We study an extensive connection between quotient forcings of Borel subsets of Polish spaces modulo a σ-ideal and quotient forcings of subsets of countable sets modulo an ideal.

We isolate a forcing which increases the value of δ12 while preserving ω₁ under the assumption that there is a precipitous ideal on ω₁ and a measurable cardinal.

With every σ-ideal I on a Polish space we associate the σ-ideal I* generated by the closed sets in I. We study the forcing notions of Borel sets modulo the respective σ-ideals I and I* and find connections between their forcing properties. To this end, we associate to a σ-ideal on a Polish space an ideal on a countable set and show how forcing properties of the forcing depend on combinatorial properties of the ideal. We also study the 1—1 or (...) constant property of σ-ideals, i.e., the property that every Borel function defined on a Borel positive set can be restricted to a positive Borel set on which it either 1—1 or constant. We prove the following dichotomy: if I is a σ-ideal generated by closed sets, then either the forcing P I adds a Cohen real, or else I has the 1—1 or constant property. (shrink)

We investigate classes of Boolean algebras related to the notion of forcing that adds Cohen reals. A Cohen algebra is a Boolean algebra that is dense in the completion of a free Boolean algebra. We introduce and study generalizations of Cohen algebras: semi-Cohen algebras, pseudo-Cohen algebras and potentially Cohen algebras. These classes of Boolean algebras are closed under completion.

Given a Polish space X and a countable collection of analytic hypergraphs on X, I consider the σ-ideal generated by Borel anticliques for the hypergraphs in the family. It turns out that many of the quotient posets are proper. I investigate the forcing properties of these posets, certain natural operations on them, and prove some related dichotomies.

There is an optimal way of increasing certain cardinal invariants of the continuum.

The cut and choose game is one of the infinitary games on a complete Boolean algebra B introduced by Jech. We prove that existence of a winning strategy for II in implies semiproperness of B. If the existence of a supercompact cardinal is consistent then so is “for every 1-distributive algebra B II has a winning strategy in ”.

We present two ways in which the model L(R) is canonical assuming the existence of large cardinals. We show that the theory of this model, with ordinal parameters, cannot be changed by small forcing; we show further that a set of ordinals in V cannot be added to L(R) by small forcing. The large cardinal needed corresponds to the consistency strength of AD L (R); roughly ω Woodin cardinals.

We isolate several large classes of definable proper forcings and show how they include many partial orderings used in practice.

Let I be an F σ -ideal on natural numbers. We characterize the ultrafilters which are generic over the model L for the poset of I -positive sets of natural numbers ordered by inclusion.

We show that in a σ‐closed forcing extension, the bounded forcing axiom for Namba forcing fails. This answers a question of J. T. Moore.

We study a generalization of the splitting number s to uncountable cardinals. We prove that $\mathfrak{s}(\kappa) > \kappa^+$ for a regular uncountable cardinal κ implies the existence of inner models with measurables of high Mitchell order. We prove that the assumption $\mathfrak{s}(\aleph_\omega) > \aleph_{\omega + 1}$ has a considerable large cardinal strength as well.

I describe several ways in which forcing arguments can be used to yield clean and conceptual proofs of nonreducibility, ergodicity and other results in the theory of analytic equivalence relations. In particular, I present simple Borel equivalence relations $E, F$ such that a natural proof of nonreducibility of $E$ to $F$ uses the independence of the Singular Cardinal Hypothesis at $\aleph_\omega$.

I describe several ways in which forcing arguments can be used to yield clean and conceptual proofs of nonreducibility, ergodicity and other results in the theory of analytic equivalence relations. In particular, I present simple Borel equivalence relationsE, Fsuch that a natural proof of nonreducibility ofEtoFuses the independence of the Singular Cardinal Hypothesis at ℵω.

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.

If [Formula: see text] is a closed Noetherian graph on a [Formula: see text]-compact Polish space with no infinite cliques, it is consistent with the choiceless set theory ZF[Formula: see text][Formula: see text][Formula: see text]DC that [Formula: see text] is countably chromatic and there is no Vitali set.

The relationship between killing ideals and adding reals by forcings is analysed.

It is proved consistent that there be a proper σ-ideal ℑ on ω 1 and an ℵ 1 -preserving poset P such that $\mathbb{P} \Vdash$ the σ-ideal generated by ℑ̌ is not proper.

It is proved consistent that there be a proper $\sigma$-ideal $\Im$ on $\omega_1$ and an $\aleph_1$-preserving poset $\mathbb{P}$ such that $\mathbb{P} \Vdash$ the $\sigma$-ideal generated by $\check{\Im}$ is not proper.

We show that all posets of uniform density ℵ 1 may have to add a Cohen real and develop some forcing machinery for obtaining this sort of result.

Certain separation problems in descriptive set theory correspond to a forcing preservation property, with a fusion type infinite game associated to it. As an application, it is consistent with the axioms of set theory that the circle.

It is consistent with ZF $\mathsf {ZF}$ set theory that the Euclidean topology on R $\mathbb {R}$ is not sequential, yet every infinite set of reals contains a countably infinite subset. This answers a question of Gutierres.

Certain set theoretical notions cannot be split into finer subnotions.

In mathematical practice certain formulas φ are believed to essentially decide all other natural properties of the object x. The purpose of this paper is to exactly quantify such a belief for four formulas φ, namely “x is a Ramsey ultrafilter”, “x is a free Souslin tree”, “x is an extendible strong Lusin set” and “x is a good diamond sequence”.

We study the possible values of the cofinality invariant for various Borel ideals on the natural numbers. We introduce the notions of a fragmented and gradually fragmented ideal and prove a dichotomy for fragmented ideals. We show that every gradually fragmented ideal has cofinality consistently strictly smaller than the cardinal invariant and produce a model where there are uncountably many pairwise distinct cofinalities of gradually fragmented ideals.

We present Woodin's proof that if there exists a measurable Woodin cardinal δ, then there is a forcing extension satisfying all $\Sigma _{2}^{2}$ sentences ϕ such that CH + ϕ holds in a forcing extension of V by a partial order in V δ . We also use some of the techniques from this proof to show that if there exists a stationary limit of stationary limits of Woodin cardinals, then in a homogeneous forcing extension there is an elementary embedding (...) j: V → M with critical point $\omega _{1}^{V}$ such that M is countably closed in the forcing extension. (shrink)

Turtles are among the most intriguing amniotes but their communication and signaling have rarely been studied. Traditionally, they have been seen as basically just silent armored ‘walking stones’ with complex physiology but no altruism, maternal care, or aesthetic perception. Recently, however, we have witnessed a radical change in the perception of turtle behavioral and cognitive skills. In our study, we start by reviewing some recent findings pertaining to various highly developed behavioral and cognitive patterns with special emphasis on turtles. Then (...) we focus on freshwater turtles and use data about their sexual behavior and size sexual dimorphism to test whether conspicuous coloration of the head is in these animals related to sexual processes. We found that absence of aggressive mating behavior is statistically associated with the presence of conspicuous coloration on turtles’ heads. It also seems that while species with female-biased SSD are characterised by conspicuously colored head ornaments, in species with male-biased SSD conspicuous coloration is absent. Unlike large females, males thus seem to be under pressure to develop conspicuous coloration and engage in non-aggressive behavior using signaling to succeed in courtship. And finally, we discuss possible roles of head color patterns in turtle communication during mating. (shrink)

The aim of this essay is to investigate various attempts at possible answers regarding the question of what is real. The basic assumption is that Kant had provided a philosophical theory of a kind such that it allows this question to be answered in a novel way. In order to show that, the philosophical conceptions of his predecessors are investigated: the metaphysical concept of A. G. Baumgarten on the one hand, the empirical concept of D. Hume on the other. The (...) main result of the investigation is that, according to Kant, we can take something to be real only insofar as it is in a relationship to ourselves. (shrink)

Appearance of Old world vipers of genus Vipera serves various purposes including crypsis and aposematism. Recent research showed that the zigzag pattern represents strong signal to predators to avoid vipers as a prey. It is also possible that vipers function within ecological community as Batesian model for numerous mimics, including other reptiles, birds, and invertebrates. It is then showed that Batesian models need to have prominent features to sustain the mimicry system. The main modulation of this system is presented here (...) as iconicity. Iconicity is treated as quantitative variable resulting from open dynamic process with multiple inputs, also including iconicity of previous states of system. Batesian mimicry is based on mimics adopting the iconicity of the model. It is an example of ecological facilitation, and as such, it is part of niche construction. Since Batesian mimicry is based on semiotic processes, it is a special case of ecological facilitation, namely semiotic facilitation. (shrink)

Tento článek usiluje o systematický popis teorie stylů myšlení a myšlenkových společenství polského mikrobiologa Ludwika Flecka. Článek se zabývá výchozím bodem jeho teorie: případovou studií tzv. Wassermanova testu. Následně je Fleckova teorie prezentována nejprve ve světle Struktury vědeckých revolucí Thomase Kuhna. Jsou zaznamenány některé podobnosti mezi oběma mysliteli. Přesto se Fleckova stanoviska od Kuhnových v některých důležitých ohledech liší. Na rozdíl od převládajícího názoru, tyto rozdíly zamezují tomu, aby byl Fleck považován za předchůdce Kuhna. Z těchto důvodů tento článek zmiňuje (...) také možnost prezentovat Fleckova stanoviska v jiném kontextu, jenž je pokládán za prospěšnější, a to francouzská epistemologie. (shrink)

Kuhn byl stoupencem tzv. „relativizovaného apriori". Apriori se nevyznačuje apodiktičností; zachovává si však stále konstitutivní funkci pro předmět poznání. Poznávající subjekty musí znát významy „paradigmatických propozic" proto, aby měli zkušenost. Zkušenost se neredukuje na vnímání. S Fleckem řečeno, pro vidění je dívání se jen nutnou, nikoli však dostatečnou podmínkou; je nadto nutné i vědět. Paradigmatické propozice se tak stávají podmínkou veškerého poznání, které je jimi „nasyceno". Různým paradigmatům respektive teoriím proto odpovídají různé zkušenosti. Paradigmata však nejsou na zkušenosti zcela nezávislá. (...) „Paradigmatické propozice" mají zkušenostní původ. Jedná se, s Poincarém řečeno, o hypotézy, které byly povzneseny do statutu principů. Jinak řečeno, role „zkušenostního vstupu" v poznání je poněkud paradoxní: vstup motivuje tu část poznání, která je na něm nezávislá a která jej podmiňuje. (shrink)

What is conventionalism in philosophy of science? Basically, it is a thesis about empirical underdetermination. According to Conventionalists, there is “a slack” between our theories and experience that is to be “lined” with conventions. As the experience does not “impose” any theory, scientists are always free to choose a theory on “softer” non-evidential grounds when facing empirical underdetermination. “Conventionalism is a philosophy of freedom,” as Édouard Le Roy put it. Yet the thing to remember is that there is no such (...) a thing as the conventionalism. Reasons for empirical underdetermination that Conventionalists state are not always the same, hence it is more convenient to talk about varieties of conventionalism. The present paper is an attempt to sketch a line between two basic variants of conventionalism which are instrumentalism and constructivism. Co je konvencionalismus? V nejobecnější rovině se jedná o tvrzení o empirické poddeterminovanosti. Podle konvencionalistů existuje "mezera" mezi zkušeností a teoriemi, která může být překonána pouze pomocí konvencí. Vědci si podle konvencionalistů mohou vybrat teorie na základě měkkých kritérií, protože teorie nikdy nejsou předepisovány zkušeností. Konvencionalimus je, jak jednou poznamenal stoupenec jeho radikální varianty, Édouard Le Roy, "filozofii svobody." Konvencionalismus by však neměl být považován za homogenní proud. Konvencionalisté se v podstatných ohledech různí v názorech na příčiny empirické poddeterminovanosti, což je i důvodem, proč je příhodnější hovořit o variantách konvencionalismu. Předložený článek je pokusem vymezit dvě základní varianty konvencionalismu: instrumentalismus a konstruktivismus. (shrink)

Článek srovnává pojetí člověka jako bytosti, situované v celkové souvislosti a vyznačující se otevřeností a sebetranscendencí, v myšlení Jana Patočky a Karla Rahnera, a to se zvláštním zřetelem na interpretaci postavy bohočlověka u obou autorů. Srovnává a dává do souvislosti Patočkův pojem "světa" a pojem "tajemství" u Karla Rahnera, jejich pojetí dějinnosti i radikálního ohrožení člověka. Konečně se zabývá interpretací postavy bohočlověka, kterou můžeme u obou autorů chápat jako klíčovou pro lidskou existenci, a zároveň si můžeme všimnout, že odlišné interpretace (...) tohoto motivu u obou autorů jsou si vnitřně překvapivě blízké. (shrink)

Dominantn pozitivistick epistemologie dat v souasn vd, kter stav na metafyzickm realismu a idelu mechanick objektivity, trp adou trhlin. Tyto nedostatky byly identifikovny v mnohch kritickch ohlasech, kter vedou k nov problematizaci ustlenho konceptu dat. Mezi kritizovanmi aspekty tohoto konceptu se opakuj varovn ohledn zasazenosti dat do kontextu jejich tvorby, jejich zprostedkovanost nebo otevenost manipulacm. Funkce dat v poslednch letech nabv na dleitosti kvli stoupajc datov nronosti vdy, a proto je dve neproblematickmu konceptu dat vnovno vce pozornosti. Objevuj se tak (...) alternativn pstupy k epistemologii dat, z nich tato prce pedstavuje smr konstruktivistick a rtorick. Prezentovan text lze dky sv kompilativn povaze, erpajc z kritick literatury zabvajc se epistemologi dat, pojmout jako syntzu a rekonfiguraci existujcch mylenek k danmu tmatu. Je mon ho rovn pijmout jako jeden z pspvk k rtorick argumentaci v diskurzu dat souasn vdy. (shrink)

We define the property of Π2-compactness of a statement Φ of set theory, meaning roughly that the hard core of the impact of Φ on combinatorics of 1 can be isolated in a canonical model for the statement Φ. We show that the following statements are Π2-compact: “dominating NUMBER = 1,” “cofinality of the meager IDEAL = 1”, “cofinality of the null IDEAL = 1”, “bounding NUMBER = 1”, existence of various types of Souslin trees and variations on uniformity of (...) measure and CATEGORY = 1. Several important new metamathematical patterns among classical statements of set theory are pointed out. (shrink)