Set Theory History. Martin Bunder. September 2015

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1 Set Theory History Martin Bunder September 2015

2 What is a set? Possible Definition A set is a collection of elements having a common property Abstraction Axiom If a(x) is a property ( y)( x)(x y a(x)) This says that for every property a(x) there is a set y that has all the elements that satisfy a(x). Notation: this y = {x a(x)} Comprehension Axiom (Cantor 1874, Frege (Begriffsschrift) 1879, 1903) ( z)(z {x a(x)}) a(z)

3 What is a set? Possible Definition A set is a collection of elements having a common property Abstraction Axiom If a(x) is a property ( y)( x)(x y a(x)) This says that for every property a(x) there is a set y that has all the elements that satisfy a(x). Notation: this y = {x a(x)} Comprehension Axiom (Cantor 1874, Frege (Begriffsschrift) 1879, 1903) ( z)(z {x a(x)}) a(z)

4 What is a set? Possible Definition A set is a collection of elements having a common property Abstraction Axiom If a(x) is a property ( y)( x)(x y a(x)) This says that for every property a(x) there is a set y that has all the elements that satisfy a(x). Notation: this y = {x a(x)} Comprehension Axiom (Cantor 1874, Frege (Begriffsschrift) 1879, 1903) ( z)(z {x a(x)}) a(z)

5 What is a set? Possible Definition A set is a collection of elements having a common property Abstraction Axiom If a(x) is a property ( y)( x)(x y a(x)) This says that for every property a(x) there is a set y that has all the elements that satisfy a(x). Notation: this y = {x a(x)} Comprehension Axiom (Cantor 1874, Frege (Begriffsschrift) 1879, 1903) ( z)(z {x a(x)}) a(z)

6 Examples of Sets {x x = a} = {a} {x x 2 = 1} = {1, 1} {x x = 1 x = 1} = {1, 1} {x x Z ( y)(y Z ( z)(z Z y = zx)} = {1, 1} x y {x x R 0 < x < 1} = (0, 1)

7 Extensionality Axiom A = B ( x)(x A x B) Sets are equal if they have the same elements. Assume if a(x) is a property of x: A = B (a(a) a(b))

8 Special Sets {x x x} = {x x R x 2 = 1} = {x x = x} = V {x x A x B} = A B {x x A x B} = A B {x x A} = A Empty Set Universal Set Union of A and B Intersection of A and B Complement of A

9 Subsets Definition A is a subset of B A B ( x)(x A x B) {x x A} = P(A)(the powerset of A) {x x } = { } {x x { } = {, { }} {x x {a, b, c}} = {, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}}

10 The Properties of Sets Come from Logic e.g. A B = B A Proof By the Comprehension Axiom: A B = {x x A x B} B A = {x x B x A} x A B x A x B x B x A by logic x B A So ( x)(x A B x B A). By the Extensionality Axiom A B = B A.

11 Similarly... A B = B A (A B) C = A (B C) A (B C) = (A B) C A (B C) = (A B) (A C) A (B C) = (A B) (A C) (A B) = A B (A B) = A B

12 Comparing Sets People Heads People Brains People Chairs Chickens Beaks Chickens Tails Definition Equivalent Sets are sets that have elements matched by a one to one mapping. A B ( f )f A B f is 1-1 and onto (a bijection) A is equivalent to B A B ( f )f A B f is 1-1 A B A B A B

13 Examples {+,, } {,, } {+,, } {,,, }

14 Defining Natural Numbers Note {+,, } {0, 1, 2} = 3. 0 = 1 = {0} = { } 2 = {0, 1} 3 = {0, 1, 2} etc. Definition If A n, n is the cardinal number of A, or n = #A. So, #{+,, } = 3.

15 Cardinal Arithmetic Definition Addition: If A B =, #A + #B = #(A B). e.g = #{0, 1, 2} + #{3, 4} = #{0, 1, 2, 3, 4} = 5. Definition Multiplication: #A #B = #(A B) where A B = {(a, b) a A, b B}. e.g. 3 2 = #{0, 1, 2} #{0, 1} = #({0, 1, 2} {0, 1}) = #{(0, 0), (0, 1), (1, 0), (1, 1), (2, 0), (2, 1)} = 6

16 Infinite Cardinal Numbers Let #{1, 2, 3, 4,... } = ℵ 0. If N = {1, 2, 3, 4,... }, E = {2, 4, 6, 8,... }. then if f (n) = 2n, f N E, f is 1-1 and onto. So N E and #E = ℵ 0. Also Q = {0, 1, 1, 1 2, 1 2, 2, 2, 1 3, 1 3, 2 3, 2 3, 3, 3,... }. So N Q and #Q = ℵ 0.

17 Some Arithmetic For n N: ℵ 0 + n = ℵ 0 nℵ 0 = ℵ 0 ℵ n 0 = ℵ 0 What about n ℵ 0? Is N [0, 1) R?

18 n ℵ 0 Assume N [0, 1). Then there is an f N [0, 1). f is 1-1 and onto So: f (1) =.a 11 a f (2) =.a 21 a f (3) =.a 31 a Let b =.b 1 b 2 b 3... a ii + 1 if a ii = 0 where b i = a ii 1 if a ii = 0 Then b f (n) for any n So N [0, 1) R #[0, 1) = #R = 2 ℵ 0. These must all be different (for 1-1) and use up all of [0, 1) (for onto)

19 Cantor #N = ℵ 0 is the smallest infinite cardinal number. ℵ 1 is the next smallest, then ℵ 2, etc. Continuum 2 ℵ 0 = ℵ 1 Hypothesis: Generalised Continuum Hypothesis: 2 ℵn = ℵ n+1 Georg Cantor ( ) Unresolved problem till 1964.

20 Cantor s Theorem A P(A) (Like the proof of N [0, 1)) Cantor s Paradox, 1899 So So V = {x x = x} P V P V But V P(V ) i.e. P(V ) V. Burali-Forti Paradox, 1897 Involves Ordinal Numbers.

21 Russel and Frege Bertrand Russell ( ) Gottlob Frege ( ) Russell sent his paradox to Frege when volume 2 of his Begriffsschrift was about to be published.

22 Russel s Paradox (1903) If R = {x x / x}, then by the Comprehension axiom: y {x a(x)} a(y) R R R / R A contradiction! (Gives: R R R R)

Hilbert at the 1900 International Congress of mathematics in Paris David Hilbert (1862-1943) What would be the major problems of interest in

23 Hilbert at the 1900 International Congress of mathematics in Paris David Hilbert ( ) What would be the major problems of interest in mathematics in the 20th century? Hilbert listed 23: 1 The Continuum hypothesis, is 2 ℵ 0 = ℵ 1? 2 Prove that the axioms of arithmetic are consistent.

24 The Hilbert Programme Aim: To ground all existing theories (including set theory) to a finite, complete set of axioms, and prove that these are consistent.

Gödel s Incompleteness Theorems 1931 Kurt Gödel (1906-1978) 1.

25 Gödel s Incompleteness Theorems 1931 Kurt Gödel ( ) 1. In any axiomatic systems, strong enough to include arithmetic, there is always an arithmetical statement that can neither be proved nor disproved. 2. A statement saying such a system is consistent can neither be proved nor disproved.

26 Ways of Beating the Paradoxes: Change the Logic

27 Intuitionistic Logic Luitzen Brouwer ( ) Arend Heyting ( ) Don t allow P P, so you don t have V V V V or R R R R.

28 Paraconsistent Logic Newton da Costa (1929 -!) Allow R R and R R, but not P P Q. (This doesn t work, Curry s Paradox, IF C = {x x x X } then X.)

29 Logic and Aritmetic based on Combinatory Logic Moses Schonfinkel ( ) Haskell Curry ( )

30 Logic and Arithmetic based on Lambda Calculus Alonzo Church ( )

31 Ways of Beating the Paradoxes: Change the Set Theory

32 Change the Set Theory Ernst Zermelo ( ) Abraham Fraenkel ( )

33 Axioms ( z)(z x z y) x = y ( y)( x)(x y x z P(x)) ( z)(x z y z) ( z)( x)(x y y w x z) Extensionality Substitution of Specification (Defines subsets of z) Pairing (defines {x, y}) Union (z is the set of the elements of the elements of w.) + Others

(1888-1977) - 1937, 1945 Gödel - 1940 Has

34 von Neumann, Bernays and Gödel John von Neumann ( ) , 1928 Paul Bernays ( ) , 1945 Gödel Has sets and classes (only sets can be elements of classes.)

35 Axioms ( z)(z X z Y ) X = Y (X, Y classes) Extensionality ( Y )(x Y a(x)) (Y is a class) comprehension Only y X if y is a set + Axioms for sets

36 Continuum Hypothesis 2 ℵ 0 = ℵ 1 Can neither be proved or disproved in NBG or ZF! Paul Cohen ( )

37 Cardinal Numbers and Measures The measure, or length, of: [m, m + n] = n #[m, m + n] = 2 ℵ 0 #[0, a] = 2 ℵ 0 if a > 0 IF #A = ℵ 0, the measure A = 0 N { 1 10 n, 2 10 n+1, 3 10 n+2,... } k can be made arbitrarily small for n large 10n+k 1 Question:If #A = 2 ℵ 0 is a measure for A > 0?

This section will take the very naive point of view that a set is a collection of objects, the collection being regarded as a single object.

This section will take the very naive point of view that a set is a collection of objects, the collection being regarded as a single object. 1.10. BASICS CONCEPTS OF SET THEORY 193 1.10 Basics Concepts of Set Theory Having learned some fundamental notions of logic, it is now a good place before proceeding to more interesting things, such as