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The relative consistency of the negation of the Axiom of Choice using permutation models

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- 1. The Relative Consistency of ¬AC Erik A. Andrejko October 21, 2004 1
- 2. The Axiom in Question The Axiom of Choice (AC) is the statement: ∀X[∀x ∈ X[x = ∅] =⇒ ∃f∀x ∈ X[f(x) ∈ x]] (AC) Zermelo added AC to his list of axioms formalizing mathematics in 1904 in order to prove Cantor’s conjecture that all sets can be well ordered. 2
- 3. The Controversy Surrounding AC The Axiom of Choice is diﬀerent in character than the other axioms of ZFC since AC implies the existence of sets without giving their construction. French mathematicians Borel, Baire and Lebesgue were suspicious of the use of AC in mathematical proofs, as was Peano who left it out of his Formulario Mathematico. The most trivial form of AC: ∀x[x = ∅ =⇒ ∃x ∈ X] (TAC) Trivial only if one considers constructive and abstract existence to be the same. For propositions P, the law of excluded middle is (LEM) ∀P[P ∨ ¬P] Theorem 1. (LEM) ⇐⇒ (TAC) The above theorem is intuitionistically valid. See [Age02]. 3
- 4. Some Paradoxes provable from ZFC The use of AC allows one to prove existence without construction. Theorem 2. (Vitali’s Theorem) There exists a non Lebesgue measurable set. Theorem 3. (Banach-Tarski Paradox) A sphere S can be decomposed into disjoint sets S=A∪B∪C∪Q ∼∼ ∼ such that A = B = C, B ∪ C = A, and Q is countable. 4
- 5. Some Theorems Equivalent to AC • Zorn’s Lemma: If every chain in a partial order P has an upper bound in P then P has a maximal element. • Tukey’s Lemma: A set X is say to have ﬁnite character if Y ∈ X ⇐⇒ ∀k ∈ Y <ω [k ∈ X] For X = ∅ with ﬁnite character then X has a ⊆-maximal element. • Basis Theorem: Every vector space has a basis. Without AC we can still consider the cardinalities of sets. We can deﬁne a class of cardinals p by consider members of the equivalence class given by x ≡ y ⇐⇒ |x| = |y| where |x| ≤ |y| if there is a 1-1 function f : x → y and |x| = |y| iﬀ |x| ≤ |y| and |y| ≤ |x|. Note that |x| ≤ |y| is a partial order. Then we deﬁne p2 = |p × p|. 5
- 6. Some Strictly Weaker Implications of AC • Prime Ideal Theorem: Every Boolean algebra has a prime ideal. Or equivalently every ideal can be extended to a prime ideal. • Ultraﬁlter Theorem: Every ﬁlter on X can be extended to an ultraﬁlter on X. • Compactness Theorem: A theory T is satisﬁable iﬀ T is ﬁnitely satisﬁable. 6
- 7. Diﬀerent Notions of Choice • Principal of Dependent Choice (DC): If R is a relation on A = ∅ such that ∀x ∈ A∃y ∈ A[xRy] then there is a sequence {xn }n<ω x0 Rx1 , x1 Rx2 , · · · , xn Rxn+1 , · · · • Selection Principal: For every family X of sets such that ∀y ∈ X[|y| ≥ 2] there exists function f such that ∀y ∈ X[∅ = f(y) X] 7
- 8. Ordering Principals • Well Ordering Principal (WOP): Every set can be well ordered. • Ordering Principal (OP): Every set can be linearly ordered. • Order Extension Principal (OEP) Every partial order P can be extended to a linear order. 8
- 9. Restriction of AC to smaller classes • AC for well ordered sets (ACW): Every set of well ordered sets has a choice function. • AC for countable sets (ACℵ0 ): Every set of countable sets has a choice function • AC for ﬁnite sets (ACF): Every set of ﬁnite sets has a choice function. • AC for pairs (AC2 ): Every set of pairs has a choice function. • Trivial AC (TAC): Every non empty set contains an element. 9
- 10. 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 Principal ww ww ww w www w w ww w Selection +3 Prime Ideal Dependant Compactness ks Principal Theorem Choice tt t ttt t tt t u} tt tt Order Extension Ordering ACℵ0 +3 ACF ks +3 r 4 Principal Principal rr rr rr rrr rr rr rrr r r AC2 ACW TAC ks +3 LEM 10
- 11. The Consistency of the Failure of Various Choice Principals The arrows on the previous slide do not reverse. We can prove this by construction models in which one holds and the negation of the other also holds. (i.e. Build a model of ZF + OP + ¬OEP to show that OP =⇒ OEP.) Theorem 4. The Prime Ideal Theorem does not imply AC. Theorem 5. The Selection Principal does not imply AC. Theorem 6. The Ordering Principal does not imply the Order Extension Principal. 11
- 12. Relative Consistency Results G¨del proved in 1930 his famous Incompleteness result that showed we cannot o prove the consistency of ZF + ¬AC within ZFC, that is Con(ZF + ¬AC) ZFC So assuming Con(ZFC) we build a model of ZF + ¬AC. Cohen showed such a construction using forcing in 1963. Fraenkel and Mostowski showed the unprovability of AC from ZF− . Theorem 7. Con(ZF− ) =⇒ Con(ZF− + ¬AC). 12
- 13. Constructing a Permutation Model By ZF− we denote the axioms of ZF minus the axiom of foundation. In particular we allow sets such as a = {a} which we will call an atom. Let A be an inﬁnite set of atoms. Deﬁne Vα (A) by induction of α as follows: V0 (A) = A Vα+1 (A) = P(Vα ) Vγ (A) for γ limit Vα (A) = γα Finally deﬁne V = Vα (A). Then we have α∈ON A = V0 (A) ⊆ V1 (A) ⊆ · · · ⊆ Vα (A) · · · ⊆ V For any x ∈ V we can assign a rank, rank(x) = least α[x ∈ Vα+1 (A)] 13
- 14. Let G be the group of permutations of A. For π ∈ G we extend π to a permutation of any x ∈ V by induction on ∈ by deﬁning π(x) = {π(y) : y ∈ x} and letting π(∅) = ∅. Then G permutes V and ﬁxes the well founded sets WF ⊆ V. Lemma 1. For all x, y ∈ V and any π ∈ G. x ∈ y ⇐⇒ π(x) ∈ π(y) That is π if an ∈-automorphism of V. From this we can prove that π({X, Y}) = {π(X), π(Y)} and so π((X, Y)) = (π(X), π(Y)) π((X, Y, Z)) = (π(X), π(Y), π(Z)) Also by induction on α it is easy to show that rank(x) = rank(πx) for all x ∈ V. 14
- 15. The Hereditarily Symmetric sets HS Let a1 , · · · , an ∈ A and deﬁne [a1 , · · · , an ] = {π ∈ G : ∀a1 , · · · , an [π(ai ) = ai ]} Call a set X ∈ V symmetric if there exists a1 , · · · , an ∈ A such that π(X) = X for all π ∈ [a1 , · · · , an ]. Deﬁne the class HS ⊆ V of hereditarily symmetric sets HS = {x ∈ V : x is symmetric and x ⊆ HS} Call a class N transitive if ∀x ∈ N[x ⊆ N] and call N almost universal if (for sets S) ∀S ⊆ N[∃Y ∈ N(S ⊆ Y)] Lemma 2. HS is transitive. Lemma 3. HS is almost universal. 15
- 16. To show that a class N |= ZF− is straight forward for most axioms of ZF− except for the axiom of Comprehension. To show N is a model of Comprehension it suﬃces to show that N is closed under G¨del Operations: o G1 (X, Y) = {X, Y} G2 (X, Y) = X Y G3 (X, Y) = X × Y G4 (X) = dom(X) G5 (X) = ∈ ∩X2 G6 (X) = {(a, b, c) : (b, c, a) ∈ X} G7 (X) = {(a, b, c) : (c, b, a) ∈ X} G8 (X) = {(a, b, c) : (a, c, b) ∈ X} Theorem 8. (ZF) If N is transitive, almost universal and closed under G¨del o Operations, then N |= ZF. Lemma 4. HS is closed under G¨del operations. o Lemma 5. HS |= ZF− . 16
- 17. HS violates AC Lemma 6. A ∈ HS. Lemma 7. Let f : ω → A be 1-1. Then f ∈ HS and so A cannot be well ordered / in HS. Theorem 9. HS |= ZF− + ¬AC. which completes the proof that Con(ZF− ) =⇒ Con(ZF− + ¬AC). In particular we have that ZF− AC. 17
- 18. Showing the Relative Consistency of ¬AC with ZF Let B be a complete boolean algebra and let MB be a boolean valued model. Let G be and M-generic ultraﬁlter on B. Then M ⊆ M[G] |= ZFC Then, as in V the universe of M[G] admits ∈-automorphism even though M does not. Let π be an automorphism of B, that is π(x · y) = π(x) · π(y) π(x + y) = π(x) + π(y) We can extend π to an automorphism of MB by induction: π(0) = 0, and let dom(π(x)) = π(dom(x)) Then let π(x)(π(y)) = π(x(y)) for all π(y) ∈ dom(x). Then π is a permutation of MB and π ﬁxes M. Then as before we can construct HS ⊆ MB . Finally, using G we can construct N ⊆ M[G] by using only names for elements of M[G] appearing in HS. Then M ⊆ N ⊆ M[G] 18
- 19. Such an N is called a symmetric extension of M. Lemma 8. N |= ZF. Finally let P = Fn(ω × ω, 2), Cohen forcing. Then we deﬁne xn to be the Cohen real xn = {m ∈ ω : ∃p ∈ G[p(n, m) = 1]} Let A be set of Cohen reals = {xn : n ω}. Then A ∈ N and A cannot be well ordered in N. To get permutations of π we can extend permutations of ω to P, then to a Boolean algebra B, then to the boolean valued model MB and ﬁnally to M[G]. From this we get that Con(ZFC) =⇒ Con(ZF + ¬AC) and so ZF AC. 19
- 20. Some Peculiarities in the Absence of AC Theorem 10. It is consistent that all uncountable cardinals can have coﬁnality ω. (see [Git80]) Theorem 11. There is a model of ZF in which the set of reals has no countable subset. Theorem 12. There is a model of ZF with the set of reals a countable union of countable sets. Theorem 13. There is a model of M |= ZF + DC in which every set of reals is Lebesgue Measurable (LM). Theorem 14. There exists a distance graph G2 in R2 such that under (ZFC) G2 has chromatic number 2, and under ZF + LM + DC the chromatic number of G2 cannot equal n ≤ ℵ0 . (see [SS04]) Theorem 15. There is a model with a vector space without a basis. Theorem 16. The axiom of choice for countable collections of countable sets does not imply that a countable union of countable sets is countable. (see [How92]) Theorem 17. (Intuitionistic Logic) The family F = {{a, b}} has no choice ˇ function. (see [FS82]) 20
- 21. References [Abi79] Alexander Abian, A simpliﬁed version of Fraenkel-Mostowski model for the independence of the axiom of choice, Rev. Roumaine Math. Pures Appl. 24 (1979), no. 4, 511–521. MR MR545072 (80k:03054) [Age02] Pierre Ageron, L’autre axiome de choix, Rev. Histoire Math. 8 (2002), no. 1, 113–140. MR MR1949810 (2003m:01037) [Bla84] Andreas Blass, Existence of bases implies the axiom of choice, Axiomatic set theory (Boulder, Colo., 1983), Contemp. Math., vol. 31, Amer. Math. Soc., Providence, RI, 1984, pp. 31–33. MR MR763890 (86a:04001) ˇ ˇc [FS82] M. P. Fourman and A. Sˇedrov, The “world’s simplest axiom of choice” fails, Manuscripta Math. 38 (1982), no. 3, 325–332. MR MR667919 (83k:03074) [Git80] M. Gitik, All uncountable cardinals can be singular, Israel J. Math. 35 (1980), no. 1-2, 61–88. MR MR576462 (81h:03096) [How92] Paul E. Howard, The axiom of choice for countable collections of countable sets does not imply the countable union theorem, Notre Dame 21
- 22. J. Formal Logic 33 (1992), no. 2, 236–243. MR MR1167981 (93e:03072) [Jec73] Thomas J. Jech, The axiom of choice, North-Holland Publishing Company, 1973. [Kun80] Kenneth Kunen, Set theory: An introduction to independance proofs, Elsevier Science Publishers B.V., 1980. [SS04] Alexander Soifer and Saharon Shelah, Axiom of choice and chromatic number: examples on the plane, J. Combin. Theory Ser. A 105 (2004), no. 2, 359–364. MR MR2046089 22

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