Beyond Counting

1
Beyond Counting

• Infinity and the Theory of Sets

Nate Jones Chelsea Landis
2
Infinity

• Basis of Method of Exhaustion
• Used to find areas of curved regions
• Underlying idea of a limit
• Foundational concept of Calculus
• Relatively new to mathematics
• I protest above all against the use of an
  infinite quantity as a completed one, which in
  mathematics is never allowed. The Infinite is
  only a matter of speaking Carl Friedrich Gauss

3

• Looking at any number we know that we can always
  add 1 to any we come up with
• Georg Cantor considered the collection of all
  counting numbers as a distinct mathematical
  object

4

• Rational and Irrational
• (Ideas around during Civil War)
• Dense in each other
• Led to idea that real numbers were evenly divided
  by the rational and irrational

5
Georg Cantor (1845-1918)

• Saw that rationals and irrationals are distinct
  entities or sets
• Tried to compare rationals and irrationals and
  tried to match them in a 1-1 correspondence
• Found that infinite sets could be compared like
  finite sets.
• Cantors concept of a set
• By a set we are to understand any collection into
  a whole of definite and separate objects of our
  intuition or our thought.

6
Cantors Results

• Not all infinite sets are the same size.
• The set of irrationals is larger than the set of
  rationals
• Set of counting numbers is the same size as the
  set of rationals
• The set of all subsets of a set is larger than
  the set itself
• The set of points within any interval of the
  number line, no matter how short, is the same
  size as the set of all points everywhere on the
  number line
• The set of all points in a plane, or in
  3-dimensional, or (n-dimensional space) for any
  natural number n is the same size as the set of
  points on a single line

7
Which infinity is greater?Counting Numbers or
Fractions?
There are just as many counting numbers as there
are fractions!
8
Leopold Kronecker

• Prominent professor at the University of Berlin
• Disagreed with Cantors ideas of infinity
• His idea was that a mathematical object does not
  exist unless it is actually constructible in a
  finite number of steps
• Looked at the set of all even numbers that can be
  written as the sum of 2 odd primes
• This was never proven
• This shows that if we cant say what elements
  belong to the set, how can we describe the set as
  a completed whole?

9
Bertrand Russell (Paradoxes in Set Theory)

• A barber in a certain village claims he shaves
  all those villagers and only those villagers who
  do not shave themselves. If his claim is true,
  does the barber shave himself?

10

• If hes in the set, he doesnt shave himself, but
  since he shaves all who dont shave themselves,
  that must mean he must shave himself, so he ISNT
  in the set.
• If he isnt in the set, then he doesnt shave
  himself, but he only shaves those who dont shave
  themselves, so he must not shave himself, so he
  IS in the set.

11
Set Theory (1874 - 1884)

• Provided a unifying approach to probability,
  geometry, algebra, etc.
• Infinite sets were based on philosophical
  assumptions. Cantor argued philosophically his
  new ideas on mathematics.
• His works were looked at by mathematicians and
  philosophers because at the same time
  philosophers were looking for a way to
  accommodate both science and religion.

12

• Mathematics can be done without first resolving
  philosophical issues.
• Unlike Cantor, modern mathematicians and
  philosophers, see the recognition of the
  separation of math and philosophy as a giant
  forward stride in the progress of human thought.

13

• Neo-Thomism
• School of philosophical thought that viewed
  religion and science as compatible
• Came about from Pope Leo XIII, in 1879, from his
  writing of Aeterni Patris
• Held that science didnt need to lead to atheism
  and materialism.
• Cantor (Catholic) claimed infinite sets dealt
  with reality, but they should not be mistaken for
  the infinite God

14
Metaphysics

• The study of being and reality
• Cantor argued that infinite collections of
  numbers had a real (not necessarily material)
  existence.
• Neo-Thomistic philosophers in Germany argued that
  because the Mind of God is all knowing, God knows
  all natural numbers, all rationals, all infinite
  decimals, etc.

15
Most Important effect of set theory in Philosophy

• Cantors investigations led to clarifications of
  logical forms, methods of proof, and errors of
  syntax. These were used to refine arguments in
  philosophy.

16
Georg Cantors Ideas
Even though from one point of view the
entire list of numbers we count with
1,2,3,4,5,....... is twice as large as the
list of even numbers 2,4,6,8,10,......., the
two lists can be matched-up in a one-to-one
fashion.
This shows the two lists are the same size,
infinite.
17
Cantors Ideas cont.

• Cantor was able to demonstrate that there are
  different sizes of infinity.
• The infinity of decimal numbers that are bigger
  than zero but smaller than one is greater than
  the infinity of counting numbers.
• Cantor Diagonalization Proof

18
There are the same number of points on a short
semicircle arc as there are on the entire
unbounded line.
19
Conclusion

• Cantors work has affected mathematics in a
  positive way. His basic set theory has provided
  a simple, unifying approach to many different
  areas of mathematics.

20
Timeline

• 1774-1784 - Cantors work on set theory
• 1879 - Pope Leo XIII wrote Aeterni Patris
• 1884 Kroneckers ideas came about
• Around 1919 - Paradoxes in Set Theory
• Early 20th century - investigation of metaphysics
  and contradictions to set theory

21
References

• Berlinghoff, William P, Gouvea, Fernando Q. Math
  through the Ages A gentle History for Teachers
  and Others. 1st edition. Farmington, Maine.
  Oxton House Publishers, 2002.
• Counting to Infinity. lthttp//scidiv.bcc.ctc.edu/M
  ath/infinity.htmlgt 11/27/06.
• Platonic Realms Minitexts. You cant get there
  from here. lthttp//www.mathacademy.com/pr/minitext
  /infinity/ gt 11/27/06.