Georg Cantor 1845 1918 Founder of modern set

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.

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.

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, R 2. 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

• • • 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 => (“aggregate”) as a collection into a whole M of definite and (8) => M~N separate objects of our (9) 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: • Intuitively a 1 -1 The double bar stands for the correspondence allows one to abstraction of the nature and interchange elements of the order of the elements. This two sets. is a definition by abstraction. • Cantor thinks of 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 N 1 of N such that N 1~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!

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: = 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 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

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 1. (N|M) • The modern notation is MN which denotes the set of all 2. functions from N to M Example: R 2 • Recall a function from N to M is a rule that associates to each 3. 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

Cantor: Contributions… • • Examples: Since A map f from {1, 2} to R is given by is values f(1) R and f(2) R R 2: ={maps from {1, 2} to R}= {(x, y)|x, y R }= Rx. R • Likewise for any set M: M 2=Mx. M, M 3=Mx. M etc. • The set of curves in R 2 is given by the maps from R to R 2. So it is (R 2)R. By the power laws (R 2)R= (R 2 XR)={(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 R 2 are given.

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)/2 n+. . Caution! There are numbers with more that one binary expansion e. g. 1. 000… = 0. 111… =1 0. 100. . . = 0. 011… =0. 10100. . . =0. 10011. . These numbers are the numbers (2 n+1)/2 m <1 and they are enumerable! • From this and the power laws it follows that the cardinality of the plane R 2 an in fact any ndimensional 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

Cantor: Contributions… § 6 The Smallest Transfinite Cardinal Number • is indeed the smallest • Also and thus transfinite number. also • For any finite n: > n (Hilbert’s Hotel at infinity) • For any other transfinite cardinal a: • Moreover <a • For the latter statement For the first statement use the enumerate the elements of definition of “<“. (N, N) in the matrix form For the second statement use A. Every transfinite aggregate T has parts with the cardinal number B. If S is a transfinite aggregate with the cardinal number and S 1 is any transfinite part of S then 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 M 1 which has cardinality 0 (pick out elements one at a time. M=MM 1 UM 1 So |M|=|MM 1|+ 0 and a+ 0=|MM 1| + 0+ 0 =|MM 1| + 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 +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 • 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: • E 1=(a 11, a 12, …, a 1 n, …) E 2=(a 21, a 22, …, a 2 n, …) … Em=(am 1, am 2, …, amn, …) … Where aij is either 0 or 1. • Now consider the sequence E 0 then the sequence E 0 is not among the En. Note: – this works in any base – this also works for any cardinal a: 2 a>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|=2 a

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 in 1 -1 correspondence with that is a) There is no subset of M which is equivalent to N. b) There is a subset N 1 of N such that N 1~M. • • 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!