9-2 Title of course: Forcing in your face until it comes out your ears I was going to use the title: "Facts about forcing which I have totally forgotten" but decided it would be too hard for me to teach that course. Prerequisites: M771 Kunen, Set Theory, Chapt 1-7. Start with a quick review of Chapt 7. P poset in M a ctble trans set model, dense subset of P, P-filter, G P-gen/M, P-names, tau^G, M[G], p ||- theta, elementary properties of ||-, existence of generic filters, truth lemma, definability lemma, M[G] |= ZFC, translates to pf of con(ZFC)=>con(ZFC+P), fin(k,2), P has ccc 9-4 Lemma P ccc, p||- t in ORD => exists A ctble p ||- t in A converse is also true Lemma P ccc, p||- t:alpha->beta => exists g:alpha->[beta]^w p ||- t(i)\in g(i) all i Thm Pccc, M[G]|= f:alpha->beta => exists g in M|= g:alpha->[beta]^w and f(i) in g(i) all i Converse almost true, there must be a maximal antichain A in P such that P has ccc beneath every element of A. Thm Pccc, M[G]|= 2^w leq k, where M|= k=|P^w|. Cor c can be omega_17, omega_{omega_1}, etc size of 2^{w_1} Solovay "The continuum can be anything it ought to be" ---------------------------------------------------------------------------- 9-9 measure and category on 2^w. m(C)=k/2^n where C clopen k number of s in 2^n inside C. Thm (in Halmos, measure thy, p54) If F a field of sets and m a ctble additive [0,1] measure (i.e., if An,A in F and A is disjoint union of An n in w then m(A)=Sum m(An)). Then m has a unique ctbly add extension to sigma-field generated by F. B = Borel(2^w)/measure zero, if G B-gen/M then there exists r in 2^w such that r in B for all B in Borel(2^w) coded in M with B in G. 9-11 Borel codes = BC, Bx=U{2^w \ Bxn : n in w}, Absoluteness of 1. x in BC_alpha 2. x in BC 3. u in Bx *4. Bx=empty 5. Bx subset By, Bx = By 6. Bx meager (Property of Baire is a sigma-algebra) *7. m(Bx)=r ---------------------------------------------------------------------------- 9-14 Did 7 above. Started on 4 above. Defined Sigma11 via operation A, proj of closed, and in 2^w proj of branches of a tree. Showed Sigma11 closed under ctble union, intersect, projection 9-16 Absoluteness of Sigma11, 4 above, Solovay characterization of random reals passed out 28: proj sets 9-18 Solovay proof of Sigma11 prop of Baire and Lebesgue measurable Kuratowski-Ulam (Fubini for category) ---------------------------------------------------------------------------- 9-21 for every eps>0 exists clopen C such that mu(C Delta B)0 for every eps>0 exists s such that mu(B inter [s])>(1-eps)mu(s). mention Leb density. measurable Tail sets have measure zero or one. every real in random real extension image of random real under Borel map. ground model reals not measurable. Steinhaus B+B contains an interval. 9-23 random real extension the grd model is meager. iteration of random real. product of random gives cohen real. non, cov meager,null in random extension. 9-25 cov(Null) in random real model. Sacks forcing. equiv to real. fusion lemma cof omega paper ---------------------------------------------------------------------------- 9-28 sacks property, minimality of sacks real 9-30 every new real is sacks and constructs the sacks real, every new meager (meas zero) set is covered by one in the ground model 10-2 splitting property, silver forcing, modified silver ---------------------------------------------------------------------------- 10-5 k-product side by side forcing perfect set forcing, cof(meager)=cof(meas zero)=w1 and c large 10-7 baumgartner: complete failure of MA 10-9 laver forcing, doesn't collapse omega1, 0 extension decides sentence ---------------------------------------------------------------------------- 10-12 laver property, no Cohen or random reals, killing SMZ in one-step 10-14 iterated forcing, finite and ctble supports 10-16 finite support ccc iteration has ccc, no new subsets of omega1 added at stages of cof omega2 ---------------------------------------------------------------------------- 10-19 consistency of MA, Bukovsky model and Stepran's every set of reals is omega1-Borel 10-21 Carlson every set omega1-Borel implies cof(c)=omega1, MA(prop K) Soulsin trees not killed by prop K 10-23 Kunen & Tall consistency of MA(prop K) + exists Souslin tree define axiom A forcing of Baumgartner ---------------------------------------------------------------------------- 10-26 Fusion lemma for axiom A - ctble support iteration, Antichain lemma succ step of induction 10-28 Antichain lemma for limit ordinals, P_w2 w2-ccc, CH preserved in intermediate models 10-30 Each Pi i exists P-pt 11-18 Griegorieff forcing P-pts 11-20 Grieg-forcing is omega-omega bounding ---------------------------------------------------------------------------- 11-23 Shelah: no P-pt in the P(U)^w model 11-25 Grieg: minimality of Silver forcing ---------------------------------------------------------------------------- 11-30 C.Gray: minimality of Laver forcing 12-02 finished minimality, Baumgartner omega1-dense orders isomorphic 12-04 Stewart Baldwin: MA -> any two omega1 dense sets of reals are homeomorphic ---------------------------------------------------------------------------- 12-07 Baumgartner Consistency of any two omega1 dense sets of reals are order isomorphic. Untangling Lemma. 12-09 Finished Baumgartner proof. (Hausdorff) Cantor space can be partitioned into omega1 Pi^0_3 sets 12-11 (Shelah,Fremlin) Cantor space can be partitioned into omega1 Pi^0_2 sets iff it can be covered by omega1 meager sets. ---------------------------------------------------------------------------- 12-14 (Groszek,Slaman) There does not exist a perfect subset of the ground model. ----------------------------------------------------------------------------