1
TO INFINITY AND BEYOND

• A DEEPER LOOK AND UNDERSTANDING

2
INFINITY DEF. AND SOME BACKGROUND

• To Infinity and Beyond the phrase coined by
  Buzz Light-year in the Disney animated film Toy
  Story is somewhat redundant
• Infinity as we know it is the beyond it is as
  far, as large, as long, etc. as our brains can
  wrap around.
• It can be defined as an unlimited extent of time,
  space, or quantity eternity boundlessness
  immensity or unlimited capacity, energy,
  excellence, or knowledge.
• Natural numbers are said to be infinite
  1,2,3,The number of points on an unending line
  are infinite. Time and space are assumed
  infinite.
• With these observations it is easy to see why our
  friend Buzzs phrase is a little off base.

3
INFINITY SOME HISTORY

• In about the sixth century B.C., the first
  acknowledgement of infinity was made by the
  Greeks. They were the first to come up with
  philosophical sciences from practical ideas.
• However, in a mathematical sense, they were not
  unable to wrap their brains around the concept
  so easily, so they sort of just dismissed the
  notion, and avoided the concept .
• It was a long time later when infinity resurfaced
  in the sixteenth century with a formula showing
  that pi can be calculated with other operations
  than geometry ones. The formula was the first to
  use infinity to express a function. (See formula
  page 1)

4
Some History cont.

• Another formula was exposed by the mathematician
  John Wallis in 1650. Wallis was born in 1616 in
  Ashford, Kent, England, and died in 1703. His
  formula included both pi and infinity as well.
  (see formula page 2)
• A third formula uncovering an infinite series was
  detected by Gregory Leibniz in 1674. (see formula
  page 3)
• All three of these discoveries were the result of
  finding an approximation for pi. Since pi is in
  essence a never ending decimal, without the help
  of infinity, we would probably still be searching
  for a way to recognize and define the expansion
  of pi.
• This was pretty much the standing point of
  infinity until later, with the genius of George
  Cantor. He published the first of many papers to
  come about the concept of infinity.

5
INFINITY BIOGRAPHY OF GEORGE CANTOR AND HIS
CONTRIBUTION

• George Cantor was a mathematician who was born in
  Russia, but lived in Germany most of his life. He
  was the son of a Danish merchant, George, and
  Maria, a Russian musician.
• He attended German schools and received his
  Doctorate from the University of Berlin in 1867.

6
CANTOR BIO. Cont.

• Cantor was a disciple of Immanuel Kent, and he
  was interested in developing a new logic of
  infinity, others than those used by astrologers,
  etc. before him.
• He assumed there was something about rational
  numbers that makes them discontinuous, and too,
  something about real numbers that made the set
  continuous.

7
Cantor cont.

• Cantor worked to come up with a dichotomy between
  these two sets of numbers, and established a
  diagonal method (shown on paper) to demonstrate
  his theory.
• He proved that the set of real numbers (R) is
  uncountable, and therefore larger than the set of
  all rational numbers (Q), with rational numbers
  being countable on the other hand.
• Cantor saw infinity as a mixture of continuous,
  as well as discontinuous sets, while both being
  infinite still at the same time.

8
Cantor cont.

• The last part of George Cantors life was not so
  productive.
• He suffered from depression, which today would be
  considered as a case of bi-polar disorder.
• He was hospitalized many times, and it was in one
  of the institutions that he passed away in 1918.
• Although Cantor did not live to be very old, and
  his innovative mathematics faced much criticism,
  he was a stronghold force in the research and
  understanding of infinity, and so is remembered
  for that.

9
Infinity Some of Cantors Work

• Cantor used a one to one correspondence method,
  between two sets of numbers that have the same
  cardinality. If you use a finite set, and an
  infinite set, the one to one correspondence acts
  in the same way.
• CORRESPONDENCE WITH A FINITE SET
• 1 2 3 4 5 cardinality of five
• 4 4 5 6 8 also cardinality of five
• (set) (set)
• CORRESPONDENCE WITH AN INFINITE SET
• Set 1 1 2 3 4 5 6...n (infinite number of
  counting numbers cardinality 0)
• Set 2 0 1 2 3 4 5...n-1... (infinite number
  of whole numbers cardinality 0)
• (set)(set)
• Since corresponding numbers represented by n in
  the first set and n-1 in the second set, no
  matter what point is chosen in set one (n), (n-1)
  will represent the correct correspondence in set
  two, and both sets have cardinality zero.

10
More of Cantors Work

• Cantors famous Diagonalization Proof shows that
  the cardinality of Reals can be proved to be
  larger than the cardinality of Natural numbers.
• First, we assume that we have found a one to one
  correspondence between the reals and the natural
  numbers between zero and one.
• Secondly, we put numbers in a list in decimal
  notation that we assume, again, to be complete.

11
Cantors Work cont.

• 1. .421341513654...
• 2. .356745784568...
• 3. .757456724253...
• 4. .558783472955...
• 5. .878936527634
• 6.
• Third, we assume our list is complete.
• Next, we choose numbers different than the first
  digit in the first number, the second digit in
  the second number, the third digit in the third
  number, and so on, on down the diagonal.
• For example we can choose our number to be
  .23452.
• This last number will be different from the
  first in the first digit, be different from the
  second digit in the second number, etc.

12
CANTORS WORK cont.

• Therefore the number will differ from all numbers
  in the list, but is clearly a number between zero
  and one and is a real number.
• Since our number is not in our list, our list is
  not complete as assumed.
• Since this is so, there cannot be any such one to
  one correspondence between the reals and the
  naturals between zero and one.
• In this manner we can map the naturals onto the
  reals but not map the reals onto the naturals.
• We can see through this that the infinity of
  decimal numbers that are bigger than zero but
  less than one is greater than the infinity of
  counting numbers.
• No matter how large a set is, we can always
  consider a set that is largerinfinity.

13
CONTRADICTION TO CANTOR

• There have been many whom have questioned George
  Cantors reasoning for his infinity findings.
  Some have recognized mistakes that Cantor did not
  acknowledge, including the following argument.

14
CONTRADICTION TO CANTOR cont.

• One of Cantors claims was that all non-finite
  countable sets have the same cardinality. (The
  contradiction starts out here)
• If we have N 0,1,2,3,4,
• There are sets
• I 0,2,4,6,8... (N x 2)
• II 0,1,4,9,16... (N squared)
• III 1,3,5,7,9... (N x 2 1)
• Claim I, II, and III have the same cardinality.
  Also goes on to claim that I, II, III are proper
  subsets of the set N.
• This is where Cantor goes wrong

15
Contradiction, cont.

• The correct interpretation is that they are not
  subsets because they have the same cardinality as
  set N.
• Cantor was unaware that given sets A,B
• If A sub B (A is a proper subset of B) then
  Cardinality (A) lt Cardinality (B)
• Cardinality (A) Cardinality (B) if and only if
  A 1-1 B (there is only one element of set A for
  every element of set B).
• Contemporary mathematicians of the time,
  including Cantor, were unable to imagine sets of
  greater cardinality than the set N.

16
CONCLUSION

• The concept of infinity does not end with Cantor,
  obviously. His results were looked over and
  studied, and met with much criticism from other
  mathematicians, as well as his peers even.
• As we all know, infinity still is prevalent today
  in physics, astronomy, cosmology, etc.
• Modern mathematics makes much use of infinite
  sequences of numbers, and will probably in time
  come up with other ways to express this concept,
  considering the exponential ways of the subject.
• Although some may consider, or even prefer, to
  not use infinity, or not try and understand its
  magnitude, we can always go beyond what infinity
  is intuitively.
• But once redundant as in To Infinity and
  Beyond, ironically, the concept itself is a
  subject of the beyond.