To Infinity and Beyond

1
To Infinity and Beyond

• The Mathematics and Philosophy of Infinity-Jeff
  Groah

2
The Infinite Improbability Drive

• Douglas Adams idea for interstellar travel on
  the Heart of Gold
• Based on quantum ideas

3
Johannes Kepler

• A parabola is an ellipse with a focus at infinity

4
Johannes Kepler

• It makes more sense to say that
• A parabola is the limit of a family of ellipses
  having a focus that tends to infinity
• We will make sense of Keplers statement by
  defining an infinity

5
Alexandroff 1-Point Compactification of the Plane
via Stereographic Projection

6
Alexandroff 1-Point Compactification of the Plane
via Stereographic Projection

• The farther out the point p is in the plane, the
  higher the point s is on the sphere
• As you send p to infinity in any direction, the
  point s tends to the north pole of the sphere
• The north pole is the point at infinity 8

7
Conics and Compactification

8
Conics to Infinity and Beyond

9
Stereographic Projection

• Known to Ptolemy (Second Century a.d.)
• Riemann used in the complex plane yielding the
  Riemann Sphere
• Riemanns use begs the question about the
  arithmetic of infinity
• Alexandrov discovered the topology of the
  one-point compactification

10
How We Count

• We line things up or make pairings

11
How We Count

• Number sense comes from the ability to make
  pairings
• These pairings are called bijections
• The ability to make bijections precedes the
  notion of number
• We can make bijections between infinite sets

12
Finite Sets

• A set S is finite if you can make a bijection
  between it and some set of the form 1,2,3,,n
• If a set S is finite, the number n, as above, is
  the size, or cardinality, of the set

13
Infinite Sets

• A set is infinite if it is not finite
• duh

14
Hilberts Hotel

• David Hilbert (1862-1943) had a very large hotel
• Rooms were arranged in a line from 1 Telegraph
  Ave. to forever
• All of the rooms were full
• Someone arrived wanting a room
• Hilbert asked each occupant to move one room
  down, making the first room available

15
Hilberts Hotel
No Vacancy, But we can fit you in.
16
Hilberts Hotel-Consequences

• Infinity plus 1 is infinity
• The Natural numbers 1, 2, and the Whole
  numbers 0, 1, 2, have the same number of
  elements

0
?
1
1
?
2
2
?
3

17
Hilberts Hotel-Consequences

• The natural numbers and the set of integers have
  the same number of elements

1
2
3
4
5
-1
0
1
2

-2

18
Hilberts Hotel-Consequences

• The natural numbers and the set of rational
  numbers have the same number of elements
• The rational numbers p/q can be equated to points
  (p,q) with integer coordinates
• The points in the plane with integer coordinates
  can be counted.

19
Are all Infinite Sets Equal?

• The real numbers are not the same size as the
  natural numbers

20
Countable Infinity

• A set the same size as the natural numbers is
  countably infinite
• ?0- The countable infinity
• If a set is either finite or countably infinite,
  then it is called countable
• The number of stars in the universe is at most
  countable
• The union of a countable number of countable sets
  is countable

21
The Stars in the Universe

22
Cantors Theorem

• If S is a set, its power set, P(S), is the set of
  all subsets of S
• P(1,2)Ø,1,2,1,2
• card(1,2)2
• card(P(1,2))4

23
Cantors Theorem

• The power set of a finite set is larger than the
  original set
• Cantors Theorem card(P(S))gt card(S)
• Cantors Theorem is valid even if the set S is
  infinite

24
Consequences of Cantors Thm

• There are an infinite number of infinities of
  larger and larger sizes
• Note The proof of Cantors Theorem is very
  simple and uses no high-powered techniques

25
Cantors Controversy

• Potential infinities was in vogue during the
  19th century until Cantor
• Cantor defined actual infinities
• Kronecker-God invented the natural numbers. Man
  invented the rest
• Brouwer rejected the actual infinities and
  started his own school of mathematical thought
• Hilbert thought Cantors invention was Paradise

26
Cantors Controversy

• Almost no one is trained in Brouwers school of
  thought
• Most mathematicians never even hear of him
• All mathematicians are trained to follow Cantor

27
The Continuum Hypothesis

• There are no infinities between the size of the
  natural numbers and the size of the real numbers
• This is an independent axiom of mathematics
• You get a different mathematics from the standard
  one if you assume this is not true
• Only marginally different

28
Ockhams Razor

• You should not multiply entities beyond necessity
• Any theory that is not testable is nonsense
• Math contains nonsense
• The nonsense in math results from the Law of the
  Excluded Middle (LEM)
• Rejecting LEM yields a very different mathematics

29
Rejecting LEM-Consequences

• Lose the ability to distinguish between
  infinities
• Lose the ability to distinguish between rational
  and irrational numbers
• Lose the ability to distinguish between open and
  closed sets
• There goes the whole specialization called
  Topology

30
Rejecting LEM-Consequences

• Many theorems remain true, but are harder to
  prove

31
Objections to Strict Constructionists /
Intuitionists

• It does not matter if math is true as long as
  math can be used to model everything in the
  universe
• The alternatives to standard math have not
  yielded anything significant that is new
• Its harder to prove standard theorems

32
Objections to Standard Math

• It is philosophical laziness to leave unaddressed
  the objections to standard math
• Alternatives to standard math should not be based
  on a logic, since there is no empirical basis for
  truth
• I.e., fuzzy logic and quantum logic do not
  provide the answer

33
Notation for Infinity

• 8
• ?0- The countable infinity
• ?1- The first uncountable infinity
• ?2 - The second uncountable infinity
• Etc.

34
Math Is Based On Nonsense

• Without math we cannot make any sense of the
  world around us