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Feb 22, 2016
Non-Euclidean geometry and consistency
Euclidean GeometryRemember we said that a mathematical system depends on its basic assumptions its axioms.
These should be self-evident.
a + b = b + a
Euclidean GeometryAxioms of Euclidean Geometry
Euclidean GeometryIt shall be possible to draw a straight line joining any two points
Euclidean Geometry2. A finite straight line may be extended without limit in either direction.
Euclidean Geometry3. It shall be possible to draw a circle with a given centre and through a given point.
Euclidean Geometry4. All right angles are equal to one another.
Euclidean Geometry5. There is just one straight line through a given point which is parallel to a given line
Non-Euclidean geometryThe 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
Non-Euclidean geometryTwo 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)
Non-Euclidean geometryPeople expected these new axioms to throw up inconsistencies.. But they didnt!
Non-Euclidean geometryAmong the theorems that can be deduced from these new axioms areAll perpendiculars to a straight line meet at one point.Two straight lines enclose an areaThe sum of the angles of a triangle are grater than 180
Do these make sense?!
All perpendiculars to a straight line meet at one point.Two straight lines enclose an areaThe sum of the angles of a triangle are grater than 180
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 areaThe sum of the angles of a triangle are grater than 180
Non-Euclidean geometryOn 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.
Although these look curved, you can be sure the airlines are following the shortest route to save money!
Straight lines in Riemannian geometryOnce we have clarified the meaning of a straight line in Riemannian geometry, we can give a meaning to the three theorems given earlier.
All perpendiculars to a straight line meet at one point.
Lines of longitude are perpendicular to the equator but meet at the North pole
Two straight lines enclose an areaAny two lines of longitude (straight lines) meet at both the North and South poles so define an area.
The sum of the angles of a triangle are greater than 180
General relativityAccording to Einsteins general theory of relativity, the Universe obeys the rules of Riemannian geometry not that of Euclid. According to Einstein, space is curved!
ConsistencyIt would seem that it is easy to have a system of mathematics that is consistent. Not so!
Set theoryAt the heart of set theory is a contradiction
Set theoryA feeling for the contradiction can be found in the following story;
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
Set theory the barber went back to his village and only shaved all the inhabitants that did not shave themselves.
Set theory Thats easy said the barber.
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.
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.
Set theory Catalogues which list themselvesCatalogues which do not list themselves
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!
Gdels incompleteness theoryKurt Gdel (1906-1978) was able to prove that it is impossible to prove that any formal system of mathematics is without contradictions.
Mathematicians certainty is an illusion!