| Set
Theory:
In
mathematics and logic: The theory of
sets i.e. of "any gathering into a whole ... of
distinct perceptual or mental objecs" or "A set is a many which allows
itself to be thought of as one". See first Set
for a clarification of sets. There are quite a few axiomatizations of
set theory. The general purpose of Set Theory is to have a set of
assumptions about sets that is sufficient to serve as a foundation for
all or most of mathematics.
The one that follows is the standard one known as ZF or
Zermelo-Frankel, after the men who thought of it. It consists of the
following set of axioms, that are here given both by name and formally.
Both the names and the axioms are standard:
ZF1: Axiom of Extensionality:
Two sets are equal precisely if they have
the same elements.
x1=x2 IFF (x3)(x3
e x1 IFF x3 e x2)
ZF2: Null Set Axiom:
There is a set without elements.
(Ex1)(x2)~(x2
e x1)
ZF3: Axiom of Pairing:
For any two sets there is a set which has
precisely these
two sets as elements.
(x1)(x2)(Ex3)(x4)(x4
e x3 IFF x4=x1 V x4=x2)
ZF4: Axiom of Unions:
For any set there is a set with all the
members of
all the members of the set.
(x1)(Ex2)(x3)(x3
e x2 IFF (Ex4)(x4 e x1 & x3
e x4))
Def: Definition of set-inclusion:
One set is included in another if all the
elements of the one
are elements of the other.
x1 inc x2 IFF (x3)(x3
e x1 --> x3 e x2)
ZF5: Power Set Axiom:
For every set there is a set which has as
elements
all the sets that are included in the set.
(x1)(Ex2)(x3)(x3
e x2 IFF x3 inc x1)
ZF6: Axiom Schema of Replacement:
For every set there is a set such that if
the first set
is by F related to the second set then for
every set A
there is a set B such that every set C is
element of A precisely if
C is related by F to B.
(x1)(Ex2)(F(x1,x2)
--> (x3)(Ex4)(x5)(x5 e x4
IFF (Ex6)(x6 e x3 & F(x6,x5)))
Def: Definition of Null Set:
Ø is a term for the set without any
elements.
x1=Ø IFF ~(Ex2)(x2
e x1)
Def: Definition of Union:
x1 = (x2 U x3)
IFF (x4)(x4 e x1 IFF x4 e x2
V x4 e x3)
ZF7: Axiom of Infinity:
There is a set which has the Null Set as an
element and
is such that union of the set and any
element of the set also
is an element of the set.
(Ex1)(Ø e x1 & (x2)(x2
e x1 --> (x1 U {x2}) e x1)
ZF8: Axiom of Foundation:
Every non-empty set has an element
which has no element in common with the
set.
(x1)(~(x1=Ø) --> (Ex2)(x2
e x1 & ~(Ex3)(x3 e x2 & x3
e x1))
Here are some comments and explanations. The Axiom Schema of
Replacement is said to be a schema because it involves an arbitrary
(schematic) function F. Ø is a unique set by the Axiom of Extensionality and it exists by the
Null Set Axiom.
The Axiom of Infinity implies that as Ø is element of the set, so are
{Ø}, {{Ø}}, {{{Ø}}} ... etc. without end. These are all distinct since
~( Ø = {Ø}) etc. because Ø has no elements and {Ø} has one element,
namely Ø.
The Axiom of Foundation prevents chains of the form (x1 e x2
& x2 e x3 & ... & xn-1 e x1).
I.o.w.: it assures that no set is element of itself nor element of any
of its elements.) For a brief explanation of what is the point of it
all - all of mathematics in one simple set of axioms and a lingua franca
for all of mathematics - see Set. There are many
good introductions to Set Theory (and also some not so good). One of the
best for the beginner must be Naive Set Theory by Paul Halmos. Those who
want a somewhat stricter and fairly complete version of ZF should look
into Suppes. Quine provides a sort of minimalist general set theory.
Mulder is a recent survey of many set theories, their uses and
foundations, and their relations to the foundations of physics and
mathematics.
Lipschutz is a fine and very clear survey and introduction.
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