Paradoxes of the Infinite Kline XXV

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Paradoxes of the InfiniteKline XXV

• Pre-May Seminar
• March 14, 2011

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Galileo Galilei (1564-1642)
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Galileo Dialogue on Two New Sciences, 1638

• Simplicio Here a difficulty presents itself
  which appears to me insoluble. Since it is clear
  that we may have one line segment longer than
  another, each containing an infinite number of
  points, we are forced to admit that, within one
  and the same class, we may have something greater
  than infinity, because the infinity of points in
  the long line segment is greater than the
  infinity of points in the short line segment.
  This assigning to an infinite quantity a value
  greater than infinity is quite beyond my
  comprehension.

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Galileos Dialogo

• Salviati This is one of the difficulties which
  arise when we attempt, with our finite minds, to
  discuss the infinite, assigning to it those
  properties which we give to the finite and
  limited but this I think is wrong, for we cannot
  speak of infinite quantities as being the one
  greater or less than or equal to another. To
  prove this I have in mind an argument, which, for
  the sake of clearness, I shall put in the form of
  questions to Simplicio who raised this difficulty.

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Galileos Dialogo

• Salviati If I should ask further how many
  squares there are, one might reply truly that
  there are as many as the corresponding number of
  roots, since every square has its own root and
  every root its own square, while no square has
  more than one root and no root more than one
  square.
• Simplicio Precisely so.

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Galileos Dialogo

• Salviati But if I inquire how many roots there
  are, it cannot be denied that there are as many
  as there are numbers because every number is a
  root of some square. This being granted we must
  say that there are as many squares as there are
  numbers because they are just as numerous as
  their roots, and all the numbers are roots. Yet
  at the outset we said there are many more numbers
  than squares, since the larger portion of them
  are not squares.

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Galileos Dialogo

• Sagredo What then must one conclude under these
  circumstances?
• Salviati So far as I see we can only infer that
  the totality of all numbers is infinite, that the
  number of squares is infinite, and that the
  number of their roots is infinite neither is the
  number of squares less than the totality of all
  numbers, nor the latter greater than the former
  and finally the attributes "equal," "greater,"
  and "less" are not applicable to infinite, but
  only to finite, quantities.

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Bernard Bolzano (1781-1848)
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Bernard Bolzano (1781-1848)

• Czech Priest

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Bernard Bolzano (1781-1848)

• Czech Priest
• 0,10,2

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Cardinality
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Cardinality

• The number of elements in a set is the
  cardinality of the set.

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Cardinality

• The number of elements in a set is the
  cardinality of the set.
• Card(S)S

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Cardinality

• The number of elements in a set is the
  cardinality of the set.
• Card(S)S
• 1,2,3a,b,c

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Cardinality

• The number of elements in a set is the
  cardinality of the set.
• Card(S)S
• 1,2,3a,b,c
• c0,1

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Cardinality

• The number of elements in a set is the
  cardinality of the set.
• Card(S)S
• 1,2,3a,b,c
• c0,1
• Lemma ca,b for any real altb.

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Cardinality

• The number of elements in a set is the
  cardinality of the set.
• Card(S)S
• 1,2,3a,b,c
• c0,1.
• Lemma ca,b for any real altb.
• Lemma Realsc.

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Richard Dedekind (1831-1916)
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Richard Dedekind (1831-1916)

• Definition of infinite sets

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Georg Cantor (1845-1918)
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?0

• 1, 2, 3, ?0

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?0

• 1, 2, 3, ?0
• 12, 22, 32, ?0

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?0

• 1, 2, 3, ?0
• 12, 22, 32, ?0
• rationals in (0,1) ?0

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?0

• 1, 2, 3, ?0
• 12, 22, 32, ?0
• rationals in (0,1) ?0
• rationals ?0

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?0

• 1, 2, 3, ?0
• 12, 22, 32, ?0
• rationals in (0,1) ?0
• rationals ?0
• algebraic numbers ?0

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?0

• 1, 2, 3, ?0
• 12, 22, 32, ?0
• rationals in (0,1) ?0
• rationals ?0
• algebraic numbers ?0
• Arithmetic ?0 ?0

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?0

• 1, 2, 3, ?0
• 12, 22, 32, ?0
• rationals in (0,1) ?0
• rationals ?0
• algebraic numbers ?0
• Arithmetic ?0 ?0
• Cardinality and Dimensionality

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Cantors Diagonal Argument
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Cantors Diagonal Argument

• (0,1)c

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Cantors Diagonal Argument

• (0,1)c
• c gt ?0

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Attacks
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Attacks

• Leopold Kronecker

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Attacks

• Leopold Kronecker
• Henri Poincare

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Attacks

• Leopold Kronecker
• Henri Poincare

Support
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Attacks

• Leopold Kronecker
• Henri Poincare

Support
David Hilbert
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Georg Cantor

• My theory stands as firm as a rock every arrow
  directed against it will return quickly to its
  archer. How do I know this? Because I have
  studied it from all sides for many years because
  I have examined all objections which have ever
  been made against the infinite numbers and above
  all because I have followed its roots, so to
  speak, to the first infallible cause of all
  created things.

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Felix Hausdorff

• Set theory is a field in which nothing is
  self-evident, whose true statements are often
  paradoxical, and whose plausible ones are false.
  
• Foundations of Set Theory (1914)

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Math May Seminar Interlaken
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Math May Seminar Interlaken
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Math May Seminar Interlaken
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Math May Seminar Interlaken
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Fun with ?0
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Fun with ?0

• Hilberts Hotel

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Fun with ?0

• Hilberts Hotel
• Bottles of Beer

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The Power Set of S
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The Power Set of S

• S1

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The Power Set of S

• S1
• S1, 2

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The Power Set of S

• S1
• S1, 2
• S1, 2, 3

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The Power Set of S

• S1
• S1, 2
• S1, 2, 3
• S2S

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The Power Set of S

• c2 ?0

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Axiom of Choice

• If p is any collection of nonempty sets A,B,,
  then there exists a set Z consisting of precisely
  one element each from A, from B, and so on for
  all sets in p.

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Continuum Hypothesis

• 1877 Cantor There is no set whose cardinality
  is strictly between that of the integers and that
  of the real numbers.
• 1900 Hilberts 1st problem
• 1908 Ernst Zermelo axiomatic set theory
• 1922 Abraham Fraenkel
• 1940 Kurt Godel
• 1963 Paul Cohen