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ZF
- 1. ZFC CHOICE ORDERS ELEMENTARY ZFC SEP Erik A. Andrejko University of Wisconsin - Madison Summer 2007 ERIK A. ANDREJKO ELEMENTARY ZFC
- 2. ZFC CHOICE ORDERS ZERMELO FRAENKEL CHOICE FIGURE: Ernst Zermelo ERIK A. ANDREJKO ELEMENTARY ZFC
- 3. ZFC CHOICE ORDERS LANGUAGE ERIK A. ANDREJKO ELEMENTARY ZFC
- 4. 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
- 5. 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
- 6. 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
- 7. 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
- 8. 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
- 9. 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
- 10. ZFC CHOICE ORDERS SCHRÖDER-BERNSTEIN THEOREM (SCHRÖDER-BERNSTEIN) B and B A then A ≈ B. If A ERIK A. ANDREJKO ELEMENTARY ZFC
- 11. 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
- 12. 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
- 13. 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