Set Theory
King Phrost
Set Theory
The branch of mathematics which deals with the formal
properties of sets as units (without regard to the nature of their individual
constituents) and the expression of other branches of mathematics in terms of
sets.
Set theory is the branch of mathematical logic that studies
sets, which can be informally described as collections of objects. Although
objects of any kind can be collected into a set, set theory, as a branch of
mathematics, is mostly concerned with those that are relevant to mathematics as
a whole. Wikipedia
Set theory is the
mathematical theory of well-determined collections, called sets, of objects
that are called members, or elements, of the set. Pure set theory deals
exclusively with sets, so the only sets under consideration are those whose
members are also sets. The theory of the hereditarily-finite sets, namely those
finite sets whose elements are also finite sets, the elements of which are also
finite, and so on, is formally equivalent to arithmetic. So, the essence of set
theory is the study of infinite sets, and therefore it can be defined as the
mathematical theory of the actual—as opposed to potential—infinite.
The notion of set is so simple that it is usually introduced
informally, and regarded as self-evident. In set theory, however, as is usual
in mathematics, sets are given axiomatically, so their existence and basic
properties are postulated by the appropriate formal axioms. The axioms of set
theory imply the existence of a set-theoretic universe so rich that all
mathematical objects can be construed as sets. Also, the formal language of
pure set theory allows one to formalize all mathematical notions and arguments.
Thus, set theory has become the standard foundation for mathematics, as every
mathematical object can be viewed as a set, and every theorem of mathematics
can be logically deduced in the Predicate Calculus from the axioms of set
theory. (Stanford Encyclopedia of Philosophy)
The origins of
set theory
Set theory, as a separate mathematical discipline, begins in
the work of Georg Cantor. One might say that set theory was born in late 1873,
when he made the amazing discovery that the linear continuum, that is, the real
line, is not countable, meaning that its points cannot be counted using the
natural numbers. So, even though the set of natural numbers and the set of real
numbers are both infinite, there are more real numbers than there are natural
numbers, which opened the door to the investigation of the different sizes of
infinity. See the entry on the early development of set theory for a discussion
of the origin of set-theoretic ideas and their use by different mathematicians
and philosophers before and around Cantor’s time. (Stanford
Encyclopedia of Philosophy)
Types of Sets
The sets are further categorised into different types, based
on elements or types of elements. These different types of sets in basic set
theory are:
Finite set: The number of elements is finite
Infinite set: The number of elements are infinite
Empty set: It has no elements
Singleton set: It has one only element
Equal set: Two sets are equal if they have same elements
Equivalent set: Two sets are equivalent if they have same
number of elements
Power set: A set of every possible subset.
Universal set: Any set that contains all the sets under
consideration.
Subset: When all the elements of set A belong to set B, then
A is subset of B
Examples Of Set Theory
In his 1874 paper Cantor considers at least two different
kinds of infinity. Before this orders of infinity did not exist but all
infinite collections were considered 'the same size'. However Cantor examines the set of algebraic real numbers, that is the
set of all real roots of equations of the form
+−1−1+−2−2+...+1+0=0anxn+an−1xn−1+an−2xn−2+...+a1x+a0=0,
where ai is
an integer. Cantor proves that the algebraic real
numbers are in one-one correspondence with the natural numbers in the following
way.
For an equation of the above form define its index to be
∣∣+∣−1∣+∣−2∣+...+∣1∣+∣0∣+∣an∣+∣an−1∣+∣an−2∣+...+∣a1∣+∣a0∣+n.
There is only
one equation of index 2, namely =0x=0. There are 3 equations
of index 3, namely
2=0,+1=0,−1=02x=0,x+1=0,x−1=0 and 2=0x2=0.
These give
roots 0, 1, -1. For each index there are only finitely many equations
and so only finitely many roots. Putting them in 1-1 correspondence
with the natural numbers is now clear but ordering them in order of index and
increasing magnitude within each index.
Given a
function φ defined on S, a set N⊆S⊆, and a distinguished element 1∈N1∈, they are as follows:
(α)(β)(γ)(δ)ϕ(N)⊂NN=ϕo{1}1∉ϕ(N)the function ϕ is
injective.(α)()⊂(β)={1}(γ)1∉()(δ)the function is injective.
Condition (β)
is crucial since it ensures minimality for the set of natural numbers, which
accounts for the validity of proofs by mathematical induction. N=ϕo{1}={1} is
read: N is the chain of singleton {1} under the
function φ, that is, the minimal closure of {1} under the function φ. In
general, one considers the chain of a set A under an arbitrary mapping γ, denoted
by γo(A)(); in his booklet Dedekind developed an interesting theory of such
chains, which allowed him to prove the Cantor-Bernstein theorem. The theory was
later generalized by Zermelo and applied by Skolem, Kuratowski, etc.
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