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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
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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 0 (aleph zero). A set with this cardinality is
called countable.
The cardinality of R is called c.
Cantor proved that
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Georg Cantor
(1845-1918)
This is a version of the continuum hypothesis.
Cantor’s 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 wellordered 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 a<b, a=b or b<a 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.
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Georg Cantor
(1845-1918)
Founder of modern set theory.
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.
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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
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Cantor:
Contributions to the Founding of the Theory of Transfinite Numbers
§1 The Conception of Power or Cardinal Number
Cantor defines a set
(7) M~N => M  N
(“aggregate”) as a collection
into a whole M of definite and
(8) M  N => M~N
separate objects of our
(9) M~ M
intuition or of our thought.
– How is this to be
Notation: for union of M and N:
understood?
(M,N)
– This shows the limitations
Modern notation MUN
of the intuitive concept of
Notation for the Cardinal or
sets and cardinals.
Power: M
• Intuitively a 1-1
The double bar stands for the
correspondence allows one to
abstraction of the nature and
interchange elements of the
the order of the elements. This
two sets.
is a definition by abstraction.
• Cantor thinks of M as a
NB. Today a set has no
special representative of the
prescribed order, modern
equivalence class which
notation for the cardinal: |M|.
consists of “units” that is
elements without any particular
M~N means that there is a 1-1
special properties.
correspondence between M
and N
• NB. There is no mention of
elements
~ is an equivalence relation
Cantor: Contributions…
§2 “Greater” and “Less” with powers
• Fix two sets M and N with
cardinals a for M and b for N
If both the conditions
a) There is no subset of M
which is equivalent to N.
b) There is a subset N1 of N
such that N1~M.
Then a < b .
• Why do we need both
conditions?
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a < b is transitive:
a < b , b < c => a < c
a < b, a > b , a = b are
mutually exclusive. So < is a
partial order.
• Cantor claims without proof:
also < is an order. That is the
trichotomy principle holds
which means that for any two
cardinals a ,b one of the
relations a < b, a > b , a =
b holds
• A proof of this statement relies
on the axiom of choice, to
which it is in fact equivalent!
Cantor: Contributions…
§3 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
1. First if M~M’ and N~N’ then
(M,N)~(M’,N’)
Thus one can define
a+b:= ( M , N )
Then since forming the union
of sets is commutative and
associative
– a+b=b+a
– a+(b+c)=a+(b+c)
2. Notation for the Cartesian
product which Cantor calls
bindings
( M . N ) : {( m, n ) | m  M , n  N }
The modern notation is MXN.
If M~M’ and N~N’ then
(M.N)~(M’.N’)
Thus one can define
a.b:= ( M .N )
Again forming the Cartesian
product is associative and
commutative on sets and
moreover distributive with respect
to the union, thus
– a.b=b.a
– a.(b.c)=a.(b.c)
– a.(b+c)=a.b+a.c
Cantor: Contributions…
§3 The Exponentiation of Powers
• 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
(N|M)
• 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 M~M’ and N~N’ then
(M|N)~(M’|N’)
Let a be the cardinality of M and b
be the cardinality of N
ab:= ( N | M )
1.
2.
3.
Now MNXMP=MNUP 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)P=MNXP 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.
1. ab.ac=ab+c, 2. ac.bc=(a.b)c, 3. (ab)c=ab.c
Cantor: Contributions…
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Examples:
Since A map f from {1,2} to R is given by is values
f(1) R and f(2)  R
R2:={maps from {1,2} to R}= {(x,y)|x,y  R }= RxR
• Likewise for any set M: M2=MxM, M3=MxMxM 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
{1,2}XR={(1,x)|x R}U{(2,y)|y R} corresponds to a
tuple of functions r(t)=(x(t),y(t)), which is how curves in
R2 are given.
Cantor: Contributions…
§3 The Exponentiation of Powers
Let 0 be the cardinality of N and
c be the cardinality of the
continuum X=[0,1]
(11)
c = 20
Use the binary expansion
x=f(1)/2+f(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
(2n+1)/2m <1 and they are
enumerable!
• From this and the power laws it
follows that the cardinality of
the plane R2 an in fact any ndimensional product of reals
Rn and even a countable
infinite product of real lines has
the same cardinality as R.
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20 20  20 0  20
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cn=c
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c0  (20 )0  20 .0
Use: For any transfinite cardinal a: a+0=a
Cantor: Contributions…
§6 The Smallest Transfinite Cardinal Number 0
• 0 is indeed the smallest
transfinite number.
• For any finite n: 0 > n
• For any other transfinite cardinal
0 <a
a:
For the first statement use the
definition of “<“.
For the second statement use
A. Every transfinite aggregate T has
parts with the cardinal number 0
B. If S is a transfinite aggregate with
the cardinal number 0 and S1 is
any transfinite part of S then
S1  0
• Also 0  1  0 and thus
0  n  0 also 0  0  0
(Hilbert’s Hotel at infinity)
• Moreover 0 .0  0
• For the latter statement
enumerate the elements of
(N,N) in the matrix form
(1,1) (1,2)
(1,3)
(2,1) (2,2)

(3,1)
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(2, n  1) 
 (3, n  2)
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(1, n) 
i.e. (1,1), (1,2), (2,1), (1,3), (2,2),
(2,1), (1,4), …, (1,n), (2,n-1),
(3,n-1),…
Cantor: Contributions…
• For any transfinite
cardinal a: a+0=a.
Choose M s.t. |M|=a. Now
M has a subset M1 which
has cardinality 0 (pick
out elements one at a
time.
M=M\M1UM1
So |M|=|M\M1|+0 and
a+0=|M\M1| +0+0
=|M\M1| +0=|M|=a
• We also get
|Z|= 0+0+1= 0
• And |Q|=0
|Q|=|Q>0|+|Q<0|+1=2|Q>0|+1
and since Q>0 is transfinite:
|Q>0|= |Q>0|+0=|NXN|=00= 0
we get
|Q|= 0 + 0 +1= 0
But: |R|=c=2o and actually c>0
as Cantor showed.
Cantor from: On an Elementary Question in
the Theory of Sets
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To show that c= 2 0  0
Cantor gives his famous “diagonal
argument”.
• Consider any enumerable subset

(En) of 2 0 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.
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• Now consider the sequence E0
0 if ann  1
bn : 
1 if ann  0
then the sequence E0 is not among
the En.
Note:
– this works in any base
– this also works for any cardinal
a: 2a>a.
• Thus one obtains an infinite
sequence of cardinals each strictly
greater than the previous .
If |M|=a then |P(M)|=|power set of M|=|set of all subsets|=2a
Summary: Sets and Cardinals
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There is the basic relation of
inclusion for sets N  M
Let a be the cardinal of N and
b be the cardinal of M then
although
Nitmight
M
happen that a = b or a < 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
a) There is no subset of M
which is equivalent to N.
b) There is a subset N1 of N
such that N1~M.
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There are three basic operations
for sets:
1. M U N
2. M X N
3. 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
1. The cardinal of MUN is a+b
2. The cardinal of M X N is ab
3. The cardinal of MN is ab
The standard laws e.g. ab+c=abac
hold as if the cardinals where
ordinary numbers!