Axiomatic set theory

Axiomatic set theory Jouko Väänänen Jouko Väänänen: Set theory

Axiom of Choice AC x y Jouko Väänänen: Set theory

Another formulation AC’ x f Jouko Väänänen: Set theory

AC AC’ Choose elements using AC Make them disjoint Return to the original sets Jouko Väänänen: Set theory

AC’  AC x f AC’ gives f Then take the range of f You get the set required to exist in AC Jouko Väänänen: Set theory

Well-ordering principle Jouko Väänänen: Set theory

WO AC Well-order the union The choice set y is the set of points z in the union such that in the particular unique (!) set that z belongs to, it is <-smallest. Jouko Väänänen: Set theory

A useful fact • If X is a set, there is no one-one class function f:On  X. • Proof. Otherwise, by the Axiom of Subset Selection, Y=f(On)X is a set. Thus we can form the function f-1:Y  On.By the Axiom of Replacement, f-1(Y) is a set of ordinals. Let  be an ordinal greater than all ordinals in f-1(Y). Then f() is in Y, so =f-1(f()) is in f-1(Y), a contradiction. Jouko Väänänen: Set theory

AC’  WO We need a function which picks an element like this From this non-empty set The point is: AC gives such a function Jouko Väänänen: Set theory

Zorn’s Lemma A maximal element .... an upper bound Every chain has an .... Jouko Väänänen: Set theory

Intuitive proof of ZL Starting from any element we keep picking larger and larger elements until we hit a maximal element. If we do not hit any such, we have anyway built a chain. By assumption, this chain has an upper bound. So we have to hit a maximal element. Jouko Väänänen: Set theory

WO Zorn’s Lemma P is first well ordered g() is chosen to be the least (in the w.o.) strict upper bound, if any g() {g(): <} Note that we use recursion on ordinals Jouko Väänänen: Set theory

An application of AC • We show that there is a non-measurable set of real numbers. • The collection of measurablesets of real numbers have the following properties: •  and all intervals [a,b] are measurable. • The complement of a measurable set is measurable. • The union of countably many measurable sets is always measurable. • If A is measurable, then A+r={x+r:xA} is measurable, what ever r is. Jouko Väänänen: Set theory

Lebesgue-measure • The Lebesgue-measurem(A) of a measurable set A is defined so that the following hold: • m()=0, m([a,b])=b-a • If the measurable sets An are disjoint, then m(nAn)=n m(An) • If A is measurable, then m(A+r)=m(A) for all reals r. Jouko Väänänen: Set theory

A non-measurable set • Define on [0,1]: xy iff x-y is rational. • This is an equivalence relation. • By AC there is a set A which has exactly one element from each -equivalence class. • We show that A is non-measurable. • Assume otherwise. Let r=m(A). Jouko Väänänen: Set theory

We derive a contradiction • [0,1] q(A+q) [-1,2], where q ranges over Q[0,1]. • 1m(q(A+q))=q m(A+q)=q r3. • If r=0, we cannot have 1  q r • If r>0, we cannot have q r3. • We have arrived at a contradiction. • So A is not measurable. QED Jouko Väänänen: Set theory

Banach-Tarski Paradox • Another application of AC: • The unit sphere in the three dimensional Euclidean space can be divided into five pieces so that by reassembling the pieces in space we get two unit spheres of the same size as the original. • The trick: the pieces are non-measurable, so we cannot say that e.g. volume has been doubled. The pieces (or at least some of them) simply do not have volume. Jouko Väänänen: Set theory

Ordinal addition   + Jouko Väänänen: Set theory

Ordinal addition formally • A=( x {0})( x {1}) • (,i)<A(,j) iff (i<j) v (i=j & <)  1  0 Jouko Väänänen: Set theory

Non-commutativity 1 +    + 1  1 has last element does not have last element  1 Jouko Väänänen: Set theory

Associativity    + + (+)+ +(+)

Cancellation laws??  +  =  + ’   =’ ? Yes  +  = ’ +    =’ ? No Jouko Väänänen: Set theory

Ordinal sum 0 1 2 0+1+2 Jouko Väänänen: Set theory

Ordinal sum 0 1  ... ... < Jouko Väänänen: Set theory

Example 1 1 1 ... ... n<1= Jouko Väänänen: Set theory

Ordinal sum <  ... ... ...   Jouko Väänänen: Set theory

Axiomatic set theory