The purpose of this paper is to outline some recent progress in descriptive inner model theory, a branch of set theory which studies descriptive set theoretic and inner model theoretic objects using tools from both areas. There are several interlaced problems that lie on the border of these two areas of set theory, but one that has been rather central for almost two decades is the conjecture known as the Mouse Set Conjecture. One particular motivation for resolving MSC is that (...) it provides grounds for solving the inner model problem which dates back to 1960s. There have been some new partial results on MSC and the methods used to prove the new instances suggest a general program for solving the full conjecture. It is then our goal to communicate the ideas of this program to the community at large. (shrink)

Building on the work of Schimmerling [Coherent sequences and threads, Adv. Math.216 89–117] and Steel [PFA implies AD L, J. Symbolic Logic70 1255–1296], we show that the failure of square principle at a singular strong limit cardinal implies that there is a nontame mouse. The proof presented is the first inductive step beyond L of the core model induction that is aimed at getting a model of ADℝ + "Θ is regular" from the failure of square at a singular strong (...) limit cardinal or PFA. (shrink)

Frege’s famous definition of number famously uses the concept of “extension”. Extensions, in the Fregean framework, are susceptible to bringing many difficulties, and, some say, even paradoxes. Therefore, neo-logicist programs want to avoid the problems and to replace the classical Fregean definition of number with Hume’s Principle. I argue that this move, even if it makes sense from a computational point of view, is at odds with Frege’s larger philosophical project. For Frege, I claim, extensions were an important part of (...) his philosophical program of logic-as-an-universal-language. This is why Frege places his project in line with Leibniz’ philosophical project of finding a lingua characterica universalis. (shrink)

Frege’s famous definition of number famously uses the concept of “extension”. Extensions, in the Fregean framework, are susceptible to bringing many difficulties, and, some say, even paradoxes. Therefore, neo-logicist programs want to avoid the problems and to replace the classical Fregean definition of number with Hume’s Principle. I argue that this move, even if it makes sense from a computational point of view, is at odds with Frege’s larger philosophical project. For Frege, I claim, extensions were an important part of (...) his philosophical program of logic-as-an-universal-language. This is why Frege places his project in line with Leibniz’ philosophical project of finding a lingua characterica universalis. (shrink)

Frege’s famous definition of number famously uses the concept of “extension”. Extensions, in the Fregean framework, are susceptible to bringing many difficulties, and, some say, even paradoxes. Therefore, neo-logicist programs want to avoid the problems and to replace the classical Fregean definition of number with Hume’s Principle. I argue that this move, even if it makes sense from a computational point of view, is at odds with Frege’s larger philosophical project. For Frege, I claim, extensions were an important part of (...) his philosophical program of logic-as-an-universal-language. This is why Frege places his project in line with Leibniz’ philosophical project of finding a lingua characterica universalis. (shrink)

I analyze in this article the different ways in which Descartes uses abstraction in the Regulæ, discussing his project of a mathematical physics, the role of the imagination, and the status of numbers. I also try to show that the doctrine of simple natures cannot be well accommodated with the theory of abstraction developed in Rule 14, having instead a greater affinity with Descartes’ later theory of abstraction and exclusion, in which imagination plays no role and everything happens at a (...) strictly intellectual level. This interpretation is supported by the recent discovery of the Cambridge manuscript, which almost certainly records an early stage of composition and in which the doctrine of simple natures is absent, thus being very probably a later development. (shrink)

We show that assuming the determinacy of all games on reals, every set of reals is \ universally baire.

We investigate the indestructibility properties of strongly compact cardinals in universes where strong compactness suffers from identity crisis. We construct an iterative poset that can be used to establish Kimchi–Magidor theorem from (in The independence between the concepts of compactness and supercompactness, circulated manuscript), i.e., that the first n strongly compact cardinals can be the first n measurable cardinals. As an application, we show that the first n strongly compact cardinals can be the first n measurable cardinals while the strong (...) compactness of each strongly compact cardinal is indestructible under Levy collapses (our theorem is actually more general, see Sect. 3). A further application is that the class of strong cardinals can be nonempty yet coincide with the class of strongly compact cardinals while strong compactness of any strongly compact cardinal κ is indestructible under κ-directed closed posets that force GCH at κ. (shrink)

During his Fall 2005 set theory seminar, Woodin asked whether V-supercompactness implies HOD-supercompactness. We show, as he predicted, that that the answer is no.

The paper propose an assemble of (un)conventional explanatory hypotheses regarding insufficiently known aspects of the genesis and evolution of the human language. The causes and mechanisms that justify the hypothesis of an original linguistic nucleus generating ethnic dialects later on have been studied here. These aspects, regarded from the interdisciplinary perspective of such sciences as semiotics and linguistics, neurology and biophotonics, psychosociology, logic and philosophy, are sustaining that the human language (word) history presupposes: an “iconicity phase” (naturalist theory), permitting an (...) essentially motivated communication, and an “arbitrary phase” (conventionalist theory). The “Babel language” myth, for which the “power of the word” means an archetype of reference, has been selected as a symbolic frame in order to describe the stages through which the human language passes on its way from Nature (“language of living“) to Culture (“language of spiritual meanings“). (shrink)

Mediated by a hermeneutic/semiotic reading of symbols, searching out and finding some similitudes between mythical constructions concerning the genesisof the world and scientific hypotheses can be considered fruitful from at least two viewpoints, since: (a) it allows the validation of a truth which is difficult to prove through other means, that early humans had intuitive-cognitive resources much more profound than appearances seem to allow; (b) it enables the utilization of some of the suggestions offered by the mythic language in the (...) scientific language context. Using such correspondences, the author proposes the integrative model of a structurally named "Cluster Universe" or the functionally named "Pulsatory expansion", able to solve some major difficulties of the cosmological thinking. (shrink)

The core of what we call transmodern turn is sustained by the shaping of an ontological model of the “Essentials Unity”, in which Human Being, the World and God should be in a non-conflictual relationship of togetherness, by a resonant / holographic mechanism of light. cognizing that the world-object and metalanguage have an objective interface, religion, philosophy and modern sciences harmonise their specific assertions through a semiotics of the "loving light” capable of proving that: syntactically, the world is governed by (...) unifying patterns which contain the semantic meanings of an objective human “language of light”; from a pragmatic point of view, biophotonically assumed, the fundamental values of human being / history can be traced in the ontologic act of signification which foregrounds the becoming of universal resonance into love. (shrink)

The goal of the present contribution is an unconventional one. It concerns not only a punctual problem - the axiology of ethnical values - assumed on the “vertical” of a single discipline, as it is usually done, but a complex of “horizontal” problems unified by an integrative methodology: that of semiotics. The great quantity of the problems involved obliged us − on a limited frame − to present them intuitively, to formulate only some hypotheses and models, for a preliminary holistic (...) knowledge. (shrink)

To understand – intuitively and analytically both – that man is a „being of light” created as a perfect „face and alikeness” and, implicitly, as a becoming measure of the cosmic „wave patterns”, is the main purpose of the present paper. To recuperate the language of the religion by the privilege of a new „living fields” science (biophotonics) means to realize rationally the millenary expectancy of the ESSENTIAL UNITY: man, cosmos, and divinity, harmoniously together.

Assuming $V=L+AD$, using methods from inner model theory, we give a new proof of the strong partition property for ${\sim}{ \delta }^{2}_{1}$. The result was originally proved by Kechris et al.

We re-prove Becker’s theorem from Becker :229–234, 1981) by showing that \}\) implies that \\vDash ``\omega _2\) is -supercompact”. Our proof uses inner model theoretic tools instead of Baire category. We also show that \ is \-strongly compact.

A set of reals is universally Baire if all of its continuous preimages in topological spaces have the Baire property. ${\sf Sealing}$ is a type of generic absoluteness condition introduced by Woodin that asserts in strong terms that the theory of the universally Baire sets cannot be changed by set forcings. The ${\sf Largest\ Suslin\ Axiom}$ is a determinacy axiom isolated by Woodin. It asserts that the largest Suslin cardinal is inaccessible for ordinal definable surjections. Let ${\sf LSA}$ - ${\sf (...) over}$ - ${\sf uB}$ be the statement that in all generic extensions there is a model of $\sf {LSA}$ whose Suslin, co-Suslin sets are the universally Baire sets. We outline the proof that over some mild large cardinal theory, $\sf {Sealing}$ is equiconsistent with $\sf {LSA}$ - $\sf {over}$ - $\sf {uB}$. In fact, we isolate an exact theory that is equiconsistent with both. As a consequence, we obtain that $\sf {Sealing}$ is weaker than the theory “ $\sf {ZFC}$ + there is a Woodin cardinal which is a limit of Woodin cardinals.” This significantly improves upon the earlier consistency proof of $\sf {Sealing}$ by Woodin. A variation of $\sf {Sealing}$, called $\sf {Tower \ Sealing}$, is also shown to be equiconsistent with $\sf {Sealing}$ over the same large cardinal theory. We also outline the proof that if V has a proper class of Woodin cardinals, a strong cardinal, and a generically universally Baire iteration strategy, then $\sf {Sealing}$ holds after collapsing the successor of the least strong cardinal to be countable. This result is complementary to the aforementioned equiconsistency result, where it is shown that $\sf {Sealing}$ holds in a generic extension of a certain minimal universe. This theorem is more general in that no minimal assumption is needed. A corollary of this is that $\sf {LSA}$ - $\sf {over}$ - $\sf {uB}$ is not equivalent to $\sf {Sealing}$. (shrink)

In this paper, we show that the failure of the unique branch hypothesis for tame iteration trees implies that in some homogenous generic extension of [Formula: see text] there is a transitive model [Formula: see text] containing [Formula: see text] such that [Formula: see text] is regular. The results of this paper significantly extend earlier works from [Non-tame mice from tame failures of the unique branch bypothesis, Canadian J. Math. 66 903–923; Core models with more Woodin cardinals, J. Symbolic Logic (...) 67 1197–1226] for tame trees. (shrink)

The debate concerning the legitimacy of awarding reparations for historical injustices focuses on the issue of finding a proper moral justification for granting reparations to the descendants of the victims of injustices which took place in the remote past. Regarding the case of Romanian communism as a more recent injustice, and analyzing the moral problems entailed by this historical lapse, within this paper I argue that overcoming such a legacy cannot be carried out, as in the case of historical injustices (...) situated more remotely in time, through the means of selective reparations, such as restitutions or compensations. For, even though they are justified from the perspective of rectificatory justice, selective reparations do not fulfill the requirements of social justice. Rather, I argue that the fall of the Romanian communist regime should have been followed by an equal distribution of all properties illegitimately seized by the state, to all adult Romanian citizens at that time, in order to attain the imperative of equal distribution of property among all citizens. The equal distribution thesis is the only way through which the Romanian society could have complied, at that moment of political and social renewal, with the requirements of justice. I also aim at explaining why other principles of justice, which either have or could have been implemented, cannot be properly justified. Finally, I analyze two main objections which could be invoked against my thesis, namely the economic efficiency objection and the legal realist objection. (shrink)

Предназначено для студентов художественных вузов и факультетов, а также для учащихся профессиональных учебных заведений художественного профиля.

kn. 1. Predystorii͡a, zasedanii͡a -- kn. 2. Khronika, portrety.