Difference between revisions of "Zermelo-Fraenkel Axioms"
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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>, \{a\} 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 == |
Revision as of 07:14, 5 August 2007
It would be very convenient indeed for set theorists if any collection of objects with a given property describable by the language of set theory could be called a set. Unfortunately, as shown by paradoxes such as Russell's Paradox, we must put some restrictions on which collections to call sets. The Zermelo Fraenkel axiom system, developed by Ernst Zermelo and Abraham Fraenkel, does precisely this.
Contents
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 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 sets and
, there exists a set with exactly those elements which belong ot at least one of the sets
and
.
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
What this precisely does I am unsure at the moment.
Statement: The relation belongs to is well-founded.
The Axiom of Choice
This allows to find a choice set for any arbitrary collection fo 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.