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Beyond Counting Infinity and the Theory of Sets Nate Jones & Chelsea Landis

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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

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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

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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

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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. Cantor’s 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.

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Cantor’s 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

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

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Leopold Kronecker Prominent professor at the University of Berlin Disagreed with Cantor’s 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 can’t say what elements belong to the set, how can we describe the set as a completed whole?

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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?

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… If he’s in the set, he doesn’t shave himself, but since he shaves all who don’t shave themselves, that must mean he must shave himself, so he ISN’T in the set. If he isn’t in the set, then he doesn’t shave himself, but he only shaves those who don’t shave themselves, so he must not shave himself, so he IS in the set.

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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.

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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.

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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 didn’t 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

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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.

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Most Important effect of set theory in Philosophy Cantor’s investigations led to clarifications of logical forms, methods of proof, and errors of syntax. These were used to refine arguments in philosophy.

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Georg Cantor’s Ideas This shows the two lists are the same size, infinite. 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.

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Cantor’s Idea’s 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

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

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Conclusion Cantor’s work has affected mathematics in a positive way. His basic set theory has provided a simple, unifying approach to many different areas of mathematics.

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Timeline 1774-1784 - Cantor’s work on set theory 1879 - Pope Leo XIII wrote Aeterni Patris 1884 – Kronecker’s ideas came about Around 1919 - Paradoxes in Set Theory Early 20 th century - investigation of metaphysics and contradictions to set theory

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References Berlinghoff, William P, Gouvea, Fernando Q. Math through the Ages A gentle History for Teachers and Others. 1 st edition. Farmington, Maine. Oxton House Publishers, 2002. Counting to Infinity. 11/27/06. Platonic Realms Minitexts. You can’t get there from here. 11/27/06.

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