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. 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 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 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.

3. Georg Cantor (1845-1918) Founder of modern set theory.
Started on the problem of the uniqueness of trigonometric expansions ( ).
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.
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

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|.
M~N means that there is a 1-1 correspondence between M and N
~ is an equivalence relation
(7) M~N =>
(8) => M~N
(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: 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
There is no subset of M which is equivalent to N.
There is a subset N1 of N such that N1~M.
Then a < b .
Why do we need both conditions?
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!

6. 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
First if M~M’ and N~N’ then (M,N)~(M’,N’)
Thus one can define
a+b:=
Then since forming the union of sets is commutative and associative
a+b=b+a
a+(b+c)=a+(b+c)
Notation for the Cartesian product which Cantor calls bindings
The modern notation is MXN.
If M~M’ and N~N’ 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.b=b.a
a.(b.c)=a.(b.c)
a.(b+c)=a.b+a.c

7. 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:=
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

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
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.

9. Cantor: Contributions… §3 The Exponentiation of Powers
Let be the cardinality of N and c be the cardinality of the continuum X=[0,1]
(11) c =
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 … = 0.111… =1
= 0.011… =0.1 =
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 n-dimensional product of reals Rn and even a countable infinite product of real lines has the same cardinality as R.
cn=c c
Use: For any transfinite cardinal a: a+0=a

10. Cantor: Contributions… §6 The Smallest Transfinite Cardinal Number
is indeed the smallest transfinite number.
For any finite n: > n
For any other transfinite cardinal a: <a
For the first statement use the definition of “<“.
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
(Hilbert’s 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+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.

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: 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

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 < 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 N1~M.
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 a+b
The cardinal of M X N is ab
The cardinal of MN is ab
The standard laws e.g. ab+c=abac hold as if the cardinals where ordinary numbers!