Jindrich Zapletal
I prove preservation theorems for countable support iteration of proper forcing concerning certain classes of capacities and submeasures. New examples of forcing notions and connections with measure theory are included.
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This is an expository paper for Chapter 6 of Proper and Improper Forcing. Now includes an exposition of Shelah's proof of preservation of Sacks property as well as omega-omega bounding.
This is an exposition of the first two sections of Chapter VI of Shelah's book Proper and Improper Forcing. It covers various preservation theorems for CS iteration of proper forcing (omega-omega bounding, Sacks property, P-point property, etc.)
This expository paper covers the first two sections of chapter VI of Shelah's book "Proper and Improper Forcing," including the preservationn (using CS iterations of proper forcings) of omega-omega bounding, Sacks property, Lavewr property, P-point property, and others.
This is an exposition of much of Sections VI.3 and XVIII.3 of "Proper and Improper Forcing", including preservations for "no random reals over V", "reals of V form a non-meager set", "every dense open set contains a dense open set in V", weak bounding, and weak $ω^ω$-bounding. The current version of part I covering Sections VI.1 and VI.2 is available from the author.
We provide a general preservation theorem for preserving selective independent families along countable support iterations. The theorem gives a general framework for a number of results in the literature concerning models in which the independence number $\mathfrak{i}$ is strictly below $\mathfrak{c}$, including iterations of Sacks forcing, Miller partition forcing, $h$-perfect tree forcings, coding with perfect trees. Moreover, applying the theorem, we show that $\mathfrak{i} = \aleph_1$ in the Miller Lite model. An important aspect of the preservation theorem is the notion of "Cohen preserva…
A preservation theorem for theories without the tree property of the first kind
本文证明了几何理论的NTP$_1$性质在可爱对和$H$-结构中保持，并给出非简单几何NTP$_1$理论的例子。
We show how to force, with finite conditions, the forcing axiom PFA(T), a relativization of PFA to proper forcing notions preserving a given Souslin tree T. The proof uses a Neeman style iteration with generalized side conditions consisting of models of two types, and a preservation theorem for such iterations. The consistency of this axiom was previously known by the standard countable support iteration, using a preservation theorem due to Miyamoto.
Preservation Theorems Through the Lens of Topology
本文提出一类可刻画保持定理存在性的拓扑空间，研究其在多种拓扑构造下的相对化问题，并将已有结果整合到新框架中以揭示证明核心。
Arboreal Categories and Equi-resource Homomorphism Preservation Theorems
本研究基于树状范畴概念提出等资源同态保持定理的公理化框架，改进了经典同态保持定理，为一阶、护卫、模态等多种逻辑片段建立了新的同态保持结果并优化了已有结论。
We give a self-contained proof of the preservation theorem for proper countable support iterations known as "tools-preservation," "Case A" or "first preservation theorem" in the literature. We do not assume that the forcings add reals.
We present preservation theorems for countable support iteration of nep forcing notions satisfying ``old reals are not Lebesgue null'' and ``old reals are not meager''. (Nep is a generalization of Suslin proper.) We also give some results for general Suslin ccc ideals.
We prove preservation theorems for $\mathcal{L}_{ω_1, G}$, the countable fragment of Vaught's closed game logic. These are direct generalizations of the theorems of Łoś-Tarski (resp. Lyndon) on sentences of $\mathcal{L}_{ω_1, ω}$ preserved by substructures (resp. homomorphic images). The solution, in $ZFC$, only uses general features and can be extended to several variants of other strong first-order logic that do not satisfy the interpolation theorem; instead, the results on infinitary definability are used. This solves an open problem dating back to 1977. Another consequence of our approach …
Preservation theorems on sparse classes revisited
该研究重新审视稀疏结构类上一阶逻辑的同态保持定理，指出原有证明中可加性假设强度不足，构造了有界树宽的反例并证明平面图类不满足该性质，同时提出用瓶颈合并条件替代可加性以恢复证明有效性。
We prove an algebraic preservation theorem for positive Horn definability in aleph-zero categorical structures. In particular, we define and study a construction which we call the periodic power of a structure, and define a periomorphism of a structure to be a homomorphism from the periodic power of the structure to the structure itself. Our preservation theorem states that, over an aleph-zero categorical structure, a relation is positive Horn definable if and only if it is preserved by all periomorphisms of the structure. We give applications of this theorem, including a new proof of the know…
We investigate a generalization of the Łoś-Tarski preservation theorem via the semantic notion of \emph{preservation under substructures modulo $k$-sized cores}. It was shown earlier that over arbitrary structures, this semantic notion for first-order logic corresponds to definability by $\exists^k\forall^*$ sentences. In this paper, we identify two properties of classes of finite structures that ensure the above correspondence. The first is based on well-quasi-ordering under the embedding relation. The second is a logic-based combinatorial property that strictly generalizes the first. We show…
The separation between two theorems in reverse mathematics is usually done by constructing a Turing ideal satisfying a theorem P and avoiding the solutions to a fixed instance of a theorem Q. Lerman, Solomon and Towsner introduced a forcing technique for iterating a computable non-reducibility in order to separate theorems over omega-models. In this paper, we present a modularized version of their framework in terms of preservation of hyperimmunity and show that it is powerful enough to obtain the same separations results as Wang did with his notion of preservation of definitions.
The Gap Forcing Theorem, a key contribution of this paper, implies essentially that after any reverse Easton iteration of closed forcing, such as the Laver preparation, every supercompactness measure on a supercompact cardinal extends a measure from the ground model. Thus, such forcing can create no new supercompact cardinals, and, if the GCH holds, neither can it increase the degree of supercompactness of any cardinal; in particular, it can create no new measurable cardinals. In a crescendo of what I call exact preservation theorems, I use this new technology to perform a kind of partial Lave…
Homomorphism Preservation Theorems for Many-Valued Structures
本文结合Rossman和Dellunde-Vidal的研究，证明了多值逻辑中有限同态保持定理的广泛适用性，并探讨其在值约束满足问题中的影响。
Simple Models of Randomization and Preservation Theorems
本文通过简单模型证明了随机化理论的NIP性质，并展示了NSOP与稳定性的关系。
Property B: A Baumgartner-style Property that Applies to Preservation of $\aleph_1$ and $\aleph_2$ under Iterations with Supports of Size $\aleph_1$
本文研究了在支持大小为$\aleph_1$的迭代中保持$\aleph_1$和$\aleph_2$的性质B，扩展到其他满足$κ^{<κ}=κ$的基数，并引入了修正的完美集forcing。
Preservation of NATP
本文证明NATP在多种模型论构造下的保持性，包括Fraïssé极限的参数化与和、稠密/共稠密扩张、一般谓词扩张及代数闭域配对扩张，并证明ACFO的模型伴随具有NATP。
$n$-variable theorems for dividing lines characterized by positive consistency-inconsistency configurations, and their applications to preservation problems
本文提出$n$-变量定理，统一处理多种分裂线，并证明在泛平凡化、可爱对、$H$-结构、向量空间、代数闭域及泛导数等扩张下保持性，同时推广到更高元数分裂线。
Finite model theory for pseudovarieties and universal algebra: preservation, definability and complexity
本文研究了有限模型理论与伪变种及普遍代数之间的关系，揭示了有限代数在公理化和可定义性方面的复杂性。
It is well known that pretameness implies the forcing theorem, and that pretameness is characterized by the preservation of the axioms of $\mathsf{ZF}^-$, that is $\mathsf{ZF}$ without the power set axiom, or equivalently, by the preservation of the axiom scheme of replacement, for class forcing over models of $\mathsf{ZF}$. We show that pretameness in fact has various other characterizations, for instance in terms of the forcing theorem, the preservation of the axiom scheme of separation, the forcing equivalence of partial orders and their dense suborders, and the existence of nice names for …
I prove forcing preservation theorems for products of definable partial orders preserving the cofinality of the meager or null ideal. Rectangular Ramsey theorems for related ideals follow from the proofs.
Weihrauch reducibility between Ramsey-type theorems and well-ordering principles at the level of $Σ^0_2$-induction: A pilot study
本文研究了有序Ramsey定理与良序保持原理在$Σ^0_2$-induction水平下的Weihrauch可约性关系，并展示了其等价性及不可比性。
Shelah shows that certain revised countable support (RCS) iterations do not add reals. His motivation is to establish the independence (relative to large cardinals) of Avraham's problem on the existence of uncountable non-constuctible sequences all of whose proper initial segments are constructible, Friedman's problem on whether every 2-coloring of $S^2_0=\{α<ω_2\colon\cf(α)=ω\}$ has an uncountable sequentially closed homogeneous subset, and existence of a precipitous normal filter on $ω_2$ with $S^2_0\in{\cal F}$. The posets which Shelah uses in these constructions are Prikry forcing, Namba f…
We prove two general results about the preservation of extendible and $C^{(n)}$-extendible cardinals under a wide class of forcing iterations (Theorems 5.4 and 7.5). As applications we give new proofs of the preservation of Vopěnka's Principle and $C^{(n)}$-extendible cardinals under Jensen's iteration for forcing the GCH, previously obtained by Brooke-Taylor and Tsaprounis, res\-pectively. We prove that $C^{(n)}$-extendible cardinals are preserved by forcing with standard Easton-support iterations for any possible $Δ_2$-definable behaviour of the power-set function on regular cardinals. We sh…
Rice's Theorem under Self-Modification: Elevation Operators and a Normal Form
本文提出语义提升算子，证明自修改系统中性质保持的不可判定性，并揭示不可验证性在提升下封闭且攀升算术层级。
Among the Ramsey-type hierarchies, namely, Ramsey's theorem, the free set, the thin set and the rainbow Ramsey theorem, only Ramsey's theorem is known to collapse in reverse mathematics. A promising approach to show the strictness of the hierarchies would be to prove that every computable instance at level n has a low_n solution. In particular, this requires effective control of iterations of the Turing jump. In this paper, we design some variants of Mathias forcing to construct solutions to cohesiveness, the Erdos-Moser theorem and stable Ramsey's theorem for pairs, while controlling their it…
The preservation theorems for semi-properness, hemi-properness, and pseudo-completeness hold for countable support iterations as well as revised countable support iterations, notwithstanding the fact that the "factor lemma" fails for the countable support versions. Example: The countable support iteration of Namba forcing over a ground model of CH does not add reals. Example: A countable support iteration built from Namba forcing and Cohen forcing, over a ground model of CH, does not collapse omega_1
Various theorems for the preservation of set-theoretic axioms under forcing are proved, regarding both forcing axioms and axioms true in the Levy-Collapse. These show in particular that certain applications of forcing axioms require to add generic countable sequences high up in the set-theoretic hierarchy even before collapsing everything down to $\aleph\_1$. Later we give applications, among them the consistency of ${\rm MM}$ with $\aleph\_ω$ not being Jonsson which answers a question raised during Oberwolfach 2005.
Regressive versions of Hindman's Theorem
本文提出相对于最小齐次性的λ-回归Hindman定理，并在反向数学中证明其首个非平凡限制等价于算术概括，且可强可计算归约底ω指数运算的保良序原理。
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