from early history to modern theories

1. HISTORY
OF
INFINITY

2. 3+4=
7
4-3=1
12/4=3
3*4=12 2^3=8

4. What is infinity?
 Infinity is a concept describing something without any
bound or larger than any natural number.
 An extremely large number of something.
 For example universe, time, natural number set

5. Philosophical background of infinity
 In philosophy and theology, infinity is explored as the
Ultimate, the Absolute, God, and Zeno's paradoxes.
 . In ethics infinity plays an important role designating that
which cannot be defined or reduced to knowledge or power.
 Pascal argues that a rational person should live as though
God exists and seeks to believe in god. If God does not
actually exist, such a person will have only a finite loss (some
pleasures, luxury, etc.), whereas they stand to receive infinite
gains and avoid infinite losses.

6. History:
Early thinking's -
 Babylonians : there was an infinite life after death
 ancient Egypt : Nun was the god of an infinite ocean
 Sanskrit word, “Aditya” corresponding to the translation
“unbounded”.
 500 BCE, Jaina : mathematics and religion as a type of
unified reality

7. Early Greek -
The concept of infinity was forced upon to Greeks
from the physical world by three traditional
observations.
 Time seems without end
 Space and time can be unendingly subdivided
 Space is without bound

8. The most ancient engagement of the idea of infinity was made by
Anaximander (610 BC – 546 BC)
 believe the universe was infinite (Apeiron meaning “unlimited”,
“boundless”, “infinite”, or “indefinite”).
 made the first map of the world.
A group of thinkers of ancient
Greece (later identified as The Atomists)
all similarly considered matter to be
made of an infinite number of “atoms”.
Anaximander

9. Zeno of Elea (490BC-430BC)
Especially known for his paradoxes that contributed to
the development of logical and mathematical rigors and
that were insoluble until the development of precise
concepts of continuity and infinity.

10. Aristotle (384BC-322BC) deals with the concept of infinity in terms of his notion
of actuality and of potentiality.
 potential infinity treats infinity as an unbounded or non-
terminating process developing over time.
 actual infinity treats the infinite as timeless and complete.
Early Indian
 The Jain mathematical text Surya Prajnapti (c. 4th–3rd century BCE) classifies
all numbers into three sets: enumerable, innumerable, and infinite. Each of
these was further subdivided into three orders:
 Enumerable: lowest, intermediate, and highest
 Innumerable: nearly innumerable, truly innumerable, and innumerably
innumerable
 Infinite: nearly infinite, truly infinite, infinitely infinite

11.  asaṃkhyāta ("countless, innumerable") and ananta ("endless,
unlimited"), between rigidly bounded and loosely bounded infinities.
Modern view
 Galileo Galilei (1564-1642)
Galileo’s paradox of the infinity
Galileo discovered something interesting about unbounded sets. if you assigned a
mapping between two such sets (in this case I will look at the sets of natural and
even numbers) you will not run out of elements from the first set before running out of
elements from the second.
1 2 3 4 5 6 7 ... n
2 4 6 8 10 12 14 ... 2*n
even numbers is a proper subset of the natural numbers.

12.  In 1655 John Wallis first used the notation ∞
 In 1699 Isaac Newton wrote about equations
with an infinite number of terms in his work.
John Wallis
Isaac Newton

13. Georg Cantor(1845-1918)
 He invented set theory
 Established the importance of one-to-one
correspondence between the members of two
sets, defined infinite and well-ordered sets, and
proved that the real numbers are more numerous
than the natural numbers.
 developed important concepts in topology and
their relation to cardinality.

14. Infinity Symbol
 The infinity symbol is a mathematical symbol representing the
concept of infinity.
 Wallis did not explain his choice of this symbol, but it has been
conjectured to be a variant form of a Roman numeral for 1,000
(originally CIƆ, also CƆ), which was sometimes used to mean
"many", or of the Greek letter ω (omega), the last letter in the
Greek alphabet.
 In mathematics, the infinity symbol is used more often to
represent a potential infinity
Symbol used by Euler to denote infinity.

15. Infinity set theory
 S can be put into one-to-one correspondence with T iff all the
elements of S can be matched with all the elements of T in a one-to-
one fashion.
 S is equinumerous with T, written S ~ T, iff there is a one-to-one
correspondence Between S and T.
 Countably Infinite and Countable Sets
 A set S is countably infinite iff it can be enumerated by an infinite
sequence.
 If T is countably infinite, then S is countably infinite iff S ~T.
 If S is finite and T is countably infinite, then S U T is countably infinite.

16.  Z = {. . . , −2, −1, 0, 1, 2, ….} is countably infinite.
 If S1, S2, . . . , Sn are countably infinite sets, then Si is countably
infinite.
 is countably infinite.
 If S and T are countably infinite, then S × T is countably infinite.
 If T is an infinite set, then T contains a countably infinite subset S.
 If T is an infinite set, then T is equinumerous with some proper subset of
itself.
 A set S is uncountably infinite/uncountable iff S is infinite but not
countably infinite.

17.  Cantor’s diagonal argument:
 Such sets are now known as uncountable
sets, and the size of infinite sets is now
treated by the theory of cardinal numbers.
 If s1, s2, … , sn, … is any enumeration of
elements from T, then there is always an
element s of Twhich corresponds to no sn in
the enumeration.
 The set T is uncountable.

18. Hilbert’s paradox of the grand hotel
 Suppose a new guess arrives and wishes to be accommodate in the hotel.
We can move the guess currently in room 1 to room 2, the guess
currently in room 2 to room 3, and so on., moving every guess from his
current room n to n+1. After this room 1 is empty and new guess can be
moved into that room. By repeating this procedure, it is possible to make
room for any finite number of new guests.
 It is also possible to accommodate a countably infinite number of new
guests: just move the person occupying room 1 to room 2, the guest
occupying room 2 to room 4, and, in general, the guest occupying
room n to room 2n, and all the odd-numbered rooms (which are
countably infinite) will be free for the new guests.
 It is possible to accommodate countably infinitely many coachloads of
countably infinite passengers each, by several different methods.

20. Constructivism
Constructivism is a philosophical viewpoint about the
nature of knowledge.
proof by contradiction is not constructively valid.

21. Mathematicians who have made major contribution to
constructivism:
 Leopold Kronecker: he was a German mathematician who worked on
number theory, algebra and logic. In algebraic number theory Kronecker
introduced the theory of divisors as an alternative to Dedekind’s theory
ideals. Kronecker also contributed to the concept of continuity,
reconstructing the form of irrational numbers in real numbers.
 L. E. J. Brower : forefather of intuitionism
 A. A. Markov: forefather of Russian school of constructivism
 Arend Heyting: formalized intuitionistic logic and theories
 Per Martin-Lof: founder of constructive type theories
 Errett Bishop: promoted a version of constructivism claimed to be
consistent with classical mathematics
 Paul Lorenzen (developed constructive analysis)

22. Mathematical applications of infinity.
Real analysis:
In real analysis the symbol ∞ called infinity, is used
to denote an unbounded limit.
X -> ∞ means that x grows without bound
X -> -∞ means the value of x is decreasing without
bound.

24. Complex analysis:
 Complex analysis the symbol ∞ called “infinity”, denotes an
unsigned infinite limit.
 X->∞ means that the magnitude |x| of x grows beyond any
assigned value.
 A point labeled ∞ can be added to the complex plane as a
topological space giving the one-point compactification of
the complex plane. When this is done, the resulting space is
one-dimensional complex manifold, or Riemann surface,
called the extended complex plane or the Riemann sphere.
 On the other hand, this kind of infinity enables division by
zero, namely z/0 = ∞ for any nonzero complex number z.

25. Cosmology
 In1584 the Italian philosophers and astronomer Giordano
Bruno proposed an unbounded universe in on the infinite
universe and worlds: “innumerable suns exist; innumerable
earths revolve around these suns in a manner similar to the
way the seven planets revolve around our sun. Living beings
inhabit these worlds”.
 Cosmologists have long sought to discover whether infinity
exists in our physical universe:
 Are there an infinite number of stars?
 Does the universe have infinite volume?
 Does space "go on forever"?