ZF

ZFC CHOICE ORDERS ELEMENTARY ZFC SEP Erik A. Andrejko University of Wisconsin - Madison Summer 2007 ERIK A. ANDREJKO ELEMENTARY ZFC

ZFC CHOICE ORDERS ZERMELO FRAENKEL CHOICE FIGURE: Ernst Zermelo ERIK A. ANDREJKO ELEMENTARY ZFC

ZFC CHOICE ORDERS LANGUAGE ERIK A. ANDREJKO ELEMENTARY ZFC

ZFC CHOICE ORDERS ZFC EXTENSIONALLY ∀z[z ∈ x ⇐⇒ z ∈ y ] =⇒ x = y That is, a set of is uniquely determine by its members. FOUNDATION ∀x[∃y(y ∈ x) =⇒ ∃y[y ∈ x ∧ ¬∃z(z ∈ x ∧ z ∈ y )]] from this we can conclude that ∀x[x ∈ x]. / COMPREHENSION For each formula ϕ without y free ∃y∀x[x ∈ y ⇐⇒ x ∈ z ∧ ϕ] Note that ϕ may mention any parameters except y . ERIK A. ANDREJKO ELEMENTARY ZFC

ZFC CHOICE ORDERS ZFC PAIRING ∃z[x ∈ z ∧ y ∈ z] That is, we may form a set {x, y } for any sets x and y . UNION ∀F∃A∀Y ∀x[x ∈ Y ∧ Y ∈ F =⇒ x ∈ A] That is, for any family F (set) of sets we may form a set A ⊃ F that contains the union of the family. Using comprehension we may form the actual set F. REPLACEMENT For each formula ϕ without y free ∀x ∈ A∃!y ϕ(x, y ) =⇒ ∃Y ∀x ∈ A∃y ∈ Y ϕ(x, y ) If we consider ϕ(x, ) to be a function f which assigns y = f (x), then the replacement scheme says that the range of the function is a set. ERIK A. ANDREJKO ELEMENTARY ZFC

ZFC CHOICE ORDERS ZFC INFINITY We deﬁne S(x) to be the successor function. That is S(x) = x ∪ {x}. Then ∃x[0 ∈ x ∧ ∀y ∈ x[S(y ) ∈ x]] That is, there exists a set x such that x is inﬁnite. POWER SET ∀x∃y∀z[z ⊆ x =⇒ z ∈ y ] That is, for any set, the powerset (set of all subsets) of x exists. CHOICE ∀A∃R[R well orders A] where R is an total order on A. That is, every set may be well ordered. ERIK A. ANDREJKO ELEMENTARY ZFC

ZFC CHOICE ORDERS ZF DEFINITION ZF All axioms except Choice. ZFC All axioms including Choice. DEFINITION ZF − ZF except foundation. Con(ZF −) =⇒ Con(ZF ) WARNING Choice is controversial. ERIK A. ANDREJKO ELEMENTARY ZFC

ZFC CHOICE ORDERS CHOICE THEOREM T.F.A.E Every set has a well order. 1 Every family of nonempty sets has a choice function. 2 Zorn’s Lemma: If every chain in a partial order P has an upper 3 bound in P then P has a maximal element. Tukey’s Lemma: A set X is say to have ﬁnite character if 4 Y ∈ X ⇐⇒ ∀k ∈ Y <ω [k ∈ X ] For X = 0 with ﬁnite character then X has a ⊆-maximal element. / Basis Theorem: Every vector space has a basis. 5 ERIK A. ANDREJKO ELEMENTARY ZFC

ZFC CHOICE ORDERS CHOICE IS COMPLICATED Relative Strength of Various Choice Principals AC KS Well Basis +3 Zorn’s ks +3 Tukey’s ks p2 = p ks +3 Ordering ks +3 Theorem Lemma Lemma w Principal www w www w w w ww ww Selection +3 Prime Ideal Dependant Compactness ks Principal Theorem Choice tt t ttt t ttt tt u} t t Order Extension +3 Ordering ACℵ0 +3 ACF ks r 4< Principal Principal rrr rrr r rrr r rr rrr r r AC2 ACW TAC ks +3 LEM ERIK A. ANDREJKO ELEMENTARY ZFC

ZFC CHOICE ORDERS SCHRÖDER-BERNSTEIN THEOREM (SCHRÖDER-BERNSTEIN) B and B A then A ≈ B. If A ERIK A. ANDREJKO ELEMENTARY ZFC

ZFC CHOICE ORDERS ORDERS DEFINITION (P, ≤) is a partial order if ≤ is transitive, anti-symmetric and reﬂexive. DEFINITION 1 If p ≤ q or q ≤ p then p and q are comparable. If there is some r such that r ≤ p and r ≤ q then p and q are 2 compatible. Denoted p ⊥ q. Otherwise p and q are incomparable. Denoted p ⊥ q. 3 A ⊆ P is an anti-chain if for all p, q ∈ A, p ⊥ q. 4 C ⊆ P is a chain if for all p, q ∈ A, p and q are comparable. 5 DEFINITION (P, ≤) is a linear order if P is a chain. ERIK A. ANDREJKO ELEMENTARY ZFC

ZFC CHOICE ORDERS ORDER TYPES DEFINITION If (P, ≤) is an order then P ∗ = (P, ≥). Order types: ω and ω ∗ , 1 Q, a dense linear order, 2 3 R DEFINITION (A, R) is well founded if every B ⊆ A has an R least element. ERIK A. ANDREJKO ELEMENTARY ZFC

ZFC CHOICE ORDERS TRANSFINITE INDUCTION ∀α[ϕ(α)] 1 ϕ(0), ϕ(α) =⇒ ϕ(α + 1), 2 For limit β , 3 ∀ξ < β [ϕ(ξ )] =⇒ ϕ(β ) ERIK A. ANDREJKO ELEMENTARY ZFC

ZF

by andrejko on Jul 16, 2007

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