Axiomatic set theory

Axiomatic set theory Jouko Väänänen Jouko Väänänen: Set theory

Every woset is isomorphic to an ordinal • Suppose (X,) is a woset. • Let E={aX : Xa is isomorphic to an ordinal}. • By Induction Principle and the previous theorem, E=X. • By the previous theorem X is isomorphic to an ordinal. Jouko Väänänen: Set theory

The ordinal of a woset • Every woset (X,) is isomorphic to a unique ordinal On(X) • Ordinals are well-ordered by  • , , ,  • ={ : <} Jouko Väänänen: Set theory

Ordinals • 0= • 1={0} • 2={0,1} • 3={0,1,2} • … • n={0,1,…,n-1} Jouko Väänänen: Set theory

Infinite ordinals • ={0,1,2,3,…,n,…} • {0,1,2,3,…,n,…, }= {} • Denote: {}= +1 • More generally: {}= +1 • Successor ordinals • {0,1,2,…,,+1,…} not a successor • Limit ordinal Jouko Väänänen: Set theory

Transfinite sequences of sets • A sequence is a function f such that dom(f) is an ordinal. • f=x :< • f()= x • Special case xn :n<={xn} Jouko Väänänen: Set theory

Paradoxes of naive set theory • Burali Forti Paradox: The collection of all ordinals is wellordered by . If it were a set, it would be the largest ordinal. But there is no largest ordinal. So what goes wrong? • Russell’s Paradox: Let R={x : xx}. We know that RR, because otherwise both RR and RR. Since RR, we have RR and RR, a contradiction. What went wrong? Jouko Väänänen: Set theory

Reasons for axiomatization • Paradoxes arise from confusion of concepts. • By stating exact axioms and rules of derivation we can (try to) avoid confusions. • We present so called Zermelo-Fraenkel axiomatization of set theory • No paradoxes have been found in this. Jouko Väänänen: Set theory

Language of set theory • We adopt the symbols: •  epsilon symbol • = identity symbol • ),( bracket symbols • vn variable symbols • wn constant symbols •  negation symbol •  disjunction symbol •  conjunction symbol •  universal quantifier symbol •  existential quantifier symbol Jouko Väänänen: Set theory

Formulas of set theory • (tt’) where t is a variable symbol of a constant symbol • (t=t’) where t is a variable symbol of a constant symbol • () negation of formula  • () disjunction of formulas  and  • () conjunction of formulas  and  • (xn) universally quantified formula • (xn) existentially quantified formula Jouko Väänänen: Set theory

Abbreviations • () abbreviates (()) • () abbreviates (() ()) • xy (z((zx)(zy))) • xy (xy (x=y)) • x={y} (z((zx)  (z=y))) • x={y,v} (z((zx)  ((z=y)(z=v))) Jouko Väänänen: Set theory

Further abbreviations • Brackets are left out if it is obvious where they should be, so we write •      instead of ((  )  ) • Or instead of (  (  )) Jouko Väänänen: Set theory

Intuitive universe of sets: The cumulative hierarchy • V0= • V+1= P(V) • V=<V for limit ordinals  V+1 V V0 Jouko Väänänen: Set theory

Zermelo-Fraenkel Axioms ZFC • Axiom of Extensionality: If two sets have the same elements, they are equal • Null Set Axiom: There is a set  with no elements • Axiom of Infinity: There is a set x such that x and {a}x whenever ax. • Power Set Axiom: If x is a set, there is a set P(x) consisting of all subsets of x. • Axiom of Union: If x is a set, there is a set x consisting of all elements of elements of x. • Axiom of Replacement: Let (vn,vm) be a formula of the language of set theory such that for all a there is a unique b with (a,b). Let x be a set. Then there is a set y consisting of exactly the elements b for which (a,b) for some a in x. • Axiom of Subset Selection: Let (vn) be a formula of the language of set. Let x be a set. Then there is a set y consisting of exactly the elements a of x for which (a) holds. • Axiom of Foundation: If x is a set, then there is a in x such that ax=. • Axiom of Choice: If x is a set of pairwise disjoint non-empty sets, then there is a set y which contains exactly one element from each set in x. Jouko Väänänen: Set theory

Axiom of Replacement b x (a,b) a y Jouko Väänänen: Set theory

Axiom of Subset Selection x y (a) a Jouko Väänänen: Set theory

Axiom of Foundation x a Jouko Väänänen: Set theory

Axiom of Choice x y Jouko Väänänen: Set theory

Classes and sets • Let (vn) be a formula of the language of set theory. • A={a : (a)} is called a class. • If (a), then a is called an element of the class A, denoted aA. • If a is a set, it is a class, a={x : xa}. • Some classes are not sets, e.g. {x:xx}, {x : x=x}, {x:x is a singleton}, etc . The are proper classes. Jouko Väänänen: Set theory

Operations on classes • AB • AB • A-B • A x B • RA x B • F:AB • Injection, bijection Jouko Väänänen: Set theory

Construction of the cumulative hierarchy • V0=, exists by the Null Set Axiom. • V+1= P(V), exists by the Power Set Axiom. • V=<V for limit ordinals , exists: Let A={V:<}. By Axiom of Replacement A exists. Now V=A. Jouko Väänänen: Set theory

Axiomatic set theory