A and no other element.
The Axiom of Replacement
This axiom allows us, given a set, to construct other sets of the same size.
Statement: Given a set and a bijective binary relation describable in the language of set theory, there is a set which consists of exactly those elements related to elements in .
The Axiom of Union
This axiom allows us to take unions of two or more sets.
Statement: Given a set , there exists a set with exactly those elements which belong to some element of .
The Axiom of Infinity
This gives us at least one infinite set.
Statement: There exists a set containing the null set, such that for all 
in , 
is also in .
The Axiom of Foundation
This makes sure no set contains itself, thus avoiding certain paradoxical situations.
Statement: The relation belongs to is well-founded.
The Axiom of Choice
This allows to find a choice set for any arbitrary collection of sets.
Statement: Given any collection of sets, there exists a set (called the choice set) containing precisely one element of each set in the collection.
There is quite a bit of controversy over this axiom, because it does not hold for infinite sets. Generally, most mathematicians are uncomfortable using it, though it is widely accepted as being used for finite sets.
