Infinity and the Theory of Sets

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Infinity and the Theory of Sets

Chapter 25 By Stephanie Lawrence
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Preview

• Zeno of Ela, a Greek mathematician poses the
  question
• Bolzano gives a definition of sets
• Pope Leo XIII gets religion involved
• The Birth of Set Theory
• Cantor publishes a six part treatise on set
  theory
• Kroeneckers Opposition
• Bertrand Russell publishes an example of a
  paradox found in Cantors theory
• Lets get Metaphysical
• Today Cantors theory is widely accepted

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Zeno of Elea

• Zenos questions on the infinite made an early
  contribution to the definition of infinity
• By the middle ages mathematicians were discussing
  comparisons of infinite sets.

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1847 Bernard Bolzano

• Bolzano defines sets as an embodiment of the idea
  or concept which we conceive when we regard the
  arrangement of its parts as a matter of
  indifference. (O'Connor, 1996)

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1879 Pope Leo XIII

• Issues a formal letter to the bishops requesting
  that the Cataholic Church revisit the study of
  Scholastic philosophy
• Neo-Thomism is a result

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1874 The Birth of Set Theory

• Unlike the majority of mathematical discoveries,
  which typically grow out of centuries of thought
  and are frequently discovered simultaneously by
  several mathematians, set theory was the
  discovery of one man.
• George Cantor publishes his first article in
  Crelles Journal. In it he considers two
  different kinds of infinity (i.e. some infinite
  sets are larger than others).
• Cantor shows that real numbers cannot be put into
  one-one correspondence with the natural numbers
  using an argument with nested intervals.

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Cantors six part treatise

• Published in Mathematische Annalen, a German
  mathematical research journal founded in 1868 by
  Alfred Clebsch und Carl Neumann and still exists
  today.

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Pigeonhole Principle

• The Pigeonhole Principle Suppose we place m
  pigeons in n pigeonholes, where m and n are
  positive integers. If m gt n, show that at least
  two pigeons must be placed in the same
  pigeonhole. (Dangello and Seyfried, 2000)
• Consider a chessboard with two of the diagonally
  opposite corners removed. Is it possible to
  cover the board with pieces of domino whose size
  is exactly two board squares? (Bogomolny, 2006)

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Chessboard Example
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Solution

• No, it's not possible. Two diagonally opposite
  squares on a chess board are of the same color.
  Therefore, when these are removed, the number of
  squares of one color exceeds by 2 the number of
  squares of another color. However, every piece of
  domino covers exactly two squares and these are
  of different colors. Every placement of domino
  pieces establishes a 1-1 correspondence between
  the set of white squares and the set of black
  squares. If the two sets have different number of
  elements, then, by the Pigeonhole Principle, no
  1-1 correspondence between the two sets is
  possible. (Bogomolny, 2006)

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Kroeneckers Opposition

• Leopold Kronecker did not believe that infinite
  sets existed because they could not be
  constructed using a finite number of steps.
• Due to his way of thinking, Cantors theories
  were incomprehensible to Kronecker.

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Bertrand Russell

• A barber in a certain village claims that he
  shaves all those villagers and only those
  villagers who do not shave themselves. If his
  claim is true, does the barber shave himself?
  (Berlinghoff and Gouvea, 2002)

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Lets get Metaphysical

• Metaphysics is a branch of philosophy that tries
  to understand the fundamental nature of reality.
• Cantor says that infinite collectinos have real
  existences, but are not necessarily material.
• The neo-Thomists buy it and Cantors theories are
  accepted among religious leaders.
• As a result, the study of mathematics is removed
  from the realm of metaphysics.

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Today

• Mathematics have resolved that valid mathematics
  do not have to revolve around the truth of
  philosophical issues, separating math from
  philosophy.
• After much scrutiny and fine tuning, Cantors
  theory of sets is widely accepted today.

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Timeline

• 450 BC Zeno of Ela, a Greek mathematician/philos
  opher whose questions on infinity presented
  paradoxes which challenged mathematicians view
  of the real world for many centuries.
• 1847 Bolzano defines sets in the following way
  an embodiment of the idea or concept which we
  conceive when we regard the arrangement of its
  parts as a matter if indifference
• 1874 The birth of Set Theory takes place when
  George Cantor publishes his first article in
  Crelles Journal. In it he considers two
  different kinds of infinity. Also, he shows that
  the real numbers cannot be put into one-one
  correspondence with the natural numbers using an
  argument with nested intervals.

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Timeline continued

• 1879 Pope Leo XIII issues the encyclical Aetemi
  Patris which instructed the Catholic church to
  study Scholastic philosophy again.
• 1879-1884 Cantor publishes a six part treatise
  on set theory declaring that infinite collections
  of numbers can be manipulated just as finite
  sets. Leopold Kronecker adamantly disagrees with
  Cantors findings because he believes that a
  mathematical object does not exist unless it can
  be constructed in a finite number of steps.
• 1919 Bertrand Russell publishes the Barber in a
  certain village example of a paradox found in
  Cantors theory of sets.
• 2006 Today Cantors set theory is widely
  accepted because mathematicians have resolved
  that valid mathematics do not revolve around the
  truth of philosophical issues.

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References

• Anderson, A (1995, Oct 3). Metaphysics Multiple
  Meanings. Retrieved December 3, 2006, from
  Webstyle.com Web site http//websyte.com/alan/met
  amul.htm
• Berlinghoff, W, Gouvea, F (2002). Math Through
  the AGes.Farmington Oxton House Publishers, LLC.
• Bogomolny, A (2006). Pigeonhole Principle.
  Retrieved December 3, 2006, from Interactive
  Mathematics Miscellany and Puzzles Web site
  http//www.cut-the-knot.org/do_you_know/pigeon.sht
  ml
• O'Connor, JJ (1996, Feb). A history of set
  theory. Retrieved December 3, 2006, from School
  of Mathematics and Statistics Web site
  http//www-groups.dcs.st-and.ac.uk/history/HistTo
  pics/Beginnings_of_set_theory.html
• Dangello, F, Seyfried, M (2000). Introductory
  Real Analysis.Boston Houghton Mifflin Company.