Khomskii Lebesgue Soloay 1 Friday 27 th November 2015 1 This slide is available on http://slideshare.net/konn/lebesguesoloay 1 / 34
Khomskii 1 2 3 4 Khomskii 2 / 34
Khomskii Solovay 3 / 34
Khomskii Lebesgue Theorem 1 (Vitali) R/Q Lebesgue R/Q R/Q R µ(r) = 0 4 / 34
Khomskii DC R [ x y (x R y) = x n n < ω (x n R x n+1 )] R/Q? Solovay-Shelah 5 / 34
Khomskii Solovay-Shelah Theorem 2 (Solovay 1970 [5]) ZF + DC + (LM) Theorem 3 (Shelah 1984 [4]) ZF + DC + LM ZFC ZFC Solovay 6 / 34
Khomskii 1 2 3 4 Khomskii 7 / 34
Khomskii Solovay Theorem 2 (Solovay 1970) κ Col(ω, < κ) HOD ω ZF + DC + LM 8 / 34
Khomskii Solovay Theorem 2 (Solovay 1970) κ Col(ω, < κ) HOD ω ZF + DC + LM 8 / 34
Khomskii Definition 3 V On ZFC ω N ω 1 ω 9 / 34
Khomskii (cont.) Definition 4 κ 2 κ := P(κ). κ = def α < κ f : α κ γ < κ [sup f < γ]. i.e. κ κ = def λ < κ 2 λ < κ. i.e. κ κ = def κ > ω κ κ κ V κ ZFC ZFC 10 / 34
Khomskii Solovay Theorem 2 (Solovay 1970) κ Col(ω, < κ) HOD ω ZF + DC + LM 11 / 34
Khomskii Solovay Theorem 2 (Solovay 1970) κ Col(ω, < κ) HOD ω ZF + DC + LM 11 / 34
Khomskii (x y M = x M) ZF Gödel L On M L V M Γ = def φ φ M ω Γ! 0 12 / 34
Khomskii HOD ω Solovay HOD ω A A OD ω = def φ(x, y) σ ω On A = { x φ(x, σ) } V Levy φ V = φ V α = φ α On V α A A HOD ω def = trcl({ A }) OD ω, i.e. A OD ω HOD ω 13 / 34
Khomskii Solovay Theorem 2 (Solovay 1970) κ Col(ω, < κ) HOD ω ZF + DC + LM 14 / 34
Khomskii Solovay Theorem 2 (Solovay 1970) κ Col(ω, < κ) HOD ω ZF + DC + LM 14 / 34
Khomskii V G V[G] On V G V[G] ω 0 15 / 34
Khomskii V G V[G] G P On P V G V[G] ω 0 15 / 34
Khomskii V G V[G] G P p P V G V[G] On P ω V G V[G] 0 15 / 34
Khomskii V G V[G] G P p P V G V[G] p P φ p P G V[G] φ On 0 ω V G V[G] x P-name ẋ V P- P V[G] 15 / 34
Khomskii V G V[G] G P p P V G V[G] p P φ p P G V[G] φ On 0 ω V G V[G] x P-name ẋ V P- V[G] V P V[G] 15 / 34
Khomskii Levy Col(ω, < κ) Col(ω, < κ) : Levy κ ω 1 On V V ω κ κ ω V 2 ω V 1 ω 0 16 / 34
Khomskii Levy Col(ω, < κ) Col(ω, < κ) : Levy κ ω 1 On κ V V[G] V ω κ ω < λ < κ ω λ ω V 2 ω V 1 ω 0 16 / 34
Khomskii Levy Col(ω, < κ) Col(ω, < κ) : Levy κ ω 1 On 0 κ = ω V[G] 1 ω V[G] V ω κ ω < λ < κ ω λ V[G] ω κ V[G] κ ω 1 16 / 34
Khomskii Theorem 2 (Solovay 1970) κ Col(ω, < κ) HOD ω ZF + DC + LM 17 / 34
Khomskii Theorem 2 (Solovay 1970) κ κ ω 1 HOD ω ZF + DC + LM 17 / 34
Khomskii Theorem 2 (Solovay 1970) κ κ ω 1 HOD ω Lebesgue Shelah Solovay 17 / 34
Khomskii Theorem 2 (Solovay 1970) κ κ ω 1 HOD ω Lebesgue Shelah Solovay V[G] HOD ω Solovay 17 / 34
Khomskii 1 2 3 4 Khomskii 18 / 34
Khomskii Solovay 1 κ ω 1 2 A HOD( ω On) V[G] φ σ ω On A = { x φ(x, σ) } 3 σ M := V[G λ] λ < κ V[G λ] λ 4 V[G λ] 5 φ, σ A M Borel B 6 Borel A 19 / 34
Khomskii Solovay 1 κ ω 1 2 A HOD( ω On) V[G] φ σ ω On A = { x φ(x, σ) } 3 σ M := V[G λ] λ < κ V[G λ] λ 4 V[G λ] 5 φ, σ A M Borel B 6 Borel A 19 / 34
Khomskii Solovay 1 κ ω 1 2 A HOD( ω On) V[G] φ σ ω On A = { x φ(x, σ) } 3 σ M := V[G λ] λ < κ V[G λ] λ 4 V[G λ] 5 φ, σ A M Borel B 6 Borel A Col(ω, < κ) 19 / 34
Khomskii Solovay κ ω 1 2 A HOD( ω On) V[G] φ σ ω On A = { x φ(x, σ) } 3 σ M := V[G λ] λ < κ V[G λ] λ 4 V[G λ] 5 φ, σ A M Borel B 6 Borel A HOD( ω On) 19 / 34
Khomskii Solovay κ ω 1 A HOD( ω On) V[G] φ σ ω On A = { x φ(x, σ) } 3 σ M := V[G λ] λ < κ V[G λ] λ 4 V[G λ] 5 φ, σ A M Borel B 6 Borel A κ σ 19 / 34
Khomskii Solovay κ ω 1 A HOD( ω On) V[G] φ σ ω On A = { x φ(x, σ) } σ M := V[G λ] λ < κ V[G λ] λ 4 V[G λ] 5 φ, σ A M Borel B 6 Borel A 19 / 34
Khomskii 4. I Definition 4 M x M def = B M [B : M Borel = x / B ]. B M B V[G] Borel k U k c(0) = 0 B c = U c(1), c(0) = 1 B c = ω ω \ B c c(0) = 2 B c = n<ω B (c)n c ω ω c (k) = c(k + 1), (c) n (k) = c( n, k ). Borel 20 / 34
Khomskii 4. II N M := V[G λ] Borel N N N V[G] M Borel (2 ℵ 0 ) M λ M κ (2 ℵ 0 ) M < κ V[G] M Borel V[G] N 21 / 34
Khomskii Solovay κ ω 1 A HOD( ω On) V[G] φ σ ω On A = { x φ(x, σ) } σ M := V[G λ] λ < κ V[G λ] λ V[G λ] 5 φ, σ A M Borel B 6 Borel A 22 / 34
Khomskii Solovay κ ω 1 A HOD( ω On) V[G] φ σ ω On A = { x φ(x, σ) } σ M := V[G λ] λ < κ V[G λ] λ V[G λ] 5 φ, σ A M Borel B 6 Borel A 22 / 34
Khomskii Borel Borel MB 23 / 34
Khomskii Borel Borel MB MB M MB M Col(ω, < κ) V[G] Col(ω, < κ) <ω µ 23 / 34
Khomskii Borel Borel MB MB M MB M Col(ω, < κ) V[G] Col(ω, < κ) <ω µ Col(ω, < κ) V[G] p 23 / 34
Khomskii Borel Borel MB MB M MB M Col(ω, < κ) V[G] Col(ω, < κ) <ω µ Col(ω, < κ) V[G] p MB MB M - ṙ 23 / 34
Khomskii Borel Borel MB MB M MB M Col(ω, < κ) V[G] Col(ω, < κ) <ω µ Col(ω, < κ) V[G] p MB MB M - ṙ x A M[x] Col(ω,<κ) φ(ṙ, ˇσ) [ ] D M D M MB Col(ω,<κ) φ(ṙ, ˇσ) 23 / 34
Khomskii Borel Borel MB MB M MB M Col(ω, < κ) V[G] Col(ω, < κ) <ω µ Col(ω, < κ) V[G] p MB MB M - ṙ x A M[x] Col(ω,<κ) φ(ṙ, ˇσ) [ ] D M D M MB Col(ω,<κ) φ(ṙ, ˇσ) B := { } D D MB M, D MB Col(ω,<κ) φ(ṙ, š) Borel Borel 23 / 34
Khomskii Solovay κ ω 1 A HOD( ω On) V[G] φ σ ω On A = { x φ(x, σ) } σ M := V[G λ] λ < κ V[G λ] λ V[G λ] φ, σ A M Borel B 6 Borel A 24 / 34
Khomskii Solovay κ ω 1 A HOD( ω On) V[G] φ σ ω On A = { x φ(x, σ) } σ M := V[G λ] λ < κ V[G λ] λ V[G λ] φ, σ A M Borel B 6 Borel A 24 / 34
Khomskii Solovay κ ω 1 A HOD( ω On) V[G] φ σ ω On A = { x φ(x, σ) } σ M := V[G λ] λ < κ V[G λ] λ V[G λ] φ, σ A M Borel B Borel A 24 / 34
Khomskii 1 2 3 4 Khomskii 25 / 34
Khomskii Baire Baire? Khomskii [2] Definition 5 I P(X) σ- 1 A I, B A = B I, 2 { A n I n < ω } n<ω A n I I + := { B B B / I }. B Borel 26 / 34
Khomskii Definition 6 (Khomskii 2012) A R I- def = B B \ I C B \ I [C \ B I (C A C A = )]. I- A 27 / 34
Khomskii Definition 6 (Khomskii 2012) A R I- def = B B \ I C B \ I [C \ B I (C A C A = )]. I- A A C C B 27 / 34
Khomskii Definition 6 (Khomskii 2012) A R I- def = B B \ I C B \ I [C \ B I (C A C A = )]. I- A A C C B 27 / 34
Khomskii Definition 6 (Khomskii 2012) A R I- def = B B \ I C B \ I [C \ B I (C A C A = )]. I- A A C C B 27 / 34
Khomskii Definition 6 (Khomskii 2012) A R I- def = B B \ I C B \ I [C \ B I (C A C A = )]. I- A A C C B 27 / 34
Khomskii σ- null := { A R A : Lebesgue }, meager := { A R A : } A A: null- A Baire A: meager- Ramsey 28 / 34
Khomskii Khomskii Theorem 7 (Khomskii 2012) I σ-i + (proper) Solovay I- I- 29 / 34
Khomskii Future Work I Solovay Wright [6] ZF + DC + BP + LM 30 / 34
Khomskii Future Work II Solovay Solovay Shelah Solovay Shelah Khomskii 31 / 34
Khomskii I Thomas Jech. Set Theory: The Third Millennium Edition, revised and expanded. Springer Monographs in Mathematics. Springer, 2002. Yurii Khomskii. Regularity Properties and Definability in the Real Number Continuum. Idealized forcing, polarized partitions, Hausdorff gaps and mad families in the projective hierarchy. English. PhD thesis. Institute for Logic, Language and Computation, Universiteit van Amsterdam, 2012. Ralf Schindler. Set Theory: Exploring Independence and Truth. Universitext. Springer International Publishing Switzerland, 2014. isbn: 9783319067247. 32 / 34
Khomskii II Saharon Shelah. Can You Take Solovay s Inaccessible Away? In: Israel Journal of Mathematics 48.1 (1984), pp. 1 47. issn: 0021-2172. doi: 10.1007/BF02760522. url: http://dx.doi.org/10.1007/bf02760522. Robert M. Solovay. A model of set theory in which every set of reals is Lebesgue measurable. In: The Annals of Mathematics. 2nd ser. 92.1 (July 1970), pp. 1 56. issn: 0003486X. doi: 10.2307/1970696. url: http://www.jstor.org/stable/1970696. 33 / 34
Khomskii III J. D. Maitland Wright. Functional Analysis for the Practical Man. English. In: Functional analysis: surveys and recent results. Proceedings of the Conference on Functional Analysis. (Paderborn, Germany). Ed. by Klaus-Dieter Bierstedt and Benno Fuchssteiner. Mathematics Studies 27. North-Holland, 1977, pp. 283 290. isbn: 9780444850577. 34 / 34