Non-Euclidean geometry and consistency

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Non-Euclidean geometry and consistency
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Euclidean Geometry

• Remember we said that a mathematical system
  depends on its basic assumptions its axioms.
• These should be self-evident.
• a b b a

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Euclidean Geometry

• Axioms of Euclidean Geometry

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Euclidean Geometry

1. It shall be possible to draw a straight line
   joining any two points

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Euclidean Geometry

• 2. A finite straight line may be extended without
  limit in either direction.

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Euclidean Geometry

• 3. It shall be possible to draw a circle with a
  given centre and through a given point.

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Euclidean Geometry

• 4. All right angles are equal to one another.

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Euclidean Geometry

• 5. There is just one straight line through a
  given point which is parallel to a given line

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Non-Euclidean geometry

• The last axiom of Euclid is not quite as self
  evident as the others.
• In the 19th century, Georg Friedrich Bernard
  Riemann came up with the idea of replacing
  Euclids axioms with their opposites

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Non-Euclidean geometry

• Two points may determine more than one line
  (instead of axiom 1)
• All lines are finite in length but endless (i.e.
  circles!) (instead of axiom 2)
• There are no parallel lines (instead of axiom 5)

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Non-Euclidean geometry

• People expected these new axioms to throw up
  inconsistencies.. But they didnt!

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Non-Euclidean geometry

• Among the theorems that can be deduced from these
  new axioms are
• All perpendiculars to a straight line meet at one
  point.
• Two straight lines enclose an area
• The sum of the angles of a triangle are grater
  than 180

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Do these make sense?!

• All perpendiculars to a straight line meet at one
  point.
• Two straight lines enclose an area
• The sum of the angles of a triangle are grater
  than 180

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Do these make sense?!

• They do if we imagine space is like the surface
  of a sphere!
• All perpendiculars to a straight line meet at one
  point.
• Two straight lines enclose an area
• The sum of the angles of a triangle are grater
  than 180

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Non-Euclidean geometry

• On the surface of a sphere, it can be shown that
  the shortest distance between two points is
  always the arc of a circle. This means in
  Riemannian geometry, a straight line will appear
  as a curve when represented in two dimensions.

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Although these look curved, you can be sure the
airlines are following the shortest route to save
money!
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Straight lines in Riemannian geometry

• Once we have clarified the meaning of a straight
  line in Riemannian geometry, we can give a
  meaning to the three theorems given earlier.

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All perpendiculars to a straight line meet at one
point.

• Lines of longitude are perpendicular to the
  equator but meet at the North pole

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Two straight lines enclose an area

• Any two lines of longitude (straight lines) meet
  at both the North and South poles so define an
  area.

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The sum of the angles of a triangle are greater
than 180
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General relativity

• According to Einsteins general theory of
  relativity, the Universe obeys the rules of
  Riemannian geometry not that of Euclid. According
  to Einstein, space is curved!

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Consistency

• It would seem that it is easy to have a system
  of mathematics that is consistent. Not so!

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Set theory

• At the heart of set theory is a contradiction

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Set theory

• A feeling for the contradiction can be found in
  the following story

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Set theory

• A barber had an affair with a princess. The king
  was very angry and wanted the barber executed.
  The princess begged for his life and the king
  agreed, provided that

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Set theory

• the barber went back to his village and only
  shaved all the inhabitants that did not shave
  themselves.

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Set theory

• Thats easy said the barber.
• Is it?

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Set theory

• Another example is to imagine catalogues in a
  library. Some catalogues are for novels, some for
  reference, poetry etc.
• The librarian notices that some catalogues list
  themselves inside, some dont.

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Set theory

• The librarian decides to make two more
  catalogues, one which lists all te catalogues
  which do list themselves, and more interestingly,
  a catalogue which lists all the catalogues which
  do not list themselves.

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Set theory

Catalogues which list themselves
Catalogues which do not list themselves
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Set theory

• Should the catalogue which lists all the
  catalogues which do not list themselves be listed
  in itself?
• If it is listed, then by definition it should
  not be listed, and if it is not listed, it should
  be listed!

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Gödels incompleteness theory

• Kurt Gödel (1906-1978) was able to prove that it
  is impossible to prove that any formal system of
  mathematics is without contradictions.

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Mathematicians certainty is an illusion!
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Thats it!

• Thanks for your attention. Good luck next year!