Georg Cantor (1845-1918)

1
Georg Cantor (1845-1918)

• Founder of modern set theory.
• Introduced the concept of cardinals.
• Two sets have the same cardinality if they are in
  1-1 correspondence.
• The cardinality of N is called (aleph
  zero). A set with this cardinality is called
  countable.
• The cardinality of R is called c.
• Cantor proved that
• Q is countable
• R is uncountable
• The algebraic numbers are countable
• The cardinality of Rn is equal to that of R
• He thought that there are only two types of
  infinite subsets of R
• Those that are countable, like the natural
  numbers
• Those that have the cardinality of R, like an
  interval
• This is a version of the continuum hypothesis.

2
Cantors Cardinals and Ordinals

• Abstracting from the particular nature and order
  of the elements of a set, we can consider two
  sets to be equivalent if there is a 1-1
  correspondence between them. Cantor defines this
  abstraction to be a cardinal.
• Question what is the relation of the cardinality
  of the real numbers and the natural numbers?
• Abstracting from the particular nature of the
  elements of a well-ordered set, we can consider
  two well-ordered sets to be equivalent if there
  is a 1-1 correspondence preserving the order
  between them. Cantor defines this abstraction to
  be an ordinal.
• He thinks of cardinals and ordinals as numbers
  and defines the usual arithmetic operations ,x,
  for them. He also believes that like numbers one
  can always compare two ordinals or two cardinals
  a and b in such a way that one of the following
  altb, ab or blta holds. This is called the
  trichotomy principle.
• This is true for ordinals, but for cardinals it
  turned out to be equivalent to a new axiom.

3
Georg Cantor (1845-1918)
Founder of modern set theory.

• Mittag-Leffler was first a supporter and then
  thought it would not be a good idea to publish
  his papers.
• Turned to philosophy, theology and history in
  1885, but back to mathematics in 1895.
• His last major papers on set theory which are
  surveys of transfinite arithmetic including the
  definitions of ordinals and cardinals appeared in
  1895 and 1897, in Mathematische Annalen
• Acknowledged at the 1897 congress by Hadamar and
  Hurwitz.
• Acknowledged by Hilbert at the 1900 congress.
• His theory was attacked by König at the
  Heidelberg congress 1904
• Depressions first appeared 1884 and became worse
  later in life.
• Had avid interest in theology and the
  Shakespeare/Bacon controversy

• Started on the problem of the uniqueness of
  trigonometric expansions (1870-1872).
• Defined real numbers as limits of rationals
  (1872)
• Showed that rational and algebraic numbers are
  countable (1873)
• Showed (1874) that there is a 1-1 correspondence
  between R, R2.
• This also holds Rn (1877) and even a countably
  infinite product of factors R.
• Formulated the continuum hypothesis (1878)
• Between 1878 and 1884 Cantor published a series
  of six papers in Mathematische Annalen designed
  to provide a basic introduction to set theory
• Founded the Deutsche Mathematiker Vereinigung
  (1890)
• His work met the skepticism of Kronecker.

4
Cantor Contributions to the Founding of the
Theory of Transfinite Numbers 1 The Conception
of Power or Cardinal Number

• Cantor defines a set (aggregate) as a
  collection into a whole M of definite and
  separate objects of our intuition or of our
  thought.
• Notation for union of M and N (M,N)
• Modern notation MUN
• Notation for the Cardinal or Power
• The double bar stands for the abstraction of the
  nature and the order of the elements. This is a
  definition by abstraction.
• NB. Today a set has no prescribed order, modern
  notation for the cardinal M.
• MN means that there is a 1-1 correspondence
  between M and N
• is an equivalence relation

• (7) MN gt
• (8) gt MN
• (9) M
• How is this to be understood?
• This shows the limitations of the intuitive
  concept of sets and cardinals.
• Intuitively a 1-1 correspondence allows one to
  interchange elements of the two sets.
• Cantor thinks of as a special
  representative of the equivalence class which
  consists of units that is elements without any
  particular special properties.
• NB. There is no mention of elements

5
Cantor Contributions2 Greater and Less
with powers

• Fix two sets M and N with cardinals a for M and
  b for N
• If both the conditions
• There is no subset of M which is equivalent to N.
• There is a subset N1 of N such that N1M.
• Then a lt b .
• Why do we need both conditions?

• a lt b is transitive
• a lt b , b lt c gt a lt c
• a lt b, a gt b , a b are mutually exclusive.
  So lt is a partial order.
• Cantor claims without proof also lt is an order.
  That is the trichotomy principle holds which
  means that for any two cardinals a ,b one of the
  relations a lt b, a gt b , a b holds
• A proof of this statement relies on the axiom of
  choice, to which it is in fact equivalent!

6
Cantor Contributions3 The addition and
Multiplication of Powers

• Fix two sets M and N. Denote their union by
  (M,N). Cantor puts the condition that M and N
  have no common elements.
• The modern notation is M U N
• First if MM and NN then (M,N)(M,N)
• Thus one can define
• ab
• Then since forming the union of sets is
  commutative and associative
• abba
• a(bc)a(bc)
• Notation for the Cartesian product which Cantor
  calls bindings

• The modern notation is MXN.
• If MM and NN then (M.N)(M.N)
• Thus one can define
• a.b
• Again forming the Cartesian product is
  associative and commutative on sets and moreover
  distributive with respect to the union, thus
• a.bb.a
• a.(b.c)a.(b.c)
• a.(bc)a.ba.c

7
Cantor Contributions3 The Exponentiation of
Powers

• Let a be the cardinality of M and b be the
  cardinality of N
• ab
• Now MNXMPMNUP since a map from NUP to M
  determines a pair of maps from N to M and from P
  to M and vice versa.
• Also MPXNP(MXN)P, since giving a map from P to
  MXN is equivalent to giving a pair of maps from P
  to M and from P to N.
• Lastly (MN)PMNXP since giving a map from NXP to
  M yields for each element p of P a map from N to
  M and vice versa given a map from P to MN we get
  an element of m for each pair of elements from P
  and N.

• Fix two sets M and N. Cantor denotes the space of
  functions from N to M, which he calls covering
  of N with M by
• (NM)
• The modern notation is MN which denotes the set
  of all functions from N to M Example R2
• Recall a function from N to M is a rule that
  associates to each element n in N an element m in
  M.
• If MM and NN then (MN)(MN)

1. ab.acabc, 2. ac.bc(a.b)c, 3. (ab)cab.c
8
Cantor Contributions

• Examples
• Since A map f from 1,2 to R is given by is
  values f(1)? R and f(2) ? R
• R2maps from 1,2 to R (x,y)x,y ? R RxR
• Likewise for any set M M2MxM, M3MxMxM etc.
• The set of curves in R2 is given by the maps from
  R to R2. So it is (R2)R.
• By the power laws
• (R2)R (R2XR)(x(t),y(t))x(t),y(t) functions
  from R to R
• Ww also know this since a function r from
  1,2XR(1,x)x ?RU(2,y)y ?R corresponds
  to a tuple of functions r(t)(x(t),y(t)), which
  is how curves in R2 are given.

9
Cantor Contributions3 The Exponentiation of
Powers

• Let be the cardinality of N and c be the
  cardinality of the continuum X0,1
• (11) c
• Use the binary expansion
• xf(1)/2f(2)/4 f(n)/2n..
• Caution! There are numbers with more that one
  binary expansion e.g. 1.000 0.111 1
• 0.100... 0.011 0.1
• 0.10100...0.10011..
• These numbers are the numbers (2n1)/2m lt1 and
  they are enumerable!

• From this and the power laws it follows that the
  cardinality of the plane R2 an in fact any
  n-dimensional product of reals Rn and even a
  countable infinite product of real lines has the
  same cardinality as R.
• cnc
• c

Use For any transfinite cardinal a a?0a
10
Cantor Contributions6 The Smallest
Transfinite Cardinal Number

• is indeed the smallest transfinite number.
• For any finite n gt n
• For any other transfinite cardinal a lta
• For the first statement use the definition of
  lt.
• For the second statement use
• Every transfinite aggregate T has parts with the
  cardinal number
• If S is a transfinite aggregate with the cardinal
  number and S1 is any transfinite part of S
  then

• Also and thus
• also
• (Hilberts Hotel at infinity)
• Moreover
• For the latter statement enumerate the elements
  of (N,N) in the matrix form
• i.e. (1,1), (1,2), (2,1), (1,3), (2,2),
  (2,1), (1,4), , (1,n), (2,n-1), (3,n-1),

11
Cantor Contributions

• For any transfinite cardinal a a?0a.
• Choose M s.t. Ma. Now M has a subset M1 which
  has cardinality ?0 (pick out elements one at a
  time.
• MM\M1UM1
• So MM\M1?0 and
• a?0M\M1 ?0?0 M\M1 ?0Ma

• We also get
• Z ?0?01 ?0
• And Q?0 QQgt0Qlt012Qgt01
• and since Qgt0 is transfinite
• Qgt0 Qgt0?0NXN?0?0 ?0
• we get
• Q ?0 ?0 1 ?0
• But Rc2?o and actually cgt?0 as Cantor
  showed.

12
Cantor from On an Elementary Question in the
Theory of Sets

• To show that c
• Cantor gives his famous diagonal argument.
• Consider any enumerable subset (En) of
  then there is at least one sequence which is not
  among the En
• E1(a11,a12,,a1n,)
• E2(a21,a22,,a2n,)
• Em(am1,am2,,amn,)
• Where aij is either 0 or 1.

• Now consider the sequence E0
• then the sequence E0 is not among the En.
• Note
• this works in any base
• this also works for any cardinal a 2agta.
• Thus one obtains an infinite sequence of
  cardinals each strictly greater than the previous
  .

If Ma then P(M)power set of Mset of all
subsets2a
13
Summary Sets and Cardinals

• There is the basic relation of inclusion for sets
• Let a be the cardinal of N and b be the cardinal
  of M then although it might happen
  that a b or a lt b
• In order to insure that we must also have that
  there is no subset of N which is in1-1
  correspondence with that is
• There is no subset of M which is equivalent to N.
• There is a subset N1 of N such that N1M.

• There are three basic operations for sets
• M U N
• M X N
• MN the space of maps of N into M
• These relations lead to addition, multiplication
  and exponentiation of cardinals.
• If the cardinal of M is a and the cardinal of N
  is b then
• The cardinal of MUN is ab
• The cardinal of M X N is ab
• The cardinal of MN is ab
• The standard laws e.g. abcabac hold as if the
  cardinals where ordinary numbers!