1. Infinity and the Theory of Sets
Chapter 25
By Stephanie Lawrence
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2. 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
Kroenecker’s Opposition
Bertrand Russell publishes an example of a paradox found in Cantor’s theory
Let’s get Metaphysical
Today Cantor’s theory is widely accepted
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3. Zeno of Elea
Zeno’s 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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4. 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)
At this time many mathematicians believed that infinite sets could not exist, but Bolzano defended this concept by giving examples of elements of an infinite set could be put in 1-1 correspondence with elements of one of its proper subsets.
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5. 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
Neo-Thomism is the thought that religions and science are compatible. An approach to science that avoided conflicts with religion.
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6. 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 Crelle’s 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.
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
an xn + an-1 xn-1 + an-2 xn a1 x + 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
|an| + |an-1| + |an-2| |a1| + |a0| + n.
There is only one equation of index 2, namely x = 0. There are 3 equations of index 3, namely
2x = 0, x + 1 = 0, x - 1 = 0 and x2 = 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.
In the same paper Cantor shows that the real numbers cannot be put into one-one correspondence with the natural numbers using an argument with nested intervals which is more complex than that used today (which is in fact due to Cantor in a later paper of 1891). Cantor now remarks that this proves a theorem due to Liouville, namely that there are infinitely many transcendental (i.e. not algebraic) numbers in each interval.
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7. Cantor’s 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.
Due to the controversy of Cantor’s work, Alfred and Carl were brave to publish his findings
Another way to look at the pigeonhole theory is with a chess board example
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8. Pigeonhole Principle
The Pigeonhole Principle: Suppose we place m pigeons in n pigeonholes, where m and n are positive integers. If m > 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)
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?
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.
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9. Chessboard Example
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10. 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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11. Kroenecker’s 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, Cantor’s theories were incomprehensible to Kronecker.
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12. 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)
Case 1:
The barber is a member of the set of all villagers who do not shave themselves
Case 2:
The barber is not a member of the set of all villagers who do not shave themselves
Which one is it?
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13. Let’s 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 Cantor’s theories are accepted among religious leaders.
As a result, the study of mathematics is removed from the realm of metaphysics.
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14. 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, Cantor’s theory of sets is widely accepted today.
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15. Timeline
450 BC Zeno of Ela, a Greek mathematician/philosopher whose questions on infinity presented paradoxes which challenged mathematicians’ view of the real world for many centuries.
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”
The birth of Set Theory takes place when George Cantor publishes his first article in Crelle’s 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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16. Timeline continued
Pope Leo XIII issues the encyclical Aetemi Patris which instructed the Catholic church to study Scholastic philosophy again.
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 Cantor’s findings because he believes that a mathematical object does not exist unless it can be constructed in a finite number of steps.
Bertrand Russell publishes the Barber in a certain village example of a paradox found in Cantor’s theory of sets.
Today Cantor’s 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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17. References
Anderson, A (1995, Oct 3). Metaphysics: Multiple Meanings. Retrieved December 3, 2006, from Webstyle.com Web site:
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:
O'Connor, JJ (1996, Feb). A history of set theory. Retrieved December 3, 2006, from School of Mathematics and Statistics Web site:
Dangello, F, & Seyfried, M (2000). Introductory Real Analysis.Boston: Houghton Mifflin Company.
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