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Axiom of extensionality
Before giving the axioms that will allow us to construct the
integers we give the axiom of set
theory that defines what we mean by '. Without this axiom there
would be little point in
defining the integers or anything else. The axiom of
extensionality tells us that sets are uniquely
defined by their members.
means and have the same truth value or are equivalent. They are
either both true or
both false. It is the same as . This axiom says a pair of sets
and areequal if and only if they have exactly the same members.
Axiom of the empty set
Before defining any structure we need an axiom that asserts the
existence of the empty set. This
axiom uses the existential quantifier ( ). means there exists
some set for which
is true. Here is any expression that includes . We also
introduce the notation
to indicate that is not a member of set .
The axiom of the empty set is as follows.
This says there exists an object that no other set belongs to.
contains nothing.
Before we can define the integers we need to give two axioms for
constructing finite sets.
Axiom of union
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A set is an arbitrary collection of objects. The axiom of union
allows us to combine the objects inmany different sets and make
them members of a single new set. It says we can go down two
levels taking not the members of a set but the members of
members of a set and combine theminto a new set.
This says for every set there exists a set that is the union of
all the members of . For
every that belongs to there must be some set such that belongs
to and belongs to
.
Axiom of infinity
The integers are defined by an axiom that asserts the existence
of a set that contains all the
integers. is defined as the set containing and having the
property that if is in then
is in . To write this compactly we define some notation. We use
the integer 0 to
represent the empty set. If we have some set then we can
construct a set containing that is
written as .
This says there exists a set that contains the empty set 0 and
for every set that belongs to
the set constructed as also belongs to .
AXIOM OF Pairing
In the formal language of the ZermeloFraenkel axioms, the axiom
reads:
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or in words:
Given anysetA and any set B, there is a set Csuch that, given
any set D, D is a member
ofCif and only ifD is equal to AorD is equal to B.
or in simpler words:
Given two sets, there is a set whose members are exactly the two
given sets.
The axiom of pairing also allows for the definition ofordered
pairs. For any sets a and b, the
ordered pairis defined by the following:
Note that this definition satisfies the condition
