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Using the method of decisive creatures (math.LO/0601083) we show the consistency of "there is no increasing omega_2 --chain of Borel sets and non(N)=non(M)= omega_2=2^omega". Hence, consistently, there are no monotone hulls for the ideal M cap N . This answers Balcerzak and Filipczak. Next we use FS iteration with partial memory to show that there may be monotone Borel hulls for the ideals M, N even if they are not generated by towers.
We investigate the class of models of a general dependent theory. We continue math.LO/0702292 in particular investigating so called "decomposition of types"; thesis is that what holds for stable theory and for Th(Q,<) hold for dependent theories. Another way to say this is: we have to look at small enough neighborhood and use reasonably definable types to analyze a type. We note the results understable without reading. First, a parallel to the "stability spectrum", the "recounting of types", that is assume lambda = lambda^{< lambda} is large enough, M a saturated model of T of cardinality lambda, let bold S_{aut}(M) be the number of complete types over M up to being conjugate, i.e. we identify p,q when some automorphism of M maps p to q . Whereas for independent T the number is 2^lambda, for dependent T the number is <= lambda moreover it is <= | alpha |^{|T|} when lambda = aleph_alpha. Second, for stable theories "lots of indiscernibility exists" a "too good indiscernible existence theorem" saying, e.g. that if the type tp (d_beta ; {d_beta : beta < alpha}) is increasing for alpha < kappa = cf(kappa) and kappa > 2^{|T|} then is indiscernible for some stationary S subseteq kappa. Third, for stable T,a model is kappa-saturated iff it is aleph_epsilon-saturated and every infinite indiscernible set (of elements) of cardinality < kappa can be increased. We prove here an analog. Fourth, for p in S(M), the number of ultrafilters on the outside definable subsets of M extending p has an absolute bound 2^{|T|} . Restricting ourselves to one phi(x, y), the number is finite, with an absolute found (well depending on T and phi).
We investigate the behaviour of concepts from dependent theories when applied to real closed fields. Our main focus is on the concept of perpendicular indiscernible sequences, a concept first introduced in section 4 of math.LO/0009056 . This is essentially the MSc. thesis of the first author written under he guidance of the second author.
This article continues Roslanowski and Shelah math.LO/9906024 and 1105.6049 We introduce here yet another property of (<lambda)-strategically complete forcing notions which implies that their lambda-support iterations do not collapse lambda^+.
We study sigma-ideals and regularity properties related to the "filter-Laver" and "dual-filter-Laver" forcing partial orders. An important innovation which enables this study is a dichotomy theorem proved recently by Miller [1]. [1] Arnold Miller, "Hechler and Laver Trees", Preprint 2012 (arXiv:1204.5198 [math.LO]).
For a cardinal lambda 5 for all alpha,beta<lambda, but (2) there is no perfect set of such eta's. The construction closely follows the one from Shelah math.LO/9802134
Given a function $f \in ω^ω$, a set $A \in [ω]^ω$ is free for $f$ if $f[A] \cap A$ is finite. For a class of functions $Γ\subseteq ω^ω$, we define $\mathfrak{ros}_Γ$ as the smallest size of a family $\mathcal{A}\subseteq [ω]^ω$ such that for every $f\inΓ$ there is a set $A \in \mathcal{A}$ which is free for $f$, and $Δ_Γ$ as the smallest size of a family $\mathcal{F}\subseteqΓ$ such that for every $A\in[ω]^ω$ there is $f\in\mathcal{F}$ such that $A$ is not free for $f$. We compare several versions of these cardinal invariants with some of the classical cardinal characteristics of the continuum. Using these notions, we partially answer some questions from arXiv:1911.01336 [math.LO] and arXiv:2004.01979 [math.GN].
Boban Velickovic asked the following question: Is there a nontrivial forcing notion with the Sacks property which is also ccc? A ``definable'' variant of this question has been answered in [Sh:480] (math.LO/9303208): Every nontrivial Souslin forcing notion which has the Sacks property has an uncountable antichain. Here we show that it is consistent that every nontrivial forcing notion which has the Sacks property has an uncountable antichain. Independently, Velickovic has also proved the consistency of this statement.
A group homomorphism eta:A-> H is called a localization of A if every homomorphism phi:A-> H can be `extended uniquely' to a homomorphism Phi:H-> H in the sense that Phi eta = phi. This categorical concepts, obviously not depending on the notion of groups, extends classical localizations as known for rings and modules. Moreover this setting has interesting applications in homotopy theory. For localizations eta:A-> H of (almost) commutative structures A often H resembles properties of A, e.g. size or satisfying certain systems of equalities and non-equalities. Perhaps the best known example is that localizations of finite abelian groups are finite abelian groups. This is no longer the case if A is a finite (non-abelian) group. Libman showed that A_n-> SO_{n-1}(R) for a natural embedding of the alternating group A_n is a localization if n even and n >= 10 . Answering an immediate question by Dror Farjoun and assuming the generalized continuum hypothesis GCH we recently showed in math.LO/9912191 that any non-abelian finite simple has arbitrarily large localizations. In this paper we want to remove GCH so that the result becomes valid in ordinary set theory. At the same time we want to generalize the statement for a larger class of A 's.
Solecki proved that the group of automorphisms of a countable structure cannot be an uncountable free abelian group. See more in Just, Shelah and Thomas math.LO/0003120 where as a by product we can say something on on uncountable structures. We prove here the following Theorem: If A is a countable model, then Aut(M) cannot be a free uncountable group.
In arXiv: math.LO/0011208 we proposed the {\sl intuitionistic or disjunctive representation of quantum logic}, i.e., a representation of the property lattice of physical systems as a complete Heyting algebra of logical propositions on these properties, where this complete Heyting algebra goes equipped with an additional operation, the {\sl operational resolution}, which identifies the properties within the logic of propositions. This representation has an important application ``towards dynamic quantum logic'', namely in describing the temporal indeterministic propagation of actual properties of physical systems. This paper can as such by conceived as an addendum to ``Quantum Logic in Intuitionistic Perspective'' that discusses spin-off and thus provides an additional motivation. We derive a quantaloidal semantics for dynamic disjunctive quantum logic and illustrate it for the particular case of a perfect (quantum) measurement.
The main result of this paper is a partial answer to [math.LO/9909115, Problem 5.5]: a finite iteration of Universal Meager forcing notions adds generic filters for many forcing notions determined by universality parameters. We also give some results concerning cardinal characteristics of the sigma-ideals determined by those universality parameters.
Given a class of finite models we would like to expand each model (allowing new elements but the old universe is a separate sort), making the expressive power of LFP (least fix point logic) and PFP (inductive logic) similar while not changing the expressive power of FO (first order logic). This continues in math.LO/9411235.
In a self-contained way, we deal with revised countable support iterated forcing for the reals. We improve theorems on preservation of the property UP, weaker than semi proper, and we hopefully improve the presentation. We continue [Sh:b, Ch.X,XI] (or see [Sh:f, Ch.X,XI]), and Gitik and Shelah [GiSh:191] and [Sh:f, Ch.XIII,XIV] and particularly Ch.XV. Concerning ``no new reals'' see lately Larson and Shelah [LrSh:746] math.LO/0011187. In particular, we fulfill some promises from [Sh:f] and give a more streamlined version.
This is the second part of math.LO/9606226.
We further investigate the class of models of a strongly dependent (first order complete) theory T, continuing math.LO/0406440. If |A|+|T|<= mu, I subseteq C, |I| >=beth_{|T|^+}(mu) then some J subseteq I of cardinality mu^+ is an indiscernible sequence over A .
We introduce more properties of forcing notions which imply that their lambda-support iterations are lambda-proper, where lambda is an inaccessible cardinal. This paper is a direct continuation of section A.2 of math.LO/0210205. As an application of our iteration result we show that it is consistent that dominating numbers associated with two normal filters on lambda are distinct.
Answering a question of the first author stated in [math.LO/0507519, 0.2] we show that limits of CS iterations of n$-Silver forcing notion have the n-localization property.