Proper forcing extensions and Solovay models

We study the preservation of the property of being a Solovay model under proper projective forcing extensions. We show that every strongly-proper forcing notion preserves this property. This yields that the consistency strength of the absoluteness of under strongly-proper forcing notions is that of the existence of an inaccessible cardinal. Further, the absoluteness of under projective strongly-proper forcing notions is consistent relative to the existence of a -Mahlo cardinal. We also show that the consistency strength of the absoluteness of under forcing extensions with -linked forcing notions is exactly that of the existence of a Mahlo cardinal, in contrast with the general ccc case, which requires a weakly-compact cardinal.Research partially supported by the research projects BFM2002-03236 of the Spanish Ministry of Science and Technology, and 2002SGR 00126 of the Generalitat de Catalunya. The second author was also partially supported by the research project GE01/HUM10, Grupos de excelencia, Principado de Asturias.Mathematics Subject Classification (2000): 03E15, 03E35     

Complete separation in the random and Cohen models

It is shown that in the model obtained by adding κ many random reals, where κ is a supercompact cardinal, every C^?-embedded subset of a first countable space (even with character smaller than κ) is C-embedded. It is also proved that if two ground model sets are completely separated after adding a random real then they were completely separated originally but CH implies that the Cohen poset does not have this property.     

Syntactic forcing models for coherent logic

We present three syntactic forcing models for coherent logic. These are based on sites whose underlying category only depends on the signature of the coherent theory, and they do not presuppose that the logic has equality. As an application we give a coherent theory $T$ and a sentence $\psi$ which is $T$-redundant (for any geometric implication $?$, possibly with equality, if $T+\psi ??$, then $T??$), yet false in the generic model of $T$. This answers in the negative a question by Wraith.     

Seasonal forcing in stochastic epidemiology models

The goal of this paper is to motivate the need and lay the foundation for the analysis of stochastic epidemiological models with seasonal forcing. We consider stochastic SIS and SIR epidemic models, where the internal noise is due to the random interactions of individuals in the population. We provide an overview of the general theoretic framework that allows one to understand noise-induced rare events, such as spontaneous disease extinction. Although there are many paths to extinction, there is one path termed the optimal path that is probabilistically most likely to occur. By extending the theory, we have identified the quasi-stationary solutions and the optimal path to extinction when seasonality in the contact rate is included in the models. Knowledge of the optimal extinction path enables one to compute the mean time to extinction, which in turn allows one to compare the effect of various control schemes, including vaccination and treatment, on the eradication of an infectious disease.     

Forcing notions in inner models

There is a partial order \({\mathbb{P}}\) preserving stationary subsets of ω ₁ and forcing that every partial order in the ground model V that collapses a sufficiently large ordinal to ω ₁ over V also collapses ω ₁ over \({V^{\mathbb{P}}}\) . The proof of this uses a coding of reals into ordinals by proper forcing discovered by Justin Moore and a symmetric extension of the universe in which the Axiom of Choice fails. Also, using one feature of the proof of the above result together with an argument involving the stationary tower it is shown that sometimes, after adding one Cohen real c, there are, for every real a in V[c], sets A and B such that c is Cohen generic over both L[A] and L[B] but a is constructible from A together with B.     

Cohen and multiple Cohen strongly summing multilinear operators

Considering the successful theory of multiple summing multilinear operators as a prototype, we introduce the classes of multiple Cohen strongly p-summing multilinear operators and polynomials. The adequacy of these classes under the viewpoint of the theory of multilinear and polynomial ideals and holomorphy types is discussed in detail.     

Homogeneously Souslin sets in small inner models

We prove that every homogeneously Souslin set is coanalytic provided that either (a) 0^long does not exist, or else (b) V = K, where K is the core model below a μ-measurable cardinal.     

Inner models with large cardinal features usually obtained by forcing

We construct a variety of inner models exhibiting features usually obtained by forcing over universes with large cardinals. For example, if there is a supercompact cardinal, then there is an inner model with a Laver indestructible supercompact cardinal. If there is a supercompact cardinal, then there is an inner model with a supercompact cardinal κ for which 2 ^κ ?=?κ ⁺, another for which 2 ^κ ?=?κ ⁺⁺ and another in which the least strongly compact cardinal is supercompact. If there is a strongly compact cardinal, then there is an inner model with a strongly compact cardinal, for which the measurable cardinals are bounded below it and another inner model W with a strongly compact cardinal κ, such that ${H^{V}_{\kappa^+} \subseteq {\rm HOD}^W}$ . Similar facts hold for supercompact, measurable and strongly Ramsey cardinals. If a cardinal is supercompact up to a weakly iterable cardinal, then there is an inner model of the Proper Forcing Axiom and another inner model with a supercompact cardinal in which GCH?+?V?=?HOD holds. Under the same hypothesis, there is an inner model with level by level equivalence between strong compactness and supercompactness, and indeed, another in which there is level by level inequivalence between strong compactness and supercompactness. If a cardinal is strongly compact up to a weakly iterable cardinal, then there is an inner model in which the least measurable cardinal is strongly compact. If there is a weakly iterable limit δ of <δ-supercompact cardinals, then there is an inner model with a proper class of Laver-indestructible supercompact cardinals. We describe three general proof methods, which can be used to prove many similar results.     

Sacks forcing,Laver forcing,and Martin's axiom

Summary In this paper we study the question assuming MA+CH does Sacks forcing or Laver forcing collapse cardinals? We show that this question is equivalent to the question of what is the additivity of Marczewski's ideals ⁰. We give a proof that it is consistent that Sacks forcing collapses cardinals. On the other hand we show that Laver forcing does not collapse cardinals.Research partially supported by NSF grant 8801139     

The forcing hull and forcing geodetic numbers of graphs

For every pair of vertices u,v in a graph, a u-v geodesic is a shortest path from u to v. For a graph G, let I_G[u,v] denote the set of all vertices lying on a u-v geodesic. Let S⊆V(G) and I_G[S] denote the union of all I_G[u,v] for all u,v∈S. A subset S⊆V(G) is a convex set of G if I_G[S]=S. A convex hull [S]_G of S is a minimum convex set containing S. A subset S of V(G) is a hull set of G if [S]_G=V(G). The hull number h(G) of a graph G is the minimum cardinality of a hull set in G. A subset S of V(G) is a geodetic set if I_G[S]=V(G). The geodetic number g(G) of a graph G is the minimum cardinality of a geodetic set in G. A subset F⊆V(G) is called a forcing hull (or geodetic) subset of G if there exists a unique minimum hull (or geodetic) set containing F. The cardinality of a minimum forcing hull subset in G is called the forcing hull number f_h(G) of G and the cardinality of a minimum forcing geodetic subset in G is called the forcing geodetic number f_g(G) of G. In the paper, we construct some 2-connected graph G with (f_h(G),f_g(G))=(0,0),(1,0), or (0,1), and prove that, for any nonnegative integers a, b, and c with a+b≥2, there exists a 2-connected graph G with (f_h(G),f_g(G),h(G),g(G))=(a,b,a+b+c,a+2b+c) or (a,2a+b,a+b+c,2a+2b+c). These results confirm a conjecture of Chartrand and Zhang proposed in [G. Chartrand, P. Zhang, The forcing hull number of a graph, J. Combin. Math. Combin. Comput. 36 (2001) 81-94].     

Maharam algebras and Cohen reals

We show that the product of any two nonatomic Maharam algebras adds a Cohen real. As a corollary of this and a result of Shelah (1994) we obtain that the product of any two nonatomic ccc Souslin forcing notions adds a Cohen real.

     

On Cohen braids

For a general connected surface M and an arbitrary braid α from the surface braid group B _n?1(M), we study the system of equations d ₁ β = … = d _n β = α, where the operation d _i is the removal of the ith strand. We prove that for M ≠ S ² and M ≠ ?P², this system of equations has a solution β ∈ B _n (M) if and only if d ₁ α = … = d _n?1 α. We call the set of braids satisfying the last system of equations Cohen braids. We study Cohen braids and prove that they form a subgroup. We also construct a set of generators for the group of Cohen braids. In the cases of the sphere and the projective plane we give some examples for a small number of strands.     

Magidor-like and radin-like forcing

We force over a model M of ZF+κ→(κ)^<γ to obtain M[G] with cf(κ)=γ. The method is reminiscent of Magidor-forcing but uses no choice. Mimicing Radin-forcing, we generalize this for strong partition cardinals κ to add a subset of κ while preserving all cardinalities, cofinalities and κ's measurability. We apply these techniques to construct models of unusual partition properties, such as ω₂→[ω₂]^ω₁ but $\text{ω}{}_2\text{?[ω}{}_2\text{]}{}^{\mathrm{\omega }}$.     

Inadmissible forcing

A structure is E-closed if it is closed under all partial E-recursive functions from V into V, a set theoretic extension of Kleene's partial recursive functions of finite type in the normal case. Let L(κ) be E-closed and ∑₁ inadmissible. Then L(κ) has reflection properties useful in the study of generic extensions of L(κ). Every set generic extension of L(κ) via countably closed forcing conditions is E-closed. A class generic construction shows: if L(κ) is countable, and inside L(κ) the greatest cardinal gc(κ), has uncountable cofinality, then there exists a T ⊆ gc(κ) such that L(κ, T) = E(T), the least E-closed set with T as a member. A partial converse is obtained via a selection theorem that implies E(X) is ∑₁ admissible when X is a set of ordinals and the greatest cardinal in the sense of E(X) has countable cofinality in E(X).     

More on trees and Cohen reals

In this paper we analyse some questions concerning trees on κ, both for the countable and the uncountable case, and the connections with Cohen reals. In particular, we provide a proof for one of the implications left open in [6, Question 5.2] about the diagram for regularity properties.