Zermelo-Fraenkel Set Theory

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1 Zermelo-Fraenkel Set Theory Zak Mesyan University of Colorado Colorado Springs

2 The Real Numbers In the 19th century attempts to prove facts about the real numbers were limited by the lack of a rigorous definition of real number. Definition (Richard Dedekind, 1872) A real number is a partition of the rational numbers into two sets, L and U, where every element of L is less than every element of U.

3 The Real Numbers In the 19th century attempts to prove facts about the real numbers were limited by the lack of a rigorous definition of real number. Definition (Richard Dedekind, 1872) A real number is a partition of the rational numbers into two sets, L and U, where every element of L is less than every element of U. Theorem (Georg Cantor, 1874) Set R of the real numbers (as defined by Dedekind) is uncountable, i.e., it cannot be put into one-to-one correspondence with N = {0, 1, 2,... }.

4 The Real Numbers In the 19th century attempts to prove facts about the real numbers were limited by the lack of a rigorous definition of real number. Definition (Richard Dedekind, 1872) A real number is a partition of the rational numbers into two sets, L and U, where every element of L is less than every element of U. Theorem (Georg Cantor, 1874) Set R of the real numbers (as defined by Dedekind) is uncountable, i.e., it cannot be put into one-to-one correspondence with N = {0, 1, 2,... }. In particular, there are infinite sets of different sizes.

5 Russell s Paradox Suppose that every set-builder formula defines a set. Then y = {x x / x} is a set.

6 Russell s Paradox Suppose that every set-builder formula defines a set. Then y = {x x / x} is a set. If y y, then, by the definition of y, we have y / y.

7 Russell s Paradox Suppose that every set-builder formula defines a set. Then y = {x x / x} is a set. If y y, then, by the definition of y, we have y / y. If y / y, then, by the definition of y, we have y y.

8 Russell s Paradox Suppose that every set-builder formula defines a set. Then y = {x x / x} is a set. If y y, then, by the definition of y, we have y / y. If y / y, then, by the definition of y, we have y y. Either way there is a contradiction.

9 Axiomatization of Set Theory To avoid paradoxes, in 1908 Ernst Zermelo proposed a system of axioms (i.e., basic assumptions) for set theory, where sets could be constructed only in certain limited ways.

10 Axiomatization of Set Theory To avoid paradoxes, in 1908 Ernst Zermelo proposed a system of axioms (i.e., basic assumptions) for set theory, where sets could be constructed only in certain limited ways. Abraham Fraenkel, Thoralf Skolem, and John von Neumann built on Zermelo s work to produce the now commonly accepted system of set theory axioms ZFC, or Zermelo-Fraenkel set theory with the Axiom of Choice (around 1930).

11 Axiomatization of Set Theory To avoid paradoxes, in 1908 Ernst Zermelo proposed a system of axioms (i.e., basic assumptions) for set theory, where sets could be constructed only in certain limited ways. Abraham Fraenkel, Thoralf Skolem, and John von Neumann built on Zermelo s work to produce the now commonly accepted system of set theory axioms ZFC, or Zermelo-Fraenkel set theory with the Axiom of Choice (around 1930). Since the ascent of set theory, all mathematical objects (e.g., natural numbers, functions, graphs, groups, topologies, etc.) have been redefined in terms of sets. So ZFC provides an axiomatic basis for all of mathematics.

12 Axiom of Extensionality x y [ z (z x z y) x = y] Intuitively: two sets are equal if they have the same elements.

13 Axiom of Extensionality x y [ z (z x z y) x = y] Intuitively: two sets are equal if they have the same elements. This allows one to define a set by describing its elements.

14 Axiom of Pairing x y z w [w z (w = x w = y)] Intuitively: given sets x and y, there exists a set z (= {x, y}) whose elements are precisely x and y.

15 Axiom of Pairing x y z w [w z (w = x w = y)] Intuitively: given sets x and y, there exists a set z (= {x, y}) whose elements are precisely x and y. This allows one to define sets consisting of complicated objects. For example, an ordered pair (x, y) is formally defined as {{x}, {x, y}}.

16 Axiom of Union x y z [z y w (w x z w)] Intuitively: for any set x, there is a set y = x whose members are precisely all the members of all the members of x. Or: the union of a set is a set.

17 Axiom of Union x y z [z y w (w x z w)] Intuitively: for any set x, there is a set y = x whose members are precisely all the members of all the members of x. Or: the union of a set is a set. For example, if x = {{1, 2}, {3}, {4, 5, 6}}, then y = {1, 2, 3, 4, 5, 6}.

18 Axiom of Union x y z [z y w (w x z w)] Intuitively: for any set x, there is a set y = x whose members are precisely all the members of all the members of x. Or: the union of a set is a set. For example, if x = {{1, 2}, {3}, {4, 5, 6}}, then y = {1, 2, 3, 4, 5, 6}. This, together with the axiom of pairing allows us to construct the familiar unions of sets, such as {1, 2} {3, 4} = {1, 2, 3, 4}.

19 Axiom of Power Set x y z [z y w (w z w x)] Intuitively: for every set x there exists a set y whose elements are the subsets of x.

20 Axiom of Power Set x y z [z y w (w z w x)] Intuitively: for every set x there exists a set y whose elements are the subsets of x. For example, the power set of x = {1, 2} is y = {, {1}, {2}, {1, 2}}.

21 Axiom of Power Set x y z [z y w (w z w x)] Intuitively: for every set x there exists a set y whose elements are the subsets of x. For example, the power set of x = {1, 2} is y = {, {1}, {2}, {1, 2}}. This axiom implies that there is no biggest set, since given any set one can construct a set with greater cardinality by taking its power set.

22 Axiom Schema of Separation z w 1 w 2... w n y x [x y (x z φ)] where φ = φ(x, z, w 1,..., w n ) is any formula in the language of set theory with free variables among x, z, w 1,..., w n. Intuitively: given a set z and a condition φ, there is a set y = {x z φ(x)} consisting of those elements of z that satisfy φ.

23 Axiom Schema of Separation z w 1 w 2... w n y x [x y (x z φ)] where φ = φ(x, z, w 1,..., w n ) is any formula in the language of set theory with free variables among x, z, w 1,..., w n. Intuitively: given a set z and a condition φ, there is a set y = {x z φ(x)} consisting of those elements of z that satisfy φ. This is a schema because it is an infinite collection of axioms one for every φ.

24 Axiom Schema of Separation z w 1 w 2... w n y x [x y (x z φ)] where φ = φ(x, z, w 1,..., w n ) is any formula in the language of set theory with free variables among x, z, w 1,..., w n. Intuitively: given a set z and a condition φ, there is a set y = {x z φ(x)} consisting of those elements of z that satisfy φ. This is a schema because it is an infinite collection of axioms one for every φ. This allows us to use set-builder notation to define new sets, such as the empty set: = {x y (x x) (x x)}, where y is any existing set.

25 Axiom Schema of Separation z w 1 w 2... w n y x [x y (x z φ)] where φ = φ(x, z, w 1,..., w n ) is any formula in the language of set theory with free variables among x, z, w 1,..., w n. Intuitively: given a set z and a condition φ, there is a set y = {x z φ(x)} consisting of those elements of z that satisfy φ. This is a schema because it is an infinite collection of axioms one for every φ. This allows us to use set-builder notation to define new sets, such as the empty set: = {x y (x x) (x x)}, where y is any existing set. This schema avoids Russell s Paradox because it only creates subsets of already existing sets.

26 Axiom of Infinity x [ x y (y x y {y} x)] Intuitively, this constructs the infinite set consisting of the following elements: { } (= { }) { } {{ }} (= {, { }}).

27 Axiom of Infinity x [ x y (y x y {y} x)] Intuitively, this constructs the infinite set consisting of the following elements: { } (= { }) { } {{ }} (= {, { }}). Labeling the sets as above as 0 =, 1 = { }, 2 = {, { }},..., we get the natural numbers!

28 Axiom of Regularity x [x y (y x z (z x (z y)))] Intuitively: every nonempty set x contains a minimal element y such that x y =.

29 Axiom of Regularity x [x y (y x z (z x (z y)))] Intuitively: every nonempty set x contains a minimal element y such that x y =. This implies that no set is an element of itself. In particular, there is no set of all sets.

30 Axiom of Regularity x [x y (y x z (z x (z y)))] Intuitively: every nonempty set x contains a minimal element y such that x y =. This implies that no set is an element of itself. In particular, there is no set of all sets. This axiom also makes induction a valid proof technique.

31 Axiom Schema of Replacement w 1... w n d [( x d!y φ(x, y, w 1,..., w n, d)) r y (y r x d φ(x, y, w 1,..., w n, d))] where φ = φ(x, y, w 1,..., w n, d) is any formula in the language of set theory with free variables among x, y, w 1,..., w n, d. Intuitively: if the expression φ represents a function f with domain d, then the range of f is a set.

32 Axiom Schema of Replacement w 1... w n d [( x d!y φ(x, y, w 1,..., w n, d)) r y (y r x d φ(x, y, w 1,..., w n, d))] where φ = φ(x, y, w 1,..., w n, d) is any formula in the language of set theory with free variables among x, y, w 1,..., w n, d. Intuitively: if the expression φ represents a function f with domain d, then the range of f is a set. This axiom is included in ZFC mostly because without it one could not construct certain desirable infinite sets.

33 Axiom of Choice x [ / x f (f : x x y x (f (y) y)] Intuitively: if x is a set whose members are all non-empty, then there exists a choice function f from x to the union of the members of x, such that f (y) y for all y x. Or: given a collection of sets, there is a function that chooses an element from each of the sets.

34 Axiom of Choice x [ / x f (f : x x y x (f (y) y)] Intuitively: if x is a set whose members are all non-empty, then there exists a choice function f from x to the union of the members of x, such that f (y) y for all y x. Or: given a collection of sets, there is a function that chooses an element from each of the sets. This is need to prove that every (infinite-dimensional) vector space has a basis, that any onto function has a right inverse, that the Cartesian product of any family of nonempty sets is nonempty, and many other commonly used mathematical facts.

35 Axiom of Choice x [ / x f (f : x x y x (f (y) y)] Intuitively: if x is a set whose members are all non-empty, then there exists a choice function f from x to the union of the members of x, such that f (y) y for all y x. Or: given a collection of sets, there is a function that chooses an element from each of the sets. This is need to prove that every (infinite-dimensional) vector space has a basis, that any onto function has a right inverse, that the Cartesian product of any family of nonempty sets is nonempty, and many other commonly used mathematical facts. This axiom was controversial for a long time because it asserts the existence of an object without explicitly constructing it.

36 Thank you!

mat.haus project Zurab Janelidze s Lectures on UNIVERSE OF SETS last updated 18 May 2017 Stellenbosch University 2017

mat.haus project Zurab Janelidze s Lectures on UNIVERSE OF SETS last updated 18 May 2017 Stellenbosch University 2017 Zurab Janelidze s Lectures on UNIVERSE OF SETS last updated 18 May 2017 Stellenbosch University 2017 Contents 1. Axiom-free universe of sets 1 2. Equality of sets and the empty set 2 3. Comprehension and