On almost precipitous ideals.
|
|
- Phebe Henry
- 5 years ago
- Views:
Transcription
1 On almost precipitous ideals. Asaf Ferber and Moti Gitik December 20, 2009 Abstract With less than 0 # two generic extensions of L are identified: one in which ℵ 1, and the other ℵ 2, is almost precipitous. This improves the consistency strength upper bound of almost precipitousness obtained in [6], and answers some questions raised there. Also, main results of [5] are generalized- assumptions on precipitousness are replaced by those on -semi precipitousness. As an application it is shown that if δ is a Woodin cardinal and there is an f : ω 1 ω 1 with f = ω 2, then after Col(ℵ 2, δ) there is a normal precipitous ideal over ℵ 1. The existence of a pseudo-precipitous ideal over a successor cardinal is shown to give an inner model with a strong cardinal. 1 Introduction Let us define some basic notions. Definition 1.1 Let κ be a regular uncountable cardinal, τ an ordinal and I a κ-complete ideal over κ. We call I τ-almost precipitous iff every generic ultrapower of I is wellfounded up to the image of τ, where as usual we force with I-positive sets, a generic object G is a V -ultrafilter and (V κ V )/G is called a generic ultrapower. Clearly, any such I is τ-almost precipitous for each τ < κ. Also, note if τ (2 κ ) + and I is τ-almost precipitous, then I is precipitous. Therefore, in fact, if I is τ-almost precipitous for every τ < (2 κ ) +, then I is precipitous. Definition 1.2 Let κ be a regular uncountable cardinal. We call κ almost precipitous iff for each τ < (2 κ ) + there is a τ-almost precipitous ideal over κ. It was shown in [6] that ℵ 1 is almost precipitous if there is an ℵ 1 -Erdős cardinal. The following questions were raised in [6]: 1
2 1. Is an ℵ 1 -Erdős cardinal needed? 2. Can cardinals above ℵ 1 be almost precipitous without there being a measurable cardinal in an inner model? We will answer both questions by constructing two generic extensions of L such that ℵ 1 will be almost precipitous in the first and ℵ 2 in the second. Some of the ideas of Donder and Levinski [1] will be crucial here. Definition 1.3 (Donder- Levinski [1]) Let κ be a cardinal and τ be a limit ordinal of cofinality above κ or τ = On. 1. κ is called τ-semi-precipitous iff there exists a forcing notion P such the following is forced by the weakest condition: There exists an elementary embedding j : V τ M such that (a) crit(j) = κ (b) M is transitive. The restriction to τ s of cofinality above κ allows to apply j to any f : κ V τ, since then f is inside V τ. 2. κ is called < λ- semi-precipitous iff it is τ-semi-precipitous for every limit ordinal τ < λ of cofinality above κ. 3. κ is called semi-precipitous iff it is τ-semi-precipitous for every limit ordinal τ of cofinality above κ. 4. κ is called -semi-precipitous iff it is On-semi-precipitous. Note if κ is a semi-precipitous, then it is not necessarily -semi-precipitous, since by Donder and Levinski [1] semi-precipitous cardinals are compatible with V = L, and -semi-precipitous cardinals imply there exists an inner model with a measurable. 5. Let P and j be as in the item 1 above. We call F = {X κ 0 P κ j (X)} 2
3 a τ-semi-precipitous filter. 1 Note that such an F is a normal filter over κ. Also, F depends on a specific name of j. Note that M in (1) of the definition may be a proper class even if τ is an ordinal. But it is easy to find then i : V τ M with M being a transitive set. Just consider U = {X κ X V and κ j(x)}. Then V κ V τ /U is well founded, since we can embedded it into M using an elementary embedding k defined by k([f] U ) = j(f)(κ). Let M be the transitive collapse of V κ V τ /U and i the corresponding elementary embedding. The paper is organized as follows. In Section 2 we examine basic connections between semi-precipitousness and almost precipitousness are made. The following result that answers a question from [6] is shown: Corollary 2.12 The following are equivalent: 1. Con( there exists an almost precipitous cardinal), 2. Con( there exists an almost precipitous cardinal with normal ideals witnessing its almost precipitousness), 3. Con(there exists < κ ++ -semi-precipitous cardinal κ). In particular the strength of existence of an almost precipitous cardinal is below 0 #. In Section 3, an almost precipitous ideal is constructed over ℵ 2 iterating Namba forcing over L and then applying a variation of a construction of [5]. In Section 4 the following is proved: Theorem 4.1 Assume that 2 κ = κ + and κ carries a λ-semi-precipitous filter for some limit ordinal λ with cof(λ) > κ. Suppose in addition that there is a forcing notion P witnessing the λ-semi-precipitous with corresponding ultrapower of V by generic ultrafilters ill founded. Then 1. if λ < κ ++, then κ is λ-almost precipitous, as witnessed by a normal filter, 2. if λ κ ++, then κ is an almost precipitous, as witnessed by normal filters. 1 M. Foreman [3] 3.37 introduced a similar notion of a pre-precipitous filter. He requires that in a generic extension V Q by a forcing notion Q there is an elementary embedding j : V M with M transitive. The filter defined by picking some Q-name t of element of M, a set Z in V and then setting F = {X Z 0 Q t j (Z)}. So, On-semi-precipitous filters are pre-precipitous. 3
4 The theorem is a kind of interplay between ill foundness and well foundness in which ill foundness helps to produce well foundness. Main results of [5] are generalized here to context of -precipitous ideals: Theorem 4.10 Assume that ℵ 1 is -semi precipitous and 2 ℵ 1 = ℵ 2. Suppose that for some forcing P witnessing this 0 P P i (ℵ 1 ) > (ℵ + 1 ) V. Then ℵ 1 is almost precipitous witnessed by normal filters. Theorem 4.12 Suppose that there is no inner model satisfying ( α o(α) = α ++ ). Assume that ℵ 1 is -semi precipitous and 2 ℵ 1 = ℵ 2. If ℵ 3 is not a limit of measurable cardinals of the core model, then there exists a normal precipitous ideal on ℵ 1. Theorem 4.13 Assume that ℵ 1 is -semi precipitous. Let P be a witnessing this forcing such that 0 P P i (ℵ 1 ) > (ℵ + 1 ) V. Then, after forcing with Col(ℵ 2, P ), there will be a normal precipitous filter on ℵ 1. As a corollary of the last theorem we deduce the following: Corollary 4.17 Suppose that δ is a Woodin cardinal and there is f : ω 1 ω 1 with f ω 2. Then in V Col(ℵ2,δ) there is a normal precipitous ideal over ℵ 1. Section 6 deals with pseudo-precipitous ideals. T. Jech [8] asked how strong is the consistency of there is a pseudo-precipitous ideal on ℵ 1? We show the following: Theorem 6.2 If there is a pseudo-precipitous ideal over a successor cardinal then there is an inner model satisfying ( α o(α) = α ++ ). In particular, an existence of precipitous ideal does not necessary imply an existence of a pseudo-precipitous one. Sections 3,4 rely on the filter construction technique of [5], but otherwise there is not extensive background knowledge required. 2 On semi-precipitous and almost precipitous ideals In this section we consider games that allow to connect semi-precipitousness and almost precipitousness and then deduce some conclusions on the strength of an existence of an almost precipitous filters. Lemma 2.1 Let F be a τ-almost precipitous normal filter over κ for some ordinal τ of cofinality above κ. Then F is τ-semi-precipitous. 4
5 Proof. Force with F +. Let i : V N = V κ V/G be the corresponding generic embedding. Set j = i τ. Then j : V τ (V i(τ) ) N. Set M = (V i(τ) ) N. We claim that M is well founded. Suppose otherwise. Then there is a sequence g n n < ω of functions such that 1. g n V 2. g n : κ V τ 3. {α < κ g n+1 (α) g n (α)} G Replace each g n by a function f n : κ τ. Set f n (α) = rank(g n (α)). Clearly, still we have {α < κ f n+1 (α) f n (α)} G. But this means that N is not well-founded below the image of τ. Contradiction. Note that the opposite direction does not necessarily hold. This is because for τ (2 κ ) +, τ-almost precipitousness implies precipitousness and hence a measurable cardinal in an inner model. By Donder and Levinski [1], it is possible to have semi-precipitous cardinals in L. The following is an analog of a game that was used in [6] with connection to almost precipitous ideals. Definition 2.2 (The game G τ (F )) Let F be a normal filter on κ and let τ > κ be an ordinal. The game G τ (F ) is defined as follows: Player I starts by picking a set A 0 in F +. Player II chooses a function f 1 : A 0 τ and either a partition B i i < ξ of A 0 into ξ < κ many pieces or a sequence B α α < κ of disjoint subsets of κ so that α<κ B α A 0. The first player then supposed to respond by picking an ordinal α 2 and a set A 2 F + which is a subset of A 0 and of one of B i s or B α s. At the next stage the second player supplies again a function f 3 : A 2 τ and either a partition B i i < ξ of A 2 into ξ < κ many pieces or a sequence B α α < κ of disjoint subsets of κ so that α<κ B α A 2. 5
6 The first player then responds by picking a set A 4 F + which is a subset of A 2 and of one of the B i s or B α s on which everywhere f 1 is either above f 3 or equal to f 3 or below f 3. In addition he picks an ordinal α 4 such that α 2, α 4 respect the order of f 1 A 4, f 3 A 4, i.e. α 2 < α 4 iff f 1 A 4 < f 3 A 4, α 2 > α 4 iff f 1 A 4 > f 3 A 4 and α 2 = α 4 iff f 1 A 4 = f 3 A 4. Intuitively, α 2n pretends to represent f 2n 1 in a generic ultrapower. The game continues in the same fashion. α 2k 0 < k n should respect the order of f 2k 1 A 2n 0 < k n. Player I wins if the game continues through infinitely many moves. Otherwise Player II wins. Clearly this game is determined. The following lemma is analogous to [6] (Lemma 3). It was proved originally for Cub κ, but the referee of the paper pointed out how to generalize the argument to arbitrary normal filter F. His suggestion is implemented below. Lemma 2.3 Let κ be a regular uncountable cardinal and F a normal filter over κ. Suppose that λ is a κ-erdős cardinal (i.e. λ is the least µ such that µ (κ) <ω 2 ). Then for each ordinal τ,κ < τ < λ, Player I has a winning strategy in the game G τ (F ). Proof. Suppose otherwise. Then Player II has a winning strategy. Let σ be a winning strategy for II. We will first find a set X λ of cardinality κ such that σ does not depend on ordinals picked by Player I from X. In order to get such X let us consider a structure Let X be a set of κ indiscernibles for A. A = H(λ),, λ, κ, P(κ), F, G τ (F ), σ. Pick now a continuous elementary chain M α α < κ of submodels of H(χ) for χ > λ big enough such that for every α < κ the following hold: 1. M α < κ, 6
7 2. σ, X M α, 3. M α κ On, 4. M β β α M α+1. Set κ α = M α κ, for each α < κ. Let E = {α < κ κ α = α}. Then E is a club. For every α < κ, the set H α = {X X F M α } is in F. Hence α<κ H α is in F as well. So H E F. Pick any α H. Then α H β for each β < α = κ α. Now, for every X M α F we have X M β F for some β < α, so X H β and hence α X. Denote M α by M. Then α = M κ and for every X M F we have α X. Let us produce an infinite play in which the second player uses σ. This will give us the desired contradiction. Consider the set S = {f(α) f M, f is a partial function from κ to τ}. Obviously, S has cardinality less than κ. Hence we can fix an order preserving function π : S X. Let one start with A 0 = κ. Consider σ(a 0 ). Clearly, σ(a 0 ) M. It consists of a function f 1 : A 0 τ and, say a sequence B ξ ξ < κ of disjoint subsets of κ so that ξ<κ B ξ A 0. Now, α A 0, hence there is ξ < α such that α B ξ. Then B ξ M, as M α. Hence, A 0 B ξ M and α A 0 B ξ. Let A 2 = A 0 B ξ. Note that A 2 C, for every C F which belongs to M, since α is in both A 2 and C. Hence, by elementarity A 2 F +, so is a legitimate move in the game. Pick α 2 = π(f 1 (α)). Consider now the answer of two which plays according to σ. It does not depend on α 2, hence it is in M. Let it be a function f 3 : A 2 τ and, say a sequence B ξ ξ < κ of disjoint subsets of κ so that ξ<κ B ξ A 2. As above find ξ < α such that α B ξ. Then B ξ M, as M α. Hence, A 2 B ξ M and α A 2 B ξ. Let A 2 = A 2 B ξ. Split it into three sets C <, C =, C > such that C < = {ν A 2 f 3 (ν) < f 1 (ν)}, C = = {ν A 2 f 3 (ν) = f 1 (ν)}, C > = {ν A 2 f 3 (ν) > f 1 (ν)}. 7
8 Clearly, α belongs to only one of them, say to C <. Set then A 4 = C <. Then, clearly, A 4 M, it is in F + and f 3 (α) < f 1 (α). Set α 4 = π(f 3 (α)). Continue further in the same fashion. The next game was introduced by Donder and Levinski in [1]. Definition 2.4 A set R is called κ-plain iff 1. R, 2. R consists of normal filters over κ, 3. for all F R and A F +, F + A R, where F +A is the filter generated by F {A}, i.e. F +A = {X κ Y F X}. Y A Definition 2.5 (The game H R (F, τ)) Let R be κ-plain, F R be a normal filter on κ and let τ > κ be an ordinal. The game H R (F, τ) is defined as follows. Set F 0 = F. Let 1 i < ω. Player I plays at stage i a pair (A i, f i ), where A i κ and f i : κ τ. Player II answers by a pair (F i, γ i ), where F i R and γ i is an ordinal. The rules are as follows: 1. For 0 i < ω, A i+1 (F i ) + 2. For 0 i < ω, F i+1 F i + A i+1 Player II wins iff for all 1 i, k n < ω : (f i < Fn f k ) (γ i < γ k ) Donder and Levinski [1] showed that the existence of a winning strategy for Player II in the game H R (F, τ) for some R, F is equivalent to κ being τ- semi precipitous. The next two lemmas deal with connections between winning strategies for the games G τ (F ) and H R (F, τ). Lemma 2.6 Suppose that Player II has a winning strategy in the game H R (F, τ), for some κ-plain R, a normal filter F R over κ and an ordinal τ. Then Player I has a winning strategy in the game G τ (F ). Proof. Let σ be a winning strategy for Player II in H R (F, τ). We define a winning strategy δ for Player I in the game G τ (F ). Let the first move according to δ be κ. Suppose that Player 8
9 II responds by a function f 1 : κ τ and a partition B 1 of κ to less then κ many subsets or a sequence B 1 = B α α < κ of κ many subsets such that α<κ B α κ. Turn to the strategy σ. Let σ(κ, f 1 ) = (F 1, γ 1 ), for some F 1 F, F 1 R and an ordinal γ 1. Now we let Player I pick A 1 (F 1 ) + such that there is a set B B 1 with A 1 B (he can always choose such an A 1 because F 1 is normal and α<κ B α (F 1 ) + ) and let the respond according to δ be (A 1, γ 1 ). Player II will now choose a function f 2 : A 1 τ and a partition B 2 of A 1 or a sequence B 2 = B α α < κ, α<κ B α A 1. Back in H R (F, τ), we consider the answer according σ of Player II to (A 1, f 2 ), i.e. σ((κ, f 1 ), (A 1, f 2 )) = (F 2, γ 2 ). Choose A 2 (F 2 ) + such that there is a set B B 2 with A 2 B (it is always possible to find such A 2 because F 2 is normal and α<κ B α (F 2 ) + ) on which either f 1 < f 2 or f 1 > f 2 or f 1 = f 2. Let the respond according to δ be (A 2, γ2), where γ2 = γ 2, unless f 1 = f 2 on A 2. If f 1 = f 2 on A 2, then set γ2 = γ 1. Note that in this case the rules of H R (F, τ) allow γ 1 γ 2, but not those of G τ (F ). Continue in a similar fashion. The play will continue for infinitely many moves. Hence Player I will always win by using the strategy δ. Lemma 2.7 Suppose that Player I has a winning strategy in the game G τ (F ), for a normal filter F over κ and an ordinal τ. Then Player II has a winning strategy in the game H R (D, τ) for some κ-plain R and D R. Proof. Let σ be a winning strategy of Player I in G τ (F ). Set J = {X κ X and any of its subsets are never used by σ}, and for every finite play t = t 1,..., t 2n via σ J t = {X κ X and any of its subsets are never used by σ in the continuation of t}. It is not hard to see that such J and J t s are normal ideals over κ. Denote by D and D t the corresponding dual filters. Pick R to be a κ-plain which includes D and all D t s. We define a winning strategy δ for Player II in the game H R (D, τ). Let (A 1, g 1 ) be the first move in H R (D, τ). Then A 1 D +. Hence σ picks a subset of A 1 in a certain play t as a move of Player I in the game G τ (F ). Continue this play, and let Player II respond by a trivial partition of A 1 consisting of A 1 itself and by the function g 1 restricted to A 1. Let 9
10 (B 1, γ 1 ) be the responce of Player I according to σ. Set t 1 = t ({A 1 }, g 1 ). Then B 1 D t1. Now we set the responce of Player II according to δ to be (D t1, γ 1 ). Continue in a similar fashion. Theorem 2.8 Suppose that λ is a κ-erdős cardinal, then κ is τ-semi precipitous for every τ < λ. Proof. By Donder and Levinski [1] the existence of a winning strategy for Player II in the game H R (F, τ) for some R, F is equivalent to κ being τ- semi precipitous. The result follows now by Lemmas 2.3,2.7. Combining the above with Theorem 17 of [6], we obtain the following: Theorem 2.9 Assume that 2 ℵ 1 = ℵ 2 and f = ω 2, for some f : ω 1 ω 1. Let τ < ℵ 3. If there is a τ-semi-precipitous filter over ℵ 1, then there is a normal τ-almost precipitous filter over ℵ 1 as well. Proof. By Donder and Levinski [1], the existence of τ-semi-precipitous filter over ℵ 1 implies that Player II has a winning strategy in the game H R (F, τ) for some ℵ 1 -plain R and a normal filter F R. Then Player I has a winning strategy in the game G τ (F ), by 2.6. Then, as it is easily seen, such a strategy will be a winning strategy in the game G τ (Cub ℵ1 ) as well. Now Theorem 17 of [6] applies. The next result is a small generalization of Donder-Levinski (Theorem 8) and it is well known for precipitous ideals. Theorem 2.10 Let κ be a τ-semi-precipitous cardinal for some τ of cofinality above κ or τ = On, as witnessed by a forcing notion P and F be a correspondent τ-semi-precipitous filter, i.e. 0 P ( there is j : V τ M, crit( j ) = κ, Mis transitive), F = {X κ 0 P κ j (X)}. Suppose that Q is a κ-c.c. forcing notion of cardinality less than τ. Then, for any generic subset G Q of Q, the following hold in V [G Q ]: 1. κ remains a τ-semi-precipitous cardinal in V [G Q ]. 2. There is a forcing notion R V such that 0 R (there is j : (V τ) V [GQ] M, j j), {X κ 0 R κ j (X)} = {X κ Y F Y X}. 10
11 Proof. Clearly (2) implies (1). So let us show (2). It is enough to find some forcing S in V Q that produces j, since then it is possible to absorb all the possibilities for such S inside Col(ω, η) for a large enough η. Let G Q be a generic subset of Q and Y be any F -positive set (in V ). Pick p P which forces κ j (Y ). Suppose that (2) does not hold. Then there is q Q such that: (*) for every generic G Q Q with q Q there is no forcing notion S V [G Q ] such that for some V [G Q ]-generic subset H of S there is G P V [G Q H] generic over V with p G P and with the embedding j = j G P which extends to some j : (V τ ) V [G Q] M. Note that (V τ ) V [G Q] = V τ [G Q ]. It follows by the cardinality assumption we made and since τ is a limit ordinal. Pick a cardinal µ to be above all possible cardinalities of models M that are produced by P, if τ is an ordinal and above all possible cardinalities of 2 j (Q), if τ = On. Let η = µ + 2 P. Consider Col(ω, η). It can be viewed as P T, where T = Col(ω, η)/p. Force with this forcing (over V ). Let G P G T be a generic set with p G P. Then M (given by G P ) is countable in V [G P G T ], so there is G V [G P G T ] which is M-generic subset of j(q) with j(q) G. Set G = {t Q j(t) G }. Recall that Q is κ-c.c. forcing and κ is the critical point of j. Hence G is V -generic subset of Q with q G. Finally we consider the forcing S = Col(ω, η)/g in V [G], i.e. S = {t Col(ω, η) d G (t d G )}, where G is a Col(ω, η)-name of G. It contradicts (*) above. By Donder and Levinski [1], the existence of 0 # implies that the first indiscernible c 0 for L is, in L, τ-semi-precipitous for each τ. They showed [1](Theorem 7) that the property κ is τ-semi-precipitous relativizes down to L. Also it is preserved under κ-c.c. forcings of cardinality κ by Now combine this with 2.9. We obtain the following: Theorem 2.11 Suppose that κ is a < κ ++ -semi-precipitous cardinal in L. Let G be a generic subset of the Levy Collapse Col(ω, < κ). Then for each τ < κ ++, κ carries a τ- almost precipitous normal ideal in L[G]. Proof. In order to apply 2.9, we need to check that there is f : ω 1 ω 1 with f = ω 2. Suppose otherwise. Then by Donder and Koepke [2] (Theorem 5.1) we will have wcc(ω 1 ) 11
12 (the weak Chang Conjecture for ω 1 ). Again by Donder and Koepke [2] (Theorem D), then (ℵ 2 ) L[G] will be almost < (ℵ 1 ) L -Erdös in L. But note that (ℵ 2 ) L[G] = (κ + ) L and in L, 2 κ = κ +. Hence, in L, we must have 2 κ (ω) 2 κ, as a particular case of 2 κ being almost < ℵ 1 -Erdös. But 2 κ (3) 2 κ. Contradiction. Corollary 2.12 The following are equivalent: 1. Con( there exists an almost precipitous cardinal), 2. Con( there exists an almost precipitous cardinal with normal ideals witnessing its almost precipitousness), 3. Con(there exists < κ ++ -semi-precipitous cardinal κ). In particular the strength of existence of an almost precipitous cardinal is below 0 #. 3 An almost precipitous ideal on ω 2 In this section we will construct a model with ℵ 2 being almost precipitous. The initial assumption will be the existence of a Mahlo cardinal κ which carries a (2 κ ) + - semi-precipitous filter F with {ν < κ ν is a regular cardinal } F. Note that in general, if κ is an inaccessible which carries a (2 κ ) + -semi-precipitous filter F, then the set {ν < κ ν is a regular cardinal } need not be in F. It is possible even to have a normal precipitous filter with {τ < κ cof(τ) = ω} inside. Actually this is possible already over the first inaccessible. On the other hand if there there is no inner model with a measurable cardinal (or actually many measurable cardinals) then each (2 κ ) + -semi-precipitous filter F over κ ℵ 3 should concentrate on {ν < κ ν = cof(ν)} If κ = ℵ 2 then a slight variation of the construction below may be used to produce an example of a (2 κ ) + -semi-precipitous filter F over κ which concentrates on {ν < κ cof(ν) = ω}. By Donder and Levinski [1] the initial assumption above is compatible with V = L. If 0 exists, then the first indiscernible will be like this in L. Assume V = L. Let P i, Qj i κ, j < κ be a Revised Countable Support iteration (see [12], Chapter 10, 1) so that for each α < κ, if α is an inaccessible cardinal (in V ), then Qα is Col(ω 1, α) 12
13 which collapses α to ℵ 1 and Qα+1 is the Namba forcing which changes the cofinality of α + (which is now ℵ 2 ) to ω. In all other cases let Qα be the trivial forcing. By [12]( Chapter 11, 4,5,6), the forcing P κ turns κ into ℵ 2, preserves ℵ 1, does not add reals and satisfies the κ -c.c. Let G be a generic subset of P κ. By 2.10, a κ-c.c. forcing of cardinality κ preserves semi-precipitousness of F. Hence F is κ ++ = ℵ 4 -semi-precipitous in L[G]. In addition, {τ < κ cof(τ) = ω 1 } F and {τ < κ cof((τ + ) V ) = ω} F. Now, there is a forcing Q in L[G] so that in L[G] Q we have a generic embedding j : L κ ++[G] M such that M is transitive and κ j(a) for every A F. By elementarity, then M is of the form L λ [G ], for some λ > κ ++, and G j(p κ ) which is L λ -generic. Note that Q κ collapses κ to (ℵ 1 ) M because it was an inaccessible cardinal, and at the very next stage its successor changes the cofinality to ω. That means that there is a function H L κ ++[G] such that j(h)(κ) : ω (ℵ 3 ) L[G] is an increasing and unbounded in (κ + ) L = (ℵ 3 ) L[G] function. Just for each inaccessible in L cardinal α set H(α) : ω (α + ) L to be the generic Namba sequence. We will use such H as a replacement of the corresponding function of [5]. Together with the fact that in the model L[G] we have a filter on ℵ 2 which is ℵ 4 semi precipitous this will allow us to construct a τ- almost precipitous filter on ℵ 2, for every τ < ℵ The construction Work in L[G]. Fix τ < κ ++. Denote by B the complete Boolean algebra RO(Q). Further by we will mean the order of B. For each p B set F p = {X κ p κ j (X)} We will use the following result of Donder and Levinski [1](Theorem 4, Claim 1): Lemma p q F p F q 2. X (F p ) + iff there is a q p, q κ j(x) 3. Let X (F p ) +, then for some q p, F q = F p + X 13
14 Proof. (1) and (2) are trivial. Let us prove (3). Suppose that X (F p ) +. Set q = κ j (X) B p. We claim that F q = F p + X. The inclusion F q F p + X is trivial. Let us show that F p + X F q. Suppose not, then there are Y (F p ) +, Y X and Z F q such that Y Z =. But Y (F p ) +, so we can find s p such that s κ j (Y ). Now, s p and s κ j (X), since Y X. Hence, s q. But then s κ j (Y ), κ j (Z), j (Z Y ) =. Contradiction. Define {A nα α < κ +, n < ω} as in [5]: A nα = {η < κ H(η)(n) = h α (η)}, where h α α < κ + is a sequence of κ + canonical functions from κ to κ (in V = L[G]). Recall that 1 B j (h α )(κ) = α. Note that here H is only cofinal and not onto, as in [5]. The following lemmas were proved in [5] and hold without changes in the present context: Lemma 3.2 For every n < ω there is an ordinal α < κ + such that A nα (F 1B ) +. Lemma 3.3 For every α < κ + and p B there is n < ω and α < β < κ + such that A nβ (F p ) +. Lemma 3.4 Let n < ω and p B. Then the set: {A nα α < κ + and A nα (F p ) + } is a maximal antichain in (F p ) +. The following is an analog of a lemma due Assaf Rinot in [5], 3.5. Lemma 3.5 Let D be a family of κ + dense subsets of B, there exists a sequence p α α < κ + such that for all Z (F 1B ) +,p Q and n < ω if Z n,p = {α < κ + A nα Z (F p ) + } has cardinality κ + then : 14
15 1. For any p B there exists α Z n,p with p p α. 2. For any D D there exists α Z n,p with p α κ j(a nα Z),p α p and p α D. Proof. Let {S i i < κ + } [κ + ] κ+ be some partition of κ +, {D α α < κ + } an enumeration of D,{q α α < κ + } an enumeration of Q and let be a well ordering of κ + κ + κ + of order type κ +. Now, fix a surjective function ϕ : κ + {(Z, n, p) ((F 1B ) +, ω, Q) Z n,p = κ + }. We would like to define a function ψ : κ + κ + κ + κ + and the sequence p α α < κ +. For that, we now define two sequences of ordinals {L α α < κ + }, {R α α < κ + } and the values of ψ and the sequence on the intervals [L α, R α ] by recursion on α < κ +. For α = 0 we set L 0 = R 0 = 0,ψ(0) = 0 and p 0 = q 0. Now, suppose that {L β, R β β < α} and ψ β<α [L β, R β ] were defined.take i to be the unique index such that α S i.let (Z, n, p) = ϕ(i) and set L α = min(κ + \ β<α [L β, R β ]), R α = min(z n,p \ L α ). Now, for each β [L α, R α ] we set ψ(β) = t,where: t = min (κ + {i} κ + ) \ ψ (Z n,p L α ). If t κ + then we set p β = q t for each β [L α, R α ].Otherwise, t = (i, δ) for some δ < κ + and because A nrα Z F p + and D δ is dense we can find some q D δ, q p, q κ j (A nrα Z) and set p β = q for each β [L α, R α ].This completes the construction. Now, we would like to check that the construction works. Fix Z F 1 + B p Q and n < ω so that Z n,p = κ +.Let i < κ + be such that ϕ(i) = Z n,p and notice that the construction insures that ψ Z n = κ + {i} κ +. (1) Let p B. There exists a t < κ + so that q t p.let α Z n be such that ψ(α) = t, so p α = q t p. (2) Let D D. There exist δ < κ + and α Z n,p such that D δ = D and ψ(α) = (i, δ).then, by the construction we have that p α D δ, p α κ j(a nα Z) and p α p. Define D = {D f f (τ κ ) V }, where D f = {p B γ On p j( ˇf)(κ) = ˇγ} and let p α α < κ + be as in lemma 3.5. We turn now to the construction of filters which will be similar to those of [5]. Start with n = 0. Let α < κ +. Consider three cases: 15
16 Case I: If {ξ < κ + A 0ξ (F 1B ) + } = κ + and p α κ j(a 0α ) then we define q <α> = p α and we associate F q<α> to the sequence < α >. Case II: If I fails but A 0α (F 1B ) + then we define q <α> = κ j(a 0α ) B and we associate F q<α> to the sequence < α >. Case III: If A 0α ˇF 1B (the dual ideal of F 1B ) then q <α> is not defined. Notice that by Lemma 3.2, there exists some α < κ + with A 0α (F 1B ) +, thus {α < κ + q α is defined } is non-empty. Definition 3.6 Set F 0 = {q α α < κ +, q α is defined }, and denote the corresponding dual ideals by I q α and I 0. Clearly, I 0 = {I q α α < κ +, q α is defined }. Also, F 0 F 1B and I 0 ˇF 1B, since each F q α F 1B and I q α ˇF 1B. Note that F 0 is a κ complete, normal and proper filter since it is an intersection of such filters and also I 0 is a κ complete, normal and proper ideal. We now describe the successor step of the construction, i.e., m = n + 1. Let σ : m κ + be a function with q σ defined and α < κ +. There are three cases: Case I: If {ξ < κ + A mξ F q + σ } = κ +, p α q σ and p α κ j(a mα ), then we define q σ α = p α and associate F qσ α to the sequence σ α. Case II: If Case I fails, but A mα (F + q σ ), then let q σ α = κ j(a mα ) B q σ, and associate F qσ α to the sequence σ α. Case III: If A mα I qσ, then q σ α is not defined. This completes the construction. Definition 3.7 Let F n+1 = {F qσ σ : n + 2 κ +, F qσ is defined }, and define the corresponding dual ideals I n+1, I qσ. Notice that each F n and I n is κ-complete, proper and normal, as an intersection of such filters and ideals respectively. Definition 3.8 Let F ω be the closure under ω intersections of n<ω F n. Let I ω = the closure under ω unions of n<ω I n. Lemma 3.9 F F 0... F n... F ω and I I 0... I n... I ω, and I ω is the dual ideal to F ω. 16
17 Lemma 3.10 Let s : m κ + with q s defined; then: 1. {α < κ + q s α is defined } = {ξ < κ + A mξ F + q s }; 2. There exists an extension σ s such that q σ is defined and {ξ < κ + A dom(σ)ξ F + q σ } = κ +. Proof. 1) is clear from the construction above. For 2), let us assume that for every extension σ s such that q σ is defined : {ξ < κ + A dom(σ)ξ F + q σ } κ. That means that Σ = {σ : n κ + n m, σ s and q σ is defined } is of cardinality less than or equal to κ, so ν = σ Σ ran(σ) is less than κ+ and q s will force that j(h)(κ) is bounded by ν, contradiction. From now on the proof that F ω is the desired filter will be almost the same as in [5]( Theorem 2.1)(just some, minor adjustments should be made because here we restricting to filters rather than to sets in [5]. A more detailed argument will be provided further in the proof of 4.1). 4 Constructing almost precipitous ideals from semiprecipitousness Suppose κ is a λ semi-precipitous cardinal for some ordinal λ which is a successor ordinal > κ or a limit one with cof(λ) > κ. Let P be a forcing notion witnessing this. Then, for each generic G P, in V [G] we have an elementary embedding j : V λ M with cp(j) = κ and M transitive. Consider U = {X κ X V, κ j(x)}. Then U is a V normal ultrafilter over κ. Let i U : V V κ V/U be the corresponding elementary embedding. Note that V κ V/U need not be well founded, but it is well founded up to the image of λ. Thus, denote V κ V/U by N. Define k : (V i(λ) ) N M in a standard fashion by setting k([f] U ) = j(f)(κ), 17
18 for each f : κ V λ, f V. Then k will be elementary embedding, and so (V i(λ) ) N is well founded. For every p P set F p = {X κ p κ j (X)}. Clearly, if G is a generic subset of P with p G and U G is the corresponding V -ultrafilter, then F p U G. Note that, if for some p P the filter F p is κ + -saturated, then each U G with p G will be generic over V for the forcing with F p -positive sets. Thus, every maximal antichain in F + p consists of at most κ many sets. Let A ν ν < κ V be such a maximal antichain. Without loss of generality we can assume that min(a ν ) > ν, for each ν < κ. Then there is ν < κ with κ j(a ν ). Hence A ν U G and we are done. It follows that in such a case N which is the ultrapower by U G is fully well founded. See, for example, T. Jech [9], Lemma on page 427. Note that in general if some forcing P produces a well founded N, then κ is -semiprecipitous. Just i and N will witness this. Our aim will be to prove the following: Theorem 4.1 Assume that 2 κ = κ + and κ carries a λ-semi-precipitous filter for some limit ordinal λ with cof(λ) > κ. Suppose in addition that there is a forcing notion P witnessing the λ-semi-precipitous with corresponding N ill founded. Then 1. if λ < κ ++, then κ is λ-almost precipitous, as witnessed by a normal filter, 2. if λ κ ++, then κ is an almost precipitous, as witnessed by normal filters. Proof. The proof will be based on an extension of the method of constructing normal filters of [5] which replaces restrictions to positive sets by restrictions to filters. An additional idea will be to use a witness of the non-well-foundedness in the construction in order to limit it to ω many steps. Let κ, τ, P be as in the statement of the theorem. Preserve the notation that we introduced above. Note also that we use here a poset rather than a Boolean algebra in the previous section. So, 0 P will replace 1 B and the order will be used in the opposite direction. Then 0 P (V i(λ) ) N is well founded and N is ill founded. 18
19 Fix a sequence g n n < ω of names of functions witnessing the ill foundedness of N, i.e. 0 P [ g n] > [ g n+1], for every n < ω. Note that, as was observed above, for every p P, the filter F p is not κ + -saturated. Fix some τ < κ ++, τ λ. We should construct a normal τ-almost precipitous filter over κ. For each p P choose a maximal antichain {A pβ β < κ + } in F p +. Let f α α < κ + enumerate all the functions from κ to τ. Fix an enumeration X α α < κ + of F 0 + P. Start now an inductive process of extending F 0P. Let n = 0. Assume for simplicity that there is a function g 0 : κ On V so that 1 P ǧ 0 = g 0. Set p = 0 P. We construct inductively a sequence of ordinals ξ β β < κ + and a sequence of conditions p β β < κ +. Let α < κ +. Case I. There is a ξ < κ + so that ξ ξ β for every β < α, and X α A p ξ F p +. Then let ξ α be the least such ξ. We would like to attach an ordinal to f ξ0α and to decide g 1. Let us pick p P such that 1. p κ j(x α A p ξ α ), 2. for some γ, p j(f ξ α )(κ) = γ, 3. there is a function g 1 : κ On, g 1 V such that p ǧ 1 = g 1. Note that then p ǧ 1 < ǧ 0, since 0 P g 1 < ǧ 0. Set p α = p. Case II. Not Case I. Then set ξ α = 0. We will leave p α undefined in this case. Note that if Case I fails then we have X α β<κ A 0P ξ τ(β) mod F 0P for a surjective τ : κ α. Set F 0 = {F p α α < κ + and p α is defined }, and denote the corresponding dual ideals by I p α and I 0. Clearly, I 0 = {I p α α < κ + and p α is defined }. Also, F 0 F 0P and I 0 ˇF 0P, since 19
20 each F p α F 0P and I p α ˇF 0P. Note that F 0 is a κ complete, normal and proper filter since it is an intersection of such filters and also I 0 is such an ideal. We now describe a successor step of the construction. Let σ : m κ + with p σ defined. Construct by induction a sequence of ordinals ξ σ β β < κ + and a sequence of conditions p σ β β < κ +. Let α < κ + : Case I. There is ξ < κ + so that ξ ξ σ β for every β < α and X α A pσ ξ F + p σ. Then let ξ σ α be the least such ξ. We would like to attach an ordinal to f ξσ α that 1. p p σ, 2. p κ j(x α A 0P ξ σ α ), 3. for some γ, p j(f ξσ α )(κ) = γ, and to decide g m. Let us pick p P such 4. there is a function g m : κ On, g m V such that p ǧ m = g m. Note that then p ǧ m < ǧ m 1, since 0 P g m < g m 1 and p σ ǧ m 1 = g m 1. Set p σ α = p. Case II. Not Case I. Then set ξ σ α = 0. We will leave p σ α undefined in this case. This completes the construction. Set F m = {F pσ α σ : m κ +, α < κ + and p σ α is defined }, and denote the corresponding dual ideals by I pσ α and I m. We will use the following: Definition 4.2 Let F ω be the closure under ω intersections of n<ω F n. Let I ω = the closure under ω unions of n<ω I n. Lemma 4.3 F 0... F n... F ω and I 0... I n... I ω, and I ω is the dual ideal to F ω. Our purpose now will be to show that we cannot continue the construction further beyond ω. Then we will be able to show that F ω is a τ -almost precipitous filter. Lemma 4.4 F + ω = {F pσ σ <ω κ +, p σ is defined}. 20
21 Proof. Let X (F ω ) + and assume that X F pσ for each σ [κ + ] <ω such that p σ is defined. Note that then there are at most κ many σ s such that X F p + σ. It is enough to argue that for every σ [κ + ] <ω such that p σ is defined, the set {ζ < κ + X A pσζ F p + σ } is of cardinality less than or equal κ. Suppose otherwise. Let α < κ + be such that X = X α. Then p σ α is defined according to Case I and X F p α. Contradiction. Set T = {σ [κ + ] <ω p σ is defined, X F p + σ } and for every σ T let B σ = {ζ < κ + X A pσ ζ F + p σ }. Then both sets are of cardinality at most κ. Let σ T. Assume that B σ = κ. The case B σ < κ is treated similar and simpler. Fix ψ σ : κ B σ, for every σ T. Note that X \ β<κ A pσψσ(β) is in the ideal I pσ. Now, let n = 0. Turn the family {A 0P ψ (γ) γ < κ} into a family of disjoint sets as follows: A 0 P ψ (0) := A 0P ψ (0) \ {0} and for each γ < κ let A 0 P ψ (γ) := A 0P ψ (γ) \ ( A 0P ψ (β) (γ + 1)). β<γ Note that β<κ A 0 P ψ (β) = {ν < κ β < ν so that ν A 0 P ψ (β)} and, because ν A 0 P ψ (β) implies that ν > β, we get that the right hand side is equal to {A 0P ψ (γ) γ < κ}. Also note that β<κ A 0 P ψ (β) = β<κ A 0P ψ (β). So {X A 0 P ψ (γ) γ < κ} is still a maximal antichain in F 0 + P mod F 0P. Set R 0 := X \ β<κ A 0 P ψ (β). Then R 0 I 0P. Turn now to n = 1. Let σ T be a sequence of the length 1. below X and X β<κ A 0 P ψ (β) 21
22 We turn the family {A pσ,ψ σ (γ) γ < κ} into a disjoint one {A p σ,ψ σ (γ) γ < κ} exactly as above. Set R σ = (X A p ξ σ ) \ ( A p σ ψ σ (γ) {δ < κ g 1 (δ) < g 0 (δ)}). γ<κ Then R σ I pσ. Define R 1 = {R σ σ T and the length of σ is 1 }. Claim 1 R 1 I 0. Proof. Suppose otherwise. Then R 1 (F 0 ) +. Note that R 1 {X A p ξ σ σ T and the length of σ is 1 } and that the right hand side is a disjoint union. Maximality of the family {X A p ξ σ σ T and the length of σ is 1 } implies that R 1 A p ξ σ F p + σ, for some σ T of the length 1. But R 1 A p ξ σ = R σ and R σ I pσ, contradiction. of the claim. Continue in a similar fashion for each n < ω. We will have R n I n 1. Set R ω := R n. n<ω Then R ω I ω and X \ R ω (F ω ) +. Now, let α X \ R ω. Using disjointness of A s we can find a sequence σ [κ + ] ω such that α n<ω(a p σ n ξ σ n {ν < κ g n+1 (ν) < g n (ν)}). Then g n+1 (α) < g n (α), for every n < ω. Contradiction. Lemma 4.5 The generic ultrapower by F ω is well founded up to the image of τ. Proof. Suppose that h n n < ω is a sequence of (F ω ) + -names of old (in V) functions from κ to τ. Let G (F ω ) + be a generic ultrafilter. Choose X 0 G and a function h 0 : κ τ, h 0 V so that X 0 F + ω ȟ 0 = h 0. Let α 0 < κ + be so that f α0 = h 0. By Lemma 4.4, we can find σ 0 [κ + ] <ω such that p σ0 is defined and X 0 F pσ0. Note that at the next stage of the construction there will be β 0 with A pσ0 α 0 F pσ, and so the value of j(f α0 )(κ) 0 β 0 22
23 will be decided. Denote this value by γ 0. Set Y 0 = X 0 A pσ0 α 0 {A pσ0 iξ σ0 i i length(σ 0)}. Assume for simplicity that Y 0 is in G (otherwise we could replace X 0 by another positive set using density). Continue below Y 0 and pick X 1 G and a function h 1 : κ τ, h 1 V such that 1. X 1 F + ω ȟ 1 = h 1, 2. {ν < κ h 1 (ν) < h 0 (ν)} X 1 or {ν < κ h 1 (ν) h 0 (ν)} X 1. If {ν < κ h 1 (ν) h 0 (ν)} X 1, then [h 0 ] G [h 1 ] G and so the sequence [h n ] G n < ω is not strictly decreasing. So, suppose that {ν < κ h 1 (ν) < h 0 (ν)} X 1. Let α 1 < κ + be so that f α1 = h 1. By Lemma 4.4, we can find σ 1 [κ + ] <ω such that F pσ1 is defined, σ 1 σ 0 and X 1 F pσ1. Again, note that at the next stage of the construction there will be β 1 with A pσ1 α 1 F pσ, and so the value of j(f α1 )(κ) will be decided. Denote this value by γ 1. Then 1 β 1 we must have γ 1 < γ 0, since p σ 1 β 1 j (h 1) = γ 1, p σ 0 β 0 j (h 0) = γ 0, p σ 1 β 1 p σ 0 β 0, p σ 1 β 1 κ j (X 1) and {ν < κ h 1 (ν) < h 0 (ν)} X 1. Continue the process for every n < ω. There must be k < m < ω such that γ k γ m and Y m G. So the sequence [h n ] G n < ω is not strictly decreasing. Let us deduce now some conclusions concerning the existence of almost precipitous filters. The following answers a question raised in [6]. Corollary 4.6 Assume 0. Let η be a regular cardinal in L. Then there is a generic extension L[G] of L such that 1. L[G] and L have the same cardinals cardinals η, 2. in L[G], η + is an almost precipitous, as witnessed by a normal filter. Proof. Let c > η be a Silver indiscernible for L. By Donder, Levinski [1](Theorem 6) c is a semi-precipitous cardinal in L. Let G Col(η, < c) be L-generic. Then by [1](Theorem 8) 23
24 or by 2.10, c = (η + ) L[G] will be a semi-precipitous cardinal in L[G]. Now the assumptions of 4.1 are satisfied, since it is impossible to have a non-trivial elementary embedding from L to a transitive model N in a generic extension of L. Note that there is no generic extension of L in which the successor of a singular in L cardinal is semi-precipitous (almost precipitous). Just if η is a singular cardinal in L and in a forcing extension L[G] we have η is preserved and η + is a semi-precipitous, then there will be unboundedly many regular L-cardinals below η that change their cofinality in L[G]. This contradicts the Jensen Covering Lemma. Corollary 4.7 Assume that there are class many Ramsey cardinals. Then every uncountable cardinal is almost precipitous, as witnessed by normal filters. Proof. It follows from 2.3 and 4.1. We would like to apply 4.1 in order to characterize almost precipitous cardinals in the model L[U]. The following is likely well known: Theorem 4.8 Assume V = L[U] with U a normal ultrafilter over κ. Then 1. U is the only -semi-precipitous filter. 2. U is the only normal precipitous filter. 3. If W is a precipitous filter, then there are n ω and a partition A k k n such that (a) for every k n, U, W + A k is an ultrafilter over κ isomorphic to a finite power of (b) W = k n (W + A k). Proof. Both (1) and (2) follow immediately from the following general fact about core models: if i : K i(k) is an embedding of a core model formed in its set generic extension, then i is an iterated ultrapower of K by its extenders, see [10] or [13](Theorem 7.4.8). The item (3) requires a little additional argument. Let W be a precipitous filter over some cardinal δ. Force with W + and let i : L[U] L[i(U)] be the corresponding embedding. Then i = i α for some α, where i α is the embedding of the α-th iterated ultrapower by U. 24
25 Claim 2 α < ω, i.e. the iteration is finite. Proof. Suppose otherwise. Then i(κ) κ ω, where κ ω = i ω (κ). Consider η = [id]. Then η is an ordinal in L[i α (U)] the α-th iterated ultrapower. Then there are m < ω, γ 1 <... < γ m η and f : κ m On, f L[U] such that η = i α (f)(κ γ1,..., κ γm ). Pick some n < ω, such that κ n {κ γ1,..., κ γm }. Now, κ n is an ordinal in a generic ultrapower by W, hence for some function g L[U] we must have i(g)(η) = κ n. But then κ n = i(g)(η) = i(g)(i(f)(κ γ1,..., κ γm )) = i(gf)(κ γ1,..., κ γm ). This is impossible since κ n {κ γ1,..., κ γm }. of the claim. Finally back in the ground model, i.e. in L[U], the set S = {ξ X W + X i = i ξ }. By the claim, S ω. Consider a maximal antichain A k k τ of W -positive sets which decide i differentely. Then τ = S and, for every k S, W + A k is an ultrafilter over κ isomorphic to a finite power of U, W = k n (W + A k). Corollary 4.9 Assume V = L[U] with U a normal ultrafilter over κ. Then 1. Every regular uncountable cardinal less than κ is almost precipitous, as witnessed by normal filters and non precipitous. 2. For each τ κ +, κ carries a normal τ-almost precipitous non precipitous filter. Proof. Let η be a regular cardinal less than κ. By 2.8, η is < κ-semi-precipitous. Note that no cardinal less than κ can be -semi precipitous, see for example [10](Lemma 3.47). Hence, η is an almost precipitous, as witnessed by a normal filter, by 4.1. This proves (1). Now, A = {η < κ η is an almost precipitous, as witnessed by a normal filter and a non-precipitous } is in U. Hence, in M κ V/U, for each τ < (κ ++ ) M there is a normal τ-almost precipitous non-precipitous filter F τ over κ. Then F τ remains such also in V, since κ M M. 25
26 We do not know if (2) remains valid once we replace τ κ + by τ < κ It is unclear to us if it is possible to have a cardinal κ which carries a normal precipitous filter, but κ is not κ + -semi-precipitous via some generic ultrafilter with illfounded ultrapower. Let us turn to the case of -semi precipitous cardinals, which was not covered by Theorem 4.1. Combining constructions of [5] with the present ones (mainly, replacing restrictions to sets by restrictions to filters; required adjustments were explained in the proof of 4.1 and also will be described below in the proof of 4.13) we obtain the following. Theorem 4.10 Assume that ℵ 1 is -semi precipitous and 2 ℵ 1 = ℵ 2. Suppose that for some forcing P witnessing this 0 P P i (ℵ 1 ) > (ℵ + 1 ) V. Then ℵ 1 is almost precipitous witnessed by normal filters. Remark 4.11 Note that, if for some limit λ κ ++ of cofinality bigger than κ there is a λ-semi-precipitous filter with a witnessing forcing producing an ill founded ultrapower, then by 4.1, κ is almost precipitous, as witnessed by normal filters, and without the additional assumptions made in the theorem above. We do not know whether -precipitousness implies always that such λ (i.e. one with ill founded ultrapower) exists. Theorem 4.12 Suppose that there is no inner model satisfying ( α that ℵ 1 is -semi precipitous and 2 ℵ 1 o(α) = α ++ ). Assume = ℵ 2. If ℵ 3 is not a limit of measurable cardinals of the core model, then there exists a normal precipitous ideal on ℵ 1. 2 The referee of the paper answered this question affirmatively. Below is his nice argument. It is enough to find arbitrary large λ and a forcing P witnessing the λ-semi-precipitousness of κ, such that the corresponding normal ultrapower N of V is ill-founded. Then 4.1 can be used to get our witnesses to almost precipitousness, but we must also make sure that the normal filter produced differs from U. Let j : V M Ult(V, U) be the U-ultrapower embedding; note that j(κ ++ ) = κ ++. Fix a limit cardinal λ of cofinality κ +, λ = j(λ). Then κ λ M λ = L λ [j(u)]. Force over M with Col(ω, λ). In M[G], there is an embedding j : M λ j(m λ) = j(m) λ, with crit(j ) = κ, since j M λ : M λ j(m) λ is an embedding of this type. Let D be the normal M λ-ultrafilter derived from j. Then D is in fact an M-ultrafilter and a V -ultrafilter. But N = Ult(M, D) cannot be well-founded, since M satisfies no cardinal less than j(κ) is -semi-precipitous, since this occurs at κ in V. Now if G is in fact V -generic, then Ult(V λ, D) (formed with all functions in κ λ) will be well founded, since κ λ M λ. Therefore also Ult(V λ, D) is well founded. But Ult(V, D) is illfounded, since Ult(M, D) is illfounded. So Col(ω, λ) is a forcing P as required. Apply now the proof of 4.1.Since P forces that the ultrapower N is illfounded, P must force that the derived ultrafilter U is not U. Pick some A κ, A U and p P which forces A U. Fix τ < κ ++, run the construction of 4.1 starting with F p instead of F 0P. This will give A F ω. Then F ω will be τ-almost precipitous, and non-precipitous. 26
27 Theorem 4.13 Assume that ℵ 1 is -semi precipitous. Let P be a witnessing this forcing such that 0 P P i (ℵ 1 ) > (ℵ + 1 ) V. Then, after forcing with Col(ℵ 2, P ), there will be a normal precipitous filter on ℵ 1. Sketch of the proof of Let P be a forcing notion witnessing -semi precipitousness such that 0 P P i (ℵ 1 ) > (ℵ + 1 ) V. Fix a function H such that for some p P p P i (H)(κ) : ω onto (κ + ) V, where here and further κ will stand for ℵ 1. Assume for simplicity that p = 0 P. Let h α α < κ + be a sequence of the canonical functions from κ to κ. For every α < κ + and n < ω set A nα = {ν H(ν)(n) = h α (ν)}. Set F p = {X κ p κ i (X)}, for each p P. Then, the following hold: Lemma 4.14 For every α < κ + and p P there is n < ω so that A nα F + p. Lemma 4.15 Let n < ω and p P. Then the set {A nα α < κ + and A nα F + p } is a maximal antichain in F + p. Denote by Col(ℵ 2, P ) = {t t is a partial function of cardinality at most ℵ 1 from ℵ 2 to P }. Let G Col(ℵ 2, P ) be a generic and C = G. We extend F 0P now as follows. Start with n = 0. If {α A 0α F 0 + P } < κ +, then set F 0 = F 0P. Suppose otherwise. Let α < κ +. If A 0α in the ideal dual to F 0P, then F α will be undefined. 27
28 If A 0α F 0 + P, then we consider F C(α). If A 0α F + C(α), then pick some p α P forcing κ i (A 0α ) and set F α = F p α. If A 0α F + C(α), then pick some p α P, p α C(α) forcing κ i (A 0α ) and set F α = F p α. Set F 0 = {F α α < κ +, F α is defined }. Let now n = 1. Fix some γ < κ + with F γ defined. If {α A 1α F + γ } < κ+, then we do nothing. Suppose that it is not the case. Let α < κ +. We define F γ,α as follows: if A 1α F + γ, then F γ,α will be undefined, if A 1α F + γ, then consider F C(α). If there is no p stronger than both C(α), p γ and forcing κ i (A 1α ), then pick some p γ,α p γ which forces κ i (A 1α ) and set F γ,α = F p γ,α. Otherwise, pick some p γ,α C(α), p γ which forces κ i (A 1α ) and set F γ,α = F p γ,α. Set F 1 = {F γ,α α, γ < κ +, F γ,α is defined }. Continue by induction and define similar filters F s, F n and conditions p s for each n < ω, s [κ + ] <ω. Finally set F ω = the closure under ω intersections of F n. The arguments like those of 4.1 transfer directly to the present context. We refer to [5] which contains more details. Let us prove the following crucial lemma. Lemma 4.16 F ω is a precipitous filter. n<ω Proof. Suppose that g n n < ω is a sequence of F + ω -names of old (in V ) functions from κ On. Let G F ω + be a generic ultrafilter. Pick a set X 0 G and a function g 0 : κ On in V such that X 0 F + ω g 0 = ǧ 0. Pick some t 0 Col(ℵ 2, P ), t 0 C such that t 0, X 0 Col(ℵ2,P ) F + ω g 0 = ǧ 0 28
29 and for some s 0 = ξ 0,..., ξ n [κ + ] <ω t 0 X 0 F s 0, moreover, for each i n, ξ i dom(t 0 ) and t 0 (ξ n ) = p s0. We shrink X 0 to a set X0 = X 0 A iξi. i n Clearly, still t 0 X 0 F s 0. + Claim 3 For each t, Y Col(ℵ 2, P ) F ω t, Y, ρ 0 On and s 0 extending s 0 such that with t, Y t 0, X 0 there are q 0, Z 0 1. q 0 (s 0( s 0 )) p s 0, 2. q 0 Col(ℵ2,P )Ž0 F s 0, 3. p s 0 P i (g 0 )(κ) = ˇρ 0. Proof. Suppose for simplicity that t, Y = t 0, X0. We know that t 0 decides F s0, t 0 (s 0 ( s 0 )) = p(s 0 ) and X 0 F s0. Find s extending s 0 of the smallest possible length such that the set B = {α A s α F s + 0 } has cardinality κ +. Remember that we do not split F s0 before getting to such s. Pick some α B\dom(t 0 ). A s α F s + 0, hence there is some p P, p p s0 which forces κ i (A s α ). Find some p P, p p and ρ 0 such that p P i (g 0 )(κ) = ρ 0. Extend now t 0 to q 0 by adding to it α, p. Let s 0 = s α and Z 0 = X 0 A s α. of the claim. By the genericity we can find q 0, Z 0, s 0 as above with q 0, Z 0 C G. Back in V [C, G], find X 1 Z 0 in G and a function g 1 : κ On in V such that X 1 F + ω g 1 = ǧ 1. Assume [g 1 ] G < [g 0 ] G. Then we can pick X 1 to be a subset of {ν g 1 (ν) < g 0 (ν)}. Proceed as above only replacing X 0 by X 1. This will define q 1, Z 1, s 1 and ρ 1 for g 1 as in the claim. Note 29
On almost precipitous ideals.
On almost precipitous ideals. Asaf Ferber and Moti Gitik July 21, 2008 Abstract We answer questions concerning an existence of almost precipitous ideals raised in [5]. It is shown that every successor
More informationSilver type theorems for collapses.
Silver type theorems for collapses. Moti Gitik May 19, 2014 The classical theorem of Silver states that GCH cannot break for the first time over a singular cardinal of uncountable cofinality. On the other
More informationStrongly compact Magidor forcing.
Strongly compact Magidor forcing. Moti Gitik June 25, 2014 Abstract We present a strongly compact version of the Supercompact Magidor forcing ([3]). A variation of it is used to show that the following
More informationExtender based forcings, fresh sets and Aronszajn trees
Extender based forcings, fresh sets and Aronszajn trees Moti Gitik August 31, 2011 Abstract Extender based forcings are studied with respect of adding branches to Aronszajn trees. We construct a model
More informationON NORMAL PRECIPITOUS IDEALS
ON NORMAL PRECIPITOUS IDEALS MOTI GITIK SCHOOL OF MATHEMATICAL SCIENCES RAYMOND AND BEVERLY SACKLER FACULTY OF EXACT SCIENCE TEL AVIV UNIVERSITY RAMAT AVIV 69978, ISRAEL Abstract. An old question of T.
More informationSy D. Friedman. August 28, 2001
0 # and Inner Models Sy D. Friedman August 28, 2001 In this paper we examine the cardinal structure of inner models that satisfy GCH but do not contain 0 #. We show, assuming that 0 # exists, that such
More informationbeing saturated Lemma 0.2 Suppose V = L[E]. Every Woodin cardinal is Woodin with.
On NS ω1 being saturated Ralf Schindler 1 Institut für Mathematische Logik und Grundlagenforschung, Universität Münster Einsteinstr. 62, 48149 Münster, Germany Definition 0.1 Let δ be a cardinal. We say
More informationA HIERARCHY OF RAMSEY-LIKE CARDINALS
A HIERARCHY OF RAMSEY-LIKE CARDINALS PETER HOLY AND PHILIPP SCHLICHT Abstract. We introduce a hierarchy of large cardinals between weakly compact and measurable cardinals, that is closely related to the
More informationAnnals of Pure and Applied Logic
Annals of Pure and Applied Logic 161 (2010) 895 915 Contents lists available at ScienceDirect Annals of Pure and Applied Logic journal homepage: www.elsevier.com/locate/apal Global singularization and
More informationGeneralization by Collapse
Generalization by Collapse Monroe Eskew University of California, Irvine meskew@math.uci.edu March 31, 2012 Monroe Eskew (UCI) Generalization by Collapse March 31, 2012 1 / 19 Introduction Our goal is
More informationContinuous images of closed sets in generalized Baire spaces ESI Workshop: Forcing and Large Cardinals
Continuous images of closed sets in generalized Baire spaces ESI Workshop: Forcing and Large Cardinals Philipp Moritz Lücke (joint work with Philipp Schlicht) Mathematisches Institut, Rheinische Friedrich-Wilhelms-Universität
More informationPhilipp Moritz Lücke
Σ 1 -partition properties Philipp Moritz Lücke Mathematisches Institut Rheinische Friedrich-Wilhelms-Universität Bonn http://www.math.uni-bonn.de/people/pluecke/ Logic & Set Theory Seminar Bristol, 14.02.2017
More informationA precipitous club guessing ideal on ω 1
on ω 1 Tetsuya Ishiu Department of Mathematics and Statistics Miami University June, 2009 ESI workshop on large cardinals and descriptive set theory Tetsuya Ishiu (Miami University) on ω 1 ESI workshop
More informationTwo Stationary Sets with Different Gaps of the Power Function
Two Stationary Sets with Different Gaps of the Power Function Moti Gitik School of Mathematical Sciences Tel Aviv University Tel Aviv 69978, Israel gitik@post.tau.ac.il August 14, 2014 Abstract Starting
More informationJanuary 28, 2013 EASTON S THEOREM FOR RAMSEY AND STRONGLY RAMSEY CARDINALS
January 28, 2013 EASTON S THEOREM FOR RAMSEY AND STRONGLY RAMSEY CARDINALS BRENT CODY AND VICTORIA GITMAN Abstract. We show that, assuming GCH, if κ is a Ramsey or a strongly Ramsey cardinal and F is a
More informationGlobal singularization and the failure of SCH
Global singularization and the failure of SCH Radek Honzik 1 Charles University, Department of Logic, Celetná 20, Praha 1, 116 42, Czech Republic Abstract We say that κ is µ-hypermeasurable (or µ-strong)
More informationADDING A LOT OF COHEN REALS BY ADDING A FEW II. 1. Introduction
ADDING A LOT OF COHEN REALS BY ADDING A FEW II MOTI GITIK AND MOHAMMAD GOLSHANI Abstract. We study pairs (V, V 1 ), V V 1, of models of ZF C such that adding κ many Cohen reals over V 1 adds λ many Cohen
More informationSTRONGLY UNFOLDABLE CARDINALS MADE INDESTRUCTIBLE
The Journal of Symbolic Logic Volume 73, Number 4, Dec. 2008 STRONGLY UNFOLDABLE CARDINALS MADE INDESTRUCTIBLE THOMAS A. JOHNSTONE Abstract. I provide indestructibility results for large cardinals consistent
More informationLARGE CARDINALS AND L-LIKE UNIVERSES
LARGE CARDINALS AND L-LIKE UNIVERSES SY D. FRIEDMAN There are many different ways to extend the axioms of ZFC. One way is to adjoin the axiom V = L, asserting that every set is constructible. This axiom
More information2. The ultrapower construction
2. The ultrapower construction The study of ultrapowers originates in model theory, although it has found applications both in algebra and in analysis. However, it is accurate to say that it is mainly
More informationNotes on getting presaturation from collapsing a Woodin cardinal
Notes on getting presaturation from collapsing a Woodin cardinal Paul B. Larson November 18, 2012 1 Measurable cardinals 1.1 Definition. A filter on a set X is a set F P(X) which is closed under intersections
More informationSHORT EXTENDER FORCING
SHORT EXTENDER FORCING MOTI GITIK AND SPENCER UNGER 1. Introduction These notes are based on a lecture given by Moti Gitik at the Appalachian Set Theory workshop on April 3, 2010. Spencer Unger was the
More informationMITCHELL S THEOREM REVISITED. Contents
MITCHELL S THEOREM REVISITED THOMAS GILTON AND JOHN KRUEGER Abstract. Mitchell s theorem on the approachability ideal states that it is consistent relative to a greatly Mahlo cardinal that there is no
More informationShort Extenders Forcings II
Short Extenders Forcings II Moti Gitik July 24, 2013 Abstract A model with otp(pcf(a)) = ω 1 + 1 is constructed, for countable set a of regular cardinals. 1 Preliminary Settings Let κ α α < ω 1 be an an
More informationChain conditions, layered partial orders and weak compactness
Chain conditions, layered partial orders and weak compactness Philipp Moritz Lücke Joint work with Sean D. Cox (VCU Richmond) Mathematisches Institut Rheinische Friedrich-Wilhelms-Universität Bonn http://www.math.uni-bonn.de/people/pluecke/
More informationThe Semi-Weak Square Principle
The Semi-Weak Square Principle Maxwell Levine Universität Wien Kurt Gödel Research Center for Mathematical Logic Währinger Straße 25 1090 Wien Austria maxwell.levine@univie.ac.at Abstract Cummings, Foreman,
More informationCharacterizing large cardinals in terms of layered partial orders
Characterizing large cardinals in terms of layered partial orders Philipp Moritz Lücke Joint work with Sean D. Cox (VCU Richmond) Mathematisches Institut Rheinische Friedrich-Wilhelms-Universität Bonn
More informationThe (λ, κ)-fn and the order theory of bases in boolean algebras
The (λ, κ)-fn and the order theory of bases in boolean algebras David Milovich Texas A&M International University david.milovich@tamiu.edu http://www.tamiu.edu/ dmilovich/ June 2, 2010 BLAST 1 / 22 The
More informationMODIFIED EXTENDER BASED FORCING
MODIFIED EXTENDER BASED FORCING DIMA SINAPOVA AND SPENCER UNGER Abstract. We analyze the modified extender based forcing from Assaf Sharon s PhD thesis. We show there is a bad scale in the extension and
More informationPERFECT TREE FORCINGS FOR SINGULAR CARDINALS
PERFECT TREE FORCINGS FOR SINGULAR CARDINALS NATASHA DOBRINEN, DAN HATHAWAY, AND KAREL PRIKRY Abstract. We investigate forcing properties of perfect tree forcings defined by Prikry to answer a question
More informationThe first author was supported by FWF Project P23316-N13.
The tree property at the ℵ 2n s and the failure of SCH at ℵ ω SY-DAVID FRIEDMAN and RADEK HONZIK Kurt Gödel Research Center for Mathematical Logic, Währinger Strasse 25, 1090 Vienna Austria sdf@logic.univie.ac.at
More informationTall, Strong, and Strongly Compact Cardinals
Tall, Strong, and Strongly Compact Cardinals Arthur W. Apter Department of Mathematics Baruch College of CUNY New York, New York 10010 USA and The CUNY Graduate Center, Mathematics 365 Fifth Avenue New
More informationLECTURE NOTES - ADVANCED TOPICS IN MATHEMATICAL LOGIC
LECTURE NOTES - ADVANCED TOPICS IN MATHEMATICAL LOGIC PHILIPP SCHLICHT Abstract. Lecture notes from the summer 2016 in Bonn by Philipp Lücke and Philipp Schlicht. We study forcing axioms and their applications.
More informationDeterminacy models and good scales at singular cardinals
Determinacy models and good scales at singular cardinals University of California, Irvine Logic in Southern California University of California, Los Angeles November 15, 2014 After submitting the title
More informationDEPTH OF BOOLEAN ALGEBRAS SHIMON GARTI AND SAHARON SHELAH
DEPTH OF BOOLEAN ALGEBRAS SHIMON GARTI AND SAHARON SHELAH Abstract. Suppose D is an ultrafilter on κ and λ κ = λ. We prove that if B i is a Boolean algebra for every i < κ and λ bounds the Depth of every
More informationGUESSING MODELS IMPLY THE SINGULAR CARDINAL HYPOTHESIS arxiv: v1 [math.lo] 25 Mar 2019
GUESSING MODELS IMPLY THE SINGULAR CARDINAL HYPOTHESIS arxiv:1903.10476v1 [math.lo] 25 Mar 2019 Abstract. In this article we prove three main theorems: (1) guessing models are internally unbounded, (2)
More informationGeneralising the weak compactness of ω
Generalising the weak compactness of ω Andrew Brooke-Taylor Generalised Baire Spaces Masterclass Royal Netherlands Academy of Arts and Sciences 22 August 2018 Andrew Brooke-Taylor Generalising the weak
More informationWähringer Strasse 25, 1090 Vienna Austria
The tree property at ℵ ω+2 with a finite gap Sy-David Friedman, 1 Radek Honzik, 2 Šárka Stejskalová 2 1 Kurt Gödel Research Center for Mathematical Logic, Währinger Strasse 25, 1090 Vienna Austria sdf@logic.univie.ac.at
More informationarxiv:math/ v1 [math.lo] 15 Jan 1991
ON A CONJECTURE OF TARSKI ON PRODUCTS OF CARDINALS arxiv:math/9201247v1 [mathlo] 15 Jan 1991 Thomas Jech 1 and Saharon Shelah 2 Abstract 3 We look at an old conjecture of A Tarski on cardinal arithmetic
More informationEaston s theorem and large cardinals from the optimal hypothesis
Easton s theorem and large cardinals from the optimal hypothesis SY-DAVID FRIEDMAN and RADEK HONZIK Kurt Gödel Research Center for Mathematical Logic, Währinger Strasse 25, 1090 Vienna Austria sdf@logic.univie.ac.at
More informationFORCING AND THE HALPERN-LÄUCHLI THEOREM. 1. Introduction This document is a continuation of [1]. It is intended to be part of a larger paper.
FORCING AND THE HALPERN-LÄUCHLI THEOREM NATASHA DOBRINEN AND DAN HATHAWAY Abstract. We will show the various effects that forcing has on the Halpern-Läuchli Theorem. We will show that the the theorem at
More informationTHE NUMBER OF UNARY CLONES CONTAINING THE PERMUTATIONS ON AN INFINITE SET
THE NUMBER OF UNARY CLONES CONTAINING THE PERMUTATIONS ON AN INFINITE SET MICHAEL PINSKER Abstract. We calculate the number of unary clones (submonoids of the full transformation monoid) containing the
More informationLOCAL CLUB CONDENSATION AND L-LIKENESS
LOCAL CLUB CONDENSATION AND L-LIKENESS PETER HOLY, PHILIP WELCH, AND LIUZHEN WU Abstract. We present a forcing to obtain a localized version of Local Club Condensation, a generalized Condensation principle
More informationNORMAL MEASURES ON A TALL CARDINAL. 1. Introduction We start by recalling the definitions of some large cardinal properties.
NORMAL MEASRES ON A TALL CARDINAL ARTHR. APTER AND JAMES CMMINGS Abstract. e study the number of normal measures on a tall cardinal. Our main results are that: The least tall cardinal may coincide with
More informationNotes to The Resurrection Axioms
Notes to The Resurrection Axioms Thomas Johnstone Talk in the Logic Workshop CUNY Graduate Center September 11, 009 Abstract I will discuss a new class of forcing axioms, the Resurrection Axioms (RA),
More informationCOMBINATORICS AT ℵ ω
COMBINATORICS AT ℵ ω DIMA SINAPOVA AND SPENCER UNGER Abstract. We construct a model in which the singular cardinal hypothesis fails at ℵ ω. We use characterizations of genericity to show the existence
More informationOn Singular Stationarity I (mutual stationarity and ideal-based methods)
On Singular Stationarity I (mutual stationarity and ideal-based methods) Omer Ben-Neria Abstract We study several ideal-based constructions in the context of singular stationarity. By combining methods
More informationarxiv: v1 [math.lo] 24 May 2009
MORE ON THE PRESSING DOWN GAME. arxiv:0905.3913v1 [math.lo] 24 May 2009 JAKOB KELLNER AND SAHARON SHELAH Abstract. We investigate the pressing down game and its relation to the Banach Mazur game. In particular
More informationStrongly Unfoldable Cardinals Made Indestructible
Strongly Unfoldable Cardinals Made Indestructible by Thomas A. Johnstone A dissertation submitted to the Graduate Faculty in Mathematics in partial fulfillment of the requirements for the degree of Doctor
More informationSUCCESSIVE FAILURES OF APPROACHABILITY
SUCCESSIVE FAILURES OF APPROACHABILITY SPENCER UNGER Abstract. Motivated by showing that in ZFC we cannot construct a special Aronszajn tree on some cardinal greater than ℵ 1, we produce a model in which
More informationCardinal arithmetic: The Silver and Galvin-Hajnal Theorems
B. Zwetsloot Cardinal arithmetic: The Silver and Galvin-Hajnal Theorems Bachelor thesis 22 June 2018 Thesis supervisor: dr. K.P. Hart Leiden University Mathematical Institute Contents Introduction 1 1
More informationON THE SINGULAR CARDINALS. A combinatorial principle of great importance in set theory is the Global principle of Jensen [6]:
ON THE SINGULAR CARDINALS JAMES CUMMINGS AND SY-DAVID FRIEDMAN Abstract. We give upper and lower bounds for the consistency strength of the failure of a combinatorial principle introduced by Jensen, Square
More informationOn the Splitting Number at Regular Cardinals
On the Splitting Number at Regular Cardinals Omer Ben-Neria and Moti Gitik January 25, 2014 Abstract Let κ,λ be regular uncountable cardinals such that κ + < λ. We construct a generic extension with s(κ)
More informationA Laver-like indestructibility for hypermeasurable cardinals
Radek Honzik Charles University, Department of Logic, Celetná 20, Praha 1, 116 42, Czech Republic radek.honzik@ff.cuni.cz The author was supported by FWF/GAČR grant I 1921-N25. Abstract: We show that if
More informationCONSECUTIVE SINGULAR CARDINALS AND THE CONTINUUM FUNCTION
CONSECUTIVE SINGULAR CARDINALS AND THE CONTINUUM FUNCTION ARTHUR W. APTER AND BRENT CODY Abstract. We show that from a supercompact cardinal κ, there is a forcing extension V [G] that has a symmetric inner
More informationTHE TREE PROPERTY AT ALL REGULAR EVEN CARDINALS
THE TREE PROPERTY AT ALL REGULAR EVEN CARDINALS MOHAMMAD GOLSHANI Abstract. Assuming the existence of a strong cardinal and a measurable cardinal above it, we construct a model of ZFC in which for every
More informationInterpolation of κ-compactness and PCF
Comment.Math.Univ.Carolin. 50,2(2009) 315 320 315 Interpolation of κ-compactness and PCF István Juhász, Zoltán Szentmiklóssy Abstract. We call a topological space κ-compact if every subset of size κ has
More informationCovering properties of derived models
University of California, Irvine June 16, 2015 Outline Background Inaccessible limits of Woodin cardinals Weakly compact limits of Woodin cardinals Let L denote Gödel s constructible universe. Weak covering
More informationBLOWING UP POWER OF A SINGULAR CARDINAL WIDER GAPS
BLOWING UP POWER OF A SINGULAR CARDINAL WIDER GAPS Moti Gitik School of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact Science Tel Aviv University Ramat Aviv 69978, Israel gitik@post.tau.ac.il
More informationLarge Cardinals with Few Measures
Large Cardinals with Few Measures arxiv:math/0603260v1 [math.lo] 12 Mar 2006 Arthur W. Apter Department of Mathematics Baruch College of CUNY New York, New York 10010 http://faculty.baruch.cuny.edu/apter
More informationMore on the Pressing Down Game
ESI The Erwin Schrödinger International Boltzmanngasse 9 Institute for Mathematical Physics A-1090 Wien, Austria More on the Pressing Down Game Jakob Kellner Saharon Shelah Vienna, Preprint ESI 2164 (2009)
More informationCOLLAPSING SUCCESSORS OF SINGULARS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 9, September 1997, Pages 2703 2709 S 0002-9939(97)03995-6 COLLAPSING SUCCESSORS OF SINGULARS JAMES CUMMINGS (Communicated by Andreas
More informationGeneric embeddings associated to an indestructibly weakly compact cardinal
Generic embeddings associated to an indestructibly weakly compact cardinal Gunter Fuchs Westfälische Wilhelms-Universität Münster gfuchs@uni-muenster.de December 4, 2008 Abstract I use generic embeddings
More informationLevel by Level Inequivalence, Strong Compactness, and GCH
Level by Level Inequivalence, Strong Compactness, and GCH Arthur W. Apter Department of Mathematics Baruch College of CUNY New York, New York 10010 USA and The CUNY Graduate Center, Mathematics 365 Fifth
More informationOn Singular Stationarity II (tight stationarity and extenders-based methods)
On Singular Stationarity II (tight stationarity and extenders-based methods) Omer Ben-Neria Abstract We study the notion of tightly stationary sets which was introduced by Foreman and Magidor in [8]. We
More informationCOMBINATORICS OF REDUCTIONS BETWEEN EQUIVALENCE RELATIONS
COMBINATORICS OF REDUCTIONS BETWEEN EQUIVALENCE RELATIONS DAN HATHAWAY AND SCOTT SCHNEIDER Abstract. We discuss combinatorial conditions for the existence of various types of reductions between equivalence
More informationHEIKE MILDENBERGER AND SAHARON SHELAH
A VERSION OF κ-miller FORCING HEIKE MILDENBERGER AND SAHARON SHELAH Abstract. Let κ be an uncountable cardinal such that 2 ω, 2 2
More informationDIAGONAL PRIKRY EXTENSIONS
DIAGONAL PRIKRY EXTENSIONS JAMES CUMMINGS AND MATTHEW FOREMAN 1. Introduction It is a well-known phenomenon in set theory that problems in infinite combinatorics involving singular cardinals and their
More informationTHE TREE PROPERTY UP TO ℵ ω+1
THE TREE PROPERTY UP TO ℵ ω+1 ITAY NEEMAN Abstract. Assuming ω supercompact cardinals we force to obtain a model where the tree property holds both at ℵ ω+1, and at ℵ n for all 2 n < ω. A model with the
More informationA relative of the approachability ideal, diamond and non-saturation
A relative of the approachability ideal, diamond and non-saturation Boise Extravaganza in Set Theory XVIII March 09, Boise, Idaho Assaf Rinot Tel-Aviv University http://www.tau.ac.il/ rinot 1 Diamond on
More informationChapter 4. Cardinal Arithmetic.
Chapter 4. Cardinal Arithmetic. 4.1. Basic notions about cardinals. We are used to comparing the size of sets by seeing if there is an injection from one to the other, or a bijection between the two. Definition.
More informationHierarchies of (virtual) resurrection axioms
Hierarchies of (virtual) resurrection axioms Gunter Fuchs August 18, 2017 Abstract I analyze the hierarchies of the bounded resurrection axioms and their virtual versions, the virtual bounded resurrection
More informationINDESTRUCTIBLE STRONG UNFOLDABILITY
INDESTRUCTIBLE STRONG UNFOLDABILITY JOEL DAVID HAMKINS AND THOMAS A. JOHNSTONE Abstract. Using the lottery preparation, we prove that any strongly unfoldable cardinal κ can be made indestructible by all
More informationANNALES ACADEMIÆ SCIENTIARUM FENNICÆ DIAMONDS ON LARGE CARDINALS
ANNALES ACADEMIÆ SCIENTIARUM FENNICÆ MATHEMATICA DISSERTATIONES 134 DIAMONDS ON LARGE CARDINALS ALEX HELLSTEN University of Helsinki, Department of Mathematics HELSINKI 2003 SUOMALAINEN TIEDEAKATEMIA Copyright
More informationBounds on coloring numbers
Ben-Gurion University, Beer Sheva, and the Institute for Advanced Study, Princeton NJ January 15, 2011 Table of contents 1 Introduction 2 3 Infinite list-chromatic number Assuming cardinal arithmetic is
More informationarxiv: v3 [math.lo] 23 Jul 2018
SPECTRA OF UNIFORMITY arxiv:1709.04824v3 [math.lo] 23 Jul 2018 YAIR HAYUT AND ASAF KARAGILA Abstract. We study some limitations and possible occurrences of uniform ultrafilters on ordinals without the
More informationEASTON FUNCTIONS AND SUPERCOMPACTNESS
EASTON FUNCTIONS AND SUPERCOMPACTNESS BRENT CODY, SY-DAVID FRIEDMAN, AND RADEK HONZIK Abstract. Suppose κ is λ-supercompact witnessed by an elementary embedding j : V M with critical point κ, and further
More informationSOME CONSEQUENCES OF REFLECTION ON THE APPROACHABILITY IDEAL
SOME CONSEQUENCES OF REFLECTION ON THE APPROACHABILITY IDEAL ASSAF SHARON AND MATTEO VIALE Abstract. We study the approachability ideal I[κ + ] in the context of large cardinals properties of the regular
More informationOn the strengths and weaknesses of weak squares
On the strengths and weaknesses of weak squares Menachem Magidor and Chris Lambie-Hanson 1 Introduction The term square refers not just to one but to an entire family of combinatorial principles. The strongest
More informationARONSZAJN TREES AND THE SUCCESSORS OF A SINGULAR CARDINAL. 1. Introduction
ARONSZAJN TREES AND THE SUCCESSORS OF A SINGULAR CARDINAL SPENCER UNGER Abstract. From large cardinals we obtain the consistency of the existence of a singular cardinal κ of cofinality ω at which the Singular
More informationarxiv:math/ v1 [math.lo] 9 Dec 2006
arxiv:math/0612246v1 [math.lo] 9 Dec 2006 THE NONSTATIONARY IDEAL ON P κ (λ) FOR λ SINGULAR Pierre MATET and Saharon SHELAH Abstract Let κ be a regular uncountable cardinal and λ > κ a singular strong
More informationOpen Problems. Problem 2. Assume PD. C 3 is the largest countable Π 1 3-set of reals. Is it true that C 3 = {x M 2 R x is. Known:
Open Problems Problem 1. Determine the consistency strength of the statement u 2 = ω 2, where u 2 is the second uniform indiscernible. Best known bounds: Con(there is a strong cardinal) Con(u 2 = ω 2 )
More informationLarge cardinals and their effect on the continuum function on regular cardinals
Large cardinals and their effect on the continuum function on regular cardinals RADEK HONZIK Charles University, Department of Logic, Celetná 20, Praha 1, 116 42, Czech Republic radek.honzik@ff.cuni.cz
More informationHod up to AD R + Θ is measurable
Hod up to AD R + Θ is measurable Rachid Atmai Department of Mathematics University of North Texas General Academics Building 435 1155 Union Circle #311430 Denton, TX 76203-5017 atmai.rachid@gmail.com Grigor
More informationPARTITIONS OF 2 ω AND COMPLETELY ULTRAMETRIZABLE SPACES
PARTITIONS OF 2 ω AND COMPLETELY ULTRAMETRIZABLE SPACES WILLIAM R. BRIAN AND ARNOLD W. MILLER Abstract. We prove that, for every n, the topological space ω ω n (where ω n has the discrete topology) can
More informationarxiv: v2 [math.lo] 21 Mar 2016
WEAK DISTRIBUTIVITY IMPLYING DISTRIBUTIVITY arxiv:1410.1970v2 [math.lo] 21 Mar 2016 DAN HATHAWAY Abstract. Let B be a complete Boolean algebra. We show that if λ is an infinite cardinal and B is weakly
More informationThe Outer Model Programme
The Outer Model Programme Peter Holy University of Bristol presenting joint work with Sy Friedman and Philipp Lücke February 13, 2013 Peter Holy (Bristol) Outer Model Programme February 13, 2013 1 / 1
More informationRVM, RVC revisited: Clubs and Lusin sets
RVM, RVC revisited: Clubs and Lusin sets Ashutosh Kumar, Saharon Shelah Abstract A cardinal κ is Cohen measurable (RVC) if for some κ-additive ideal I over κ, P(κ)/I is forcing isomorphic to adding λ Cohen
More informationLarge cardinals and the Continuum Hypothesis
Large cardinals and the Continuum Hypothesis RADEK HONZIK Charles University, Department of Logic, Celetná 20, Praha 1, 116 42, Czech Republic radek.honzik@ff.cuni.cz Abstract. This is a survey paper which
More informationarxiv: v2 [math.lo] 13 Feb 2014
A LOWER BOUND FOR GENERALIZED DOMINATING NUMBERS arxiv:1401.7948v2 [math.lo] 13 Feb 2014 DAN HATHAWAY Abstract. We show that when κ and λ are infinite cardinals satisfying λ κ = λ, the cofinality of the
More informationUPWARD STABILITY TRANSFER FOR TAME ABSTRACT ELEMENTARY CLASSES
UPWARD STABILITY TRANSFER FOR TAME ABSTRACT ELEMENTARY CLASSES JOHN BALDWIN, DAVID KUEKER, AND MONICA VANDIEREN Abstract. Grossberg and VanDieren have started a program to develop a stability theory for
More informationChromatic number of infinite graphs
Chromatic number of infinite graphs Jerusalem, October 2015 Introduction [S] κ = {x S : x = κ} [S]
More informationON SCH AND THE APPROACHABILITY PROPERTY
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Xxxx XXXX, Pages 000 000 S 0002-9939(XX)0000-0 ON SCH AND THE APPROACHABILITY PROPERTY MOTI GITIK AND ASSAF SHARON (Communicated by
More informationAxiomatization of generic extensions by homogeneous partial orderings
Axiomatization of generic extensions by homogeneous partial orderings a talk at Colloquium on Mathematical Logic (Amsterdam Utrecht) May 29, 2008 (Sakaé Fuchino) Chubu Univ., (CRM Barcelona) (2008 05 29
More informationCardinal characteristics at κ in a small u(κ) model
Cardinal characteristics at κ in a small u(κ) model A. D. Brooke-Taylor a, V. Fischer b,, S. D. Friedman b, D. C. Montoya b a School of Mathematics, University of Bristol, University Walk, Bristol, BS8
More informationarxiv: v1 [math.lo] 26 Mar 2014
A FRAMEWORK FOR FORCING CONSTRUCTIONS AT SUCCESSORS OF SINGULAR CARDINALS arxiv:1403.6795v1 [math.lo] 26 Mar 2014 JAMES CUMMINGS, MIRNA DŽAMONJA, MENACHEM MAGIDOR, CHARLES MORGAN, AND SAHARON SHELAH Abstract.
More informationClosed Maximality Principles: Implications, Separations and Combinations
Closed Maximality Principles: Implications, Separations and Combinations Gunter Fuchs Institut für Mathematische Logik und Grundlagenforschung Westfälische Wilhelms-Universität Münster Einsteinstr. 62
More informationThe Resurrection Axioms
The Resurrection Axioms Thomas Johnstone New York City College of Technology, CUNY and Kurt Gödel Research Center, Vienna tjohnstone@citytech.cuny.edu http://www.logic.univie.ac.at/~tjohnstone/ Young Set
More informationSHIMON GARTI AND SAHARON SHELAH
(κ, θ)-weak NORMALITY SHIMON GARTI AND SAHARON SHELAH Abstract. We deal with the property of weak normality (for nonprincipal ultrafilters). We characterize the situation of Q λ i/d = λ. We have an application
More informationarxiv: v2 [math.lo] 26 Feb 2014
RESURRECTION AXIOMS AND UPLIFTING CARDINALS arxiv:1307.3602v2 [math.lo] 26 Feb 2014 JOEL DAVID HAMKINS AND THOMAS A. JOHNSTONE Abstract. We introduce the resurrection axioms, a new class of forcing axioms,
More information