We show that many countable support iterations of proper forcings preserve Souslin trees. We establish sufficient conditions in terms of games and we draw connections to other preservation properties. We present a proof of preservation properties in countable support iterations in the so-called Case A that does not need a division into forcings that add reals and those who do not.

I prove several natural preservation theorems for the countable support iteration. This solves a question of Rosłanowski regarding the preservation of localization properties and greatly simplifies the proofs in the area.

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.

It is shown to be consistent that countable, Fréchet,α 1-spaces are first countable. The result is obtained by using a countable support iteration of proper partial orders of lengthω 2.

Let T be an $\omega_{1}-Souslin$ tree. We show the property of forcing notions; "is $\lbrace\omega_{1}\rbrace-semi-proper$ and preserves T" is preserved by a new kind of revised countable support iteration of arbitrary length. As an application we have a forcing axiom which is compatible with the existence of an $\omega_{1}-Souslin$ tree for preorders as wide as possible.

Using countable support iterations of S-proper posets, we show that the existence of a definable wellorder of the reals is consistent with each of the following: , and.

We give a simplified treatment of revised countable support (RCS) forcing iterations, previously considered by Shelah (see [Sh, Chap. X]). In particular we prove the fundamental theorem of semi-proper forcing, which is due to Shelah: any RCS iteration of semi-proper posets is semi-proper.

There is a proper countable support iteration of length ω adding no new reals at finite stages and adding a Sacks real in the limit.

In 2002, Yorioka introduced the σ-ideal ${{\mathcal {I}}_f}$ for strictly increasing functions f from ω into ω to analyze the cofinality of the strong measure zero ideal. For each f, we study the cardinal coefficients (the additivity, covering number, uniformity and cofinality) of ${{\mathcal {I}}_f}$.

We present an approach to forcing with finite sequences of models that uses models of two types. This approach builds on earlier work of Friedman and Mitchell on forcing to add clubs in cardinals larger than $\aleph_{1}$, with finite conditions. We use the two-type approach to give a new proof of the consistency of the proper forcing axiom. The new proof uses a finite support forcing, as opposed to the countable support iteration in the standard proof. (...) The distinction is important since a proof using finite supports is more amenable to generalizations to cardinals greater than $\aleph_{1}$. (shrink)

We present an exposition of Section VI.1 and most of Section VI.2 from Shelah’s book Proper and Improper Forcing. These sections offer proofs of the preservation under countable support iteration of proper forcing of various properties, including proofs that ωω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\omega^\omega}$$\end{document} -bounding, the Sacks property, the Laver property, and the P-point property are preserved by countable support iteration of proper forcing.

We present an exposition of much of Sections VI.3 and XVIII.3 from Shelah's book Proper and Improper Forcing. This covers numerous preservation theorems for countable support iterations of proper forcing, including preservation of the property “no new random reals over V ”, the property “reals of the ground model form a non-meager set”, the property “every dense open set contains a dense open set of the ground model”, and preservation theorems related to the weak bounding property, the weak (...) ωω -bounding property, and the property “the set of reals of the ground model has positive outer measure”. (shrink)

For f, g $ \in \omega ^\omega $ let $c_{f,g}^\forall $ be the minimal number of uniform g-splitting trees (or: Slaloms) to cover the uniform f-splitting tree, i.e., for every branch v of the f-tree, one of the g-trees contains v. $c_{f,g}^\exists $ is the dual notion: For every branch v, one of the g-trees guesses v(m) infinitely often. It is consistent that $c_{f \in ,g \in }^\exists = c_{f \in ,g \in }^\forall = k_ \in $ for N₁ many (...) pairwise different cardinals $k_ \in $ and suitable pairs $(f_{ \in ,g \in } ).$ For the proof we use creatures with sufficient bigness and halving. We show that the lim-inf creature forcing satisfies fusion and pure decision. We introduce decisiveness and use it to construct a variant of the countable support iteration of such forcings, which still satisfies fusion and pure decision. (shrink)

We introduce a class of proper posets which is preserved under countable support iterations, includes \(\omega ^\omega \) -bounding, Cohen, Miller, and Mathias posets associated to filters with the Hurewicz covering properties, and has the property that the ground model reals remain splitting and unbounded in corresponding extensions. Our results may be considered as a possible path towards solving variations of the famous Roitman problem.

We define the notion of Souslin forcing, and we prove that some properties are preserved under iteration. We define a weaker form of Martin's axiom, namely MA(Γ + ℵ 0 ), and using the results on Souslin forcing we show that MA(Γ + ℵ 0 ) is consistent with the existence of a Souslin tree and with the splitting number s = ℵ 1 . We prove that MA(Γ + ℵ 0 ) proves the additivity of measure. Also we (...) introduce the notion of proper Souslin forcing, and we prove that this property is preserved under countable support iterated forcing. We use these results to show that ZFC + there is an inaccessible cardinal is equiconsistent with ZFC + the Borel conjecture + Σ 1 2 -measurability. (shrink)

The Theme. Strong forcing axioms like Martin’s Maximum give a reasonably satisfactory structural analysis of $H(\omega _2)$. A broad program in modern Set Theory is searching for strong forcing axioms beyond $\omega _1$. In other words, one would like to figure out the structural properties of taller initial segments of the universe. However, the classical techniques of forcing iterations seem unable to bypass the obstacles, as the resulting forcings axioms beyond $\omega _1$ have not thus far been strong enough! However, (...) with his celebrated work on generalised side conditions, I. Neeman introduced us to a novel paradigm to iterate forcings. In particular, he could, among other things, reprove the consistency of the Proper Forcing Axiom using an iterated forcing with finite supports. In 2015, using his technology of virtual models, Veličković built up an iteration of semi-proper forcings with finite supports, hence reproving the consistency of Martin’s Maximum, an achievement leading to the notion of a virtual model.In this thesis, we are interested in constructing forcing notions with finitely many virtual models as side conditions to preserve three uncountable cardinals. The thesis constitutes six chapters and three appendices that amount to 118 pages, where Section 1 is devoted to preliminaries, and Section 2 is a warm-up about the scaffolding poset of a proper forcing. In Section 3, we present the general theory of virtual models in the context of forcing with sets of models of two types, where we, e.g., define the “meet” between two virtual models and prove its properties.The main results are joint with Boban Veličković, and partly appeared in Guessing models and the approachability ideal, J. Math. Log. 21 (2021).Pure Side Conditions. In Section 4, we use two types of virtual models (countable and large non-transitive ones induced by a supercompact cardinal, which we call Magidor models) to construct our forcing with pure side conditions. The forcing covertly uses a third type of models that are transitive. We also add decorations to the conditions to add many clubs in the generic $\omega _2$. In contrast to Neeman’s method, we do not have a single chain, but $\alpha $ -chains, for an ordinal $\alpha $ with $V_\alpha \prec V_\lambda $. Thus, starting from suitable large cardinals $\kappa <\lambda $, we construct a forcing notion whose conditions are finite sets of virtual models described earlier. The forcing is strongly proper, preserves $\kappa $, and has the $\lambda $ -Knaster property. The relevant quotients of the forcing are strongly proper, which helps us prove strong guessing model principles. The construction is generalisable to a ${<}\mu $ -closed forcing, for any given cardinal $\mu $ with $\mu ^{<\mu }=\mu <\kappa $.The Iteration Theorem. In Section 5, we use the forcing with pure side conditions to iterate a subclass of proper and $\aleph _2$ -c.c. forcings and obtain a forcing axiom at the level of $\aleph _2$. The iterable class is closely related to Asperó–Mota’s forcing axiom for finitely proper forcings.Guessing Model Principles. Section 6 encompasses the main parts of the thesis. We prove the consistency of the guessing principle $\mathrm {GMP}^+(\omega _3,\omega _1)$ that states for any cardinal ${\theta \geq \omega _3}$, the set of $\aleph _2$ -sized elementary submodels M of $H(\theta )$, which are the union of an $\omega _1$ -continuous $\in $ -chain of $\omega _1$ -guessing, I.C. models is stationary in $\mathcal P_{\omega _3}(H(\theta ))$. The consistency and consequences of this principle are demonstrated in the following diagram. We also prove that one can obtain the above guessing models in a way that the $\omega _1$ -sized $\omega _1$ -guessing models remain $\omega _1$ -guessing model in any outer transitive model with the same $\omega _1$, and we denote this principle by $\rm{SGMP}^+(\omega_3,\omega_1)$.In the following diagram, $\mathrm{TP}$ stands for the tree property; $w\mathrm{KH}$ stands for the weak Kurepa Hypothesis; $\mathrm{MP}$ stands for Mitchell property, i.e., the approachability ideal is trivial modulo the nonstationary ideal; $\mathrm{AP}$ stands for the approachability property; $\mathrm {AMTP}(\kappa ^+)$ states that if $2^\kappa <\aleph _{\kappa ^+}$, then every forcing which adds a new subset of $\kappa ^+$ whose initial segments are in the ground model, collapses some cardinal $\leq 2^{\kappa }$. The dotted arrow denotes the relative consistency, while others are logical implications.Appendices. Appendix A includes merely the above diagram. Appendix B presents a proof of the Mapping Reflection Principle with finite conditions under $\mathrm {PFA}$. Appendix C contains open problems. Finally, the thesis’s bibliography consists of 42 items.Abstract prepared by Rahman MohammadpourE-mail: [email protected]: https://theses.hal.science/tel-03209264. (shrink)

We prove that the property "P doesn't make the old reals Lebesgue null" is preserved under countable support iterations of proper forcings, under the additional assumption that the forcings are nep (a generalization of Suslin proper) in an absolute way. We also give some results for general Suslin ccc ideals.

We show that after forcing with a countable support iteration or a finite product of Sacks or splitting forcing over L, every analytic hypergraph on a Polish space admits a $\mathbf {\Delta }^1_2$ maximal independent set. This extends an earlier result by Schrittesser (see [25]). As a main application we get the consistency of $\mathfrak {r} = \mathfrak {u} = \mathfrak {i} = \omega _2$ together with the existence of a $\Delta ^1_2$ ultrafilter, a $\Pi ^1_1$ maximal (...) independent family, and a $\Delta ^1_2$ Hamel basis. This solves open problems of Brendle, Fischer, and Khomskii [5] and the author [23]. We also show in ZFC that $\mathfrak {d} \leq \mathfrak {i}_{cl}$, addressing another question from [5]. (shrink)

The Covering Property Axiom, which attempts to capture some of the combinatorics of the Sacks model, the model obtained from by countable support iteration of length of the Sacks forcing, seems to miss a Suslin tree. We add a diamond polish to the axiom to remedy this.

We introduce a class of forcing notions, called forcing notions of type S, which contains among other Sacks forcing, Prikry-Silver forcing and their iterations and products with countable supports. We construct and investigate some formalism suitable for this forcing notions, which allows all standard tricks for iterations or products with countable supports of Sacks forcing. On the other hand it does not involve internal combinatorial structure of conditions of iterations or products. We prove that the class of forcing (...) notions of type S is closed under products and certain iterations with countable supports. (shrink)

A model of ZFC is constructed in which the distributivity cardinal h is 2 ℵ 0 = ℵ 2 , and in which there are no ω 2 -towers in [ω] ω . As an immediate corollary, it follows that any base-matrix tree in this model has no cofinal branches. The model is constructed via a form of iterated Mathias forcing, in which a mixture of finite and countable supports is used.

We show that Hechler’s forcings for adding a tower and for adding a mad family can be represented as finite support iterations of Mathias forcings with respect to filters and that these filters are $${\mathcal {B}}$$ B -Canjar for any countably directed unbounded family $${\mathcal {B}}$$ B of the ground model. In particular, they preserve the unboundedness of any unbounded scale of the ground model. Moreover, we show that $${\mathfrak {b}}=\omega _1$$ b = ω 1 in every extension by (...) the above forcing notions. (shrink)

We study two ideals which are naturally associated to independent families. The first of them, denoted \, is characterized by a diagonalization property which allows along a cofinal sequence of stages along a finite support iteration to adjoin a maximal independent family. The second ideal, denoted \\), originates in Shelah’s proof of \ in Shelah, 433–443, 1992). We show that for every independent family \, \\subseteq \mathcal {J}_\mathcal {A}\) and define a class of maximal independent families, to which (...) we refer as densely maximal, for which the two ideals coincide. Building upon the techniques of Shelah we characterize Sacks indestructibility for such families in terms of properties of \\) and devise a countably closed poset which adjoins a Sacks indestructible densely maximal independent family. (shrink)

We consider the existence of coherent families of finite-to-one functions on countable subsets of an uncountable cardinal κ. The existence of such families for κ implies the existence of a winning 2-tactic for player TWO in the countable-finite game on κ. We prove that coherent families exist on κ = ωn, where n ∈ ω, and that they consistently exist for every cardinal κ. We also prove that iterations of Axiom A forcings with countable supports are Axiom (...) A. (shrink)

Any finite support iteration of posets with precalibre ℵ 1 which has the length of cofinality greater than ω 1 yields a model for the dual Borel conjecture in which the real line is covered by ℵ 1 strong measure zero sets.

This article continues Rosłanowski and Shelah (Int J Math Math Sci 28:63–82, 2001; Quaderni di Matematica 17:195–239, 2006; Israel J Math 159:109–174, 2007; 2011; Notre Dame J Formal Logic 52:113–147, 2011) and we introduce here a new property of (<λ)-strategically complete forcing notions which implies that their λ-support iterations do not collapse λ + (for a strongly inaccessible cardinal λ).

We consider of constructing normal ultrafilters in extensions are here Easton support iterations of Prikry-type forcing notions. New ways presented. It turns out that, in contrast with other supports, seemingly unrelated measures or extenders can be involved here.

We give some applications of mixed support forcing iterations to the topics of disjoint stationary sequences and internally approachable sets. In the first half of the paper we study the combinatorial content of the idea of a disjoint stationary sequence, including its relation to adding clubs by forcing, the approachability ideal, canonical structure, the proper forcing axiom, and properties related to internal approachability. In the second half of the paper we present some consistency results related to these ideas. We (...) construct a model in which a disjoint stationary sequence exists at the successor of an arbitrary regular uncountable cardinal. We also construct models in which the properties of being internally stationary, internally club, and internally approachable are distinct. (shrink)

The dissertation under comment is a contribution to the area of Set Theory concerned with the interactions between the method of Forcing and the so-called Large Cardinal axioms.The dissertation is divided into two thematic blocks. In Block I we analyze the large-cardinal hierarchy between the first supercompact cardinal and Vopěnka’s Principle. In turn, Block II is devoted to the investigation of some problems arising from Singular Cardinal Combinatorics.We commence Part I by investigating the Identity Crisis phenomenon in the region comprised (...) between the first supercompact cardinal and Vopěnka’s Principle. As a result, we generalize Magidor’s classical theorems [2] to this higher region of the large-cardinal hierarchy. Also, our analysis allows to settle all the questions that were left open in [1]. Finally, we conclude Part I by presenting a general theory of preservation of $C^{}$ -extendible cardinals under class forcing iterations. From this analysis we derive several applications. For instance, our arguments are used to show that an extendible cardinal is consistent with “ $^{\mathrm {HOD}}<\lambda ^+$, for every regular cardinal $\lambda $.” In particular, if Woodin’s HOD Conjecture holds, and therefore it is provable in ZFC + “There exists an extendible cardinal” that above the first extendible cardinal every singular cardinal $\lambda $ is singular in HOD and $^{\textrm {{HOD}}}=\lambda ^+$, there may still be no agreement at all between V and HOD about successors of regular cardinals.In Part II and Part III we analyse the relationship between the Singular Cardinal Hypothesis with other relevant combinatorial principles at the level of successors of singular cardinals. Two of these are the Tree Property and the Reflection of Stationary sets, which are central in Infinite Combinatorics.Specifically, Part II is devoted to prove the consistency of the Tree Property at both $\kappa ^+$ and $\kappa ^{++}$, whenever $\kappa $ is a strong limit singular cardinal witnessing an arbitrary failure of the SCH. This generalizes the main result of [3] in two senses: it allows arbitrary cofinalities for $\kappa $ and arbitrary failures for the SCH.In the last part of the dissertation we introduce the notion of $\Sigma $ -Prikry forcing. This new concept allows an abstract and uniform approach to the theory of Prikry-type forcings and encompasses several classical examples of Prikry-type forcing notions, such as the classical Prikry forcing, the Gitik-Sharon poset, or the Extender Based Prikry forcing, among many others.Our motivation in this part of the dissertation is to prove an iteration theorem at the level of the successor of a singular cardinal. Specifically, we aim for a theorem asserting that every $\kappa ^{++}$ -length iteration with support of size $\leq \kappa $ has the $\kappa ^{++}$ -cc, provided the iterates belong to a relevant class of $\kappa ^{++}$ -cc forcings. While there are a myriad of works on this vein for regular cardinals, this contrasts with the dearth of investigations in the parallel context of singular cardinals. Our main contribution is the proof that such a result is available whenever the class of forcings under consideration is the family of $\Sigma $ -Prikry forcings. Finally, and as an application, we prove that it is consistent—modulo large cardinals—the existence of a strong limit cardinal $\kappa $ with countable cofinality such that $\mathrm {SCH}_\kappa $ fails and every finite family of stationary subsets of $\kappa ^+$ reflects simultaneously. (shrink)

Let M be a short extender mouse. We prove that if $E\in M$ and $M\models $ “E is a countably complete short extender whose support is a cardinal $\theta $ and $\mathcal {H}_\theta \subseteq \mathrm {Ult}(V,E)$ ”, then E is in the extender sequence $\mathbb {E}^M$ of M. We also prove other related facts, and use them to establish that if $\kappa $ is an uncountable cardinal of M and $\kappa ^{+M}$ exists in M then $(\mathcal {H}_{\kappa ^+})^M$ satisfies (...) the Axiom of Global Choice. We prove that if M satisfies the Power Set Axiom then $\mathbb {E}^M$ is definable over the universe of M from the parameter $X=\mathbb {E}^M\!\upharpoonright \!\aleph _1^M$, and M satisfies “Every set is $\mathrm {OD}_{\{X\}}$ ”. We also prove various local versions of this fact in which M has a largest cardinal, and a version for generic extensions of M. As a consequence, for example, the minimal proper class mouse with a Woodin limit of Woodin cardinals models “ $V=\mathrm {HOD}$ ”. This adapts to many other similar examples. We also describe a simplified approach to Mitchell–Steel fine structure, which does away with the parameters $u_n$. (shrink)

De Finetti would claim that we can make sense of a draw in which each positive integer has equal probability of winning. This requires a uniform probability distribution over the natural numbers, violating countable additivity. Countable additivity thus appears not to be a fundamental constraint on subjective probability. It does, however, seem mandated by Dutch Book arguments similar to those that support the other axioms of the probability calculus as compulsory for subjective interpretations. These two lines of (...) reasoning can be reconciled through a slight generalization of the Dutch Book framework. Countable additivity may indeed be abandoned for de Finetti's lottery, but this poses no serious threat to its adoption in most applications of subjective probability. Introduction The de Finetti lottery Two objections to equiprobability 3.1 The ‘No random mechanism’ argument 3.2 The Dutch Book argument Equiprobability and relative betting quotients The re-labelling paradox 5.1 The paradox 5.2 Resolution: from symmetry to relative probability Beyond the de Finetti lottery. (shrink)

We work out the details of a schema for a mixed support forcing iteration, which generalizes the Mitchell model [7] with no Aronszajn trees on ω2.

Iterated conditionals of the form If p, then if q, r are an important topic in philosophical logic. In recent years, psychologists have gained much knowledge about how people understand simple conditionals, but there are virtually no published psychological studies of iterated conditionals. This paper presents experimental evidence from a study comparing the iterated form, If p, then if q, r with the “imported,” noniterated form, If p and q, then r, using a probability evaluation task and a truth‐table task, (...) and taking into account qualitative individual differences. This allows us to critically contrast philosophical and psychological approaches that make diverging predictions regarding the interpretation of these forms. The results strongly support the probabilistic Adams conditional and the “new paradigm” that takes this conditional as a starting point. (shrink)

Iterated conditionals of the form If p, then if q, r are an important topic in philosophical logic. In recent years, psychologists have gained much knowledge about how people understand simple conditionals, but there are virtually no published psychological studies of iterated conditionals. This paper presents experimental evidence from a study comparing the iterated form, If p, then if q, r with the “imported,” noniterated form, If p and q, then r, using a probability evaluation task and a truth-table task, (...) and taking into account qualitative individual differences. This allows us to critically contrast philosophical and psychological approaches that make diverging predictions regarding the interpretation of these forms. The results strongly support the probabilistic Adams conditional and the “new paradigm” that takes this conditional as a starting point. (shrink)

The prequel to this paper introduced the topic of iteration principles in epistemology and surveyed some arguments in support of them. In this sequel, I'll consider two influential families of objection to iteration principles. The first turns on the idea that they lead to some variety of skepticism, and the second turns on ‘margin for error’ considerations adduced by Timothy Williamson.

Brouwer’s view on induction has relatively recently been characterised as one on which it is not only intuitive (as expected) but functional, by van Dalen. He claims that Brouwer’s ‘Ur-intuition’ also yields the recursor. Appealing to Husserl’s phenomenology, I offer an analysis of Brouwer’s view that supports this characterisation and claim, even if assigning the primary role to the iterator instead. Contrasts are drawn to accounts of induction by Poincaré, Heyting, and Kreisel. On the phenomenological side, the analysis provides an (...) alternative to that in Tieszen’s Mathematical Intuition, and confirms a view of Gödel on his Dialectica Interpretation. (shrink)

In Part I of this series [A. Poveda, A. Rinot and D. Sinapova, Sigma-Prikry forcing I: The axioms, Canad. J. Math. 73(5) (2021) 1205–1238], we introduced a class of notions of forcing which we call [Formula: see text]-Prikry, and showed that many of the known Prikry-type notions of forcing that center around singular cardinals of countable cofinality are [Formula: see text]-Prikry. We showed that given a [Formula: see text]-Prikry poset [Formula: see text] and a [Formula: see text]-name for a (...) non-reflecting stationary set [Formula: see text], there exists a corresponding [Formula: see text]-Prikry poset that projects to [Formula: see text] and kills the stationarity of [Formula: see text]. In this paper, we develop a general scheme for iterating [Formula: see text]-Prikry posets and, as an application, we blow up the power of a countable limit of Laver-indestructible supercompact cardinals, and then iteratively kill all non-reflecting stationary subsets of its successor. This yields a model in which the singular cardinal hypothesis fails and simultaneous reflection of finite families of stationary sets holds. (shrink)

Journal of Mathematical Logic, Volume 22, Issue 03, December 2022. In Part I of this series [A. Poveda, A. Rinot and D. Sinapova, Sigma-Prikry forcing I: The axioms, Canad. J. Math. 73(5) (2021) 1205–1238], we introduced a class of notions of forcing which we call [math]-Prikry, and showed that many of the known Prikry-type notions of forcing that center around singular cardinals of countable cofinality are [math]-Prikry. We showed that given a [math]-Prikry poset [math] and a [math]-name for a (...) non-reflecting stationary set [math], there exists a corresponding [math]-Prikry poset that projects to [math] and kills the stationarity of [math]. In this paper, we develop a general scheme for iterating [math]-Prikry posets and, as an application, we blow up the power of a countable limit of Laver-indestructible supercompact cardinals, and then iteratively kill all non-reflecting stationary subsets of its successor. This yields a model in which the singular cardinal hypothesis fails and simultaneous reflection of finite families of stationary sets holds. (shrink)

We consider a classM of Boolean algebras with strictly positive, finitely additive measures. It is shown thatM is closed under iterations with finite support and that the forcing via such an algebra does not destroy the Lebesgue measure structure from the ground model. Also, we deduce a simple characterization of Martin's Axiom reduced to the classM.

In this paper, we discuss what we argue is a newly observed use of nouns like woman, man, and lawyer, in the sort of morphosyntax characteristic of count nouns. We argue that the relevant data constitutes normative uses of the relevant nouns, and we build an analysis on the assumption that such nouns are polysemous between descriptive and normative senses (Leslie 2015), using the formal account of polysemy in Pustejovsky (1998), and the analysis of count- ability in Rothstein (2010). In (...) doing so, we provide evidence in support of the aforementioned kinds of analyses of social terms and countability, as opposed to others which do not seem to be able to account for the data in as straight forward a way. (shrink)

If M is a countable transitive model of $ZFC+MA_{\aleph_{1}}$ , then for every real x there is a unique shortest iteration $j: M \rightarrow N$ with $x \in N$ , or none at all.

The iterative conception of set commonly is regarded as supporting the axioms of Zermelo-Fraenkel set theory (ZF). This paper presents a modified version of the iterative conception of set and explores the consequences of that modified version for set theory. The modified conception maintains most of the features of the iterative conception of set, but allows for some non-wellfounded sets. It is suggested that this modified iterative conception of set supports the axioms of Quine's set theory NF.

This manuscript investigates fixed point of single-valued Hardy-Roger’s type F -contraction globally as well as locally in a convex b -metric space. The paper, using generalized Mann’s iteration, iterates fixed point of the abovementioned contraction; however, the third axiom of the F -contraction is removed, and thus the mapping F is relaxed. An important approach used in the article is, though a subset closed ball of a complete convex b -metric space is not necessarily complete, the convergence of the (...) Cauchy sequence is confirmed in the subset closed ball. The results further lead us to some important corollaries, and examples are produced in support of our main theorems. The paper most importantly presents application of our results in finding solution to the integral equations. (shrink)

We show that there are proper forcings based upon countable trees of creatures that specialise a given Aronszajn tree.

The present paper argues that there is a knowledge norm for conversational implicature: one may conversationally implicate p only if one knows p. Linguistic data about the cancellation behavior of implicatures and the ways they are challenged and criticized by speakers is presented to support the thesis. The knowledge norm for implicature is then used to present a new consideration in favor of the KK thesis. It is argued that if implicature and assertion have knowledge norms, then assertion requires (...) not only knowledge but iterated knowledge: knowing that you know that you know that... you know. Such a condition on permissible assertion is argued to be plausible only if the KK thesis is true. (shrink)