On the Imperial Order of Saint Prince Alexander Nevsky and the Phenomenon of the Repeated Companionage in the Order Corporation in Russia in the 18th Century

В статье собраны и проанализированы те немногие сведения об императорском Ордене Святого Благоверного Князя Александра Невского, что содержались в Установлении о российских орденах 1797 г. Этот юридический памятник обязан своим появлением неосуществленному проекту императора Павла I учредить единственный в стране Российский Кавалерский Орден, «классом» которого предстояло стать дотоле суверенной институции Св. князя Александра. Поскольку Установление 1797 г. не имело цели восполнить отсутствие на тот момент статутов орденов Свв. Андрея Первозванного и Александра Невского, то чересчур лаконичные и отрывочные упоминания обоих фалеронимов в документе оставило много лакун в описании орденской организации. В частности, Орден Св. Александра, задуманный императором Петром I как первая универсальная специализированная награда столь высокого ранга за заслуги, не получил ясно сформулированных квалификационных требований к потенциальным реципиентам. После отказа государя Павла Петровича от затеи с Российским Кавалерским Орденом, орденская институция Св. князя Александра Невского продолжила свое суверенное существование вплоть до последних дней существования исторической России, однако полноценного уставного документа так и не обрела. Социальные пертурбации эпохи «дворцовых бурь», приводившие к череде поражения в правах представителей политической элиты, приводили к неоднократному повторному вступлению некоторых из них в один и тот же орден. В статье этот феномен отечественной фалеристики рассматривается на примере троекратного пожалования инсигний датского Ордена Слона князю Василию Владимировичу Долгорукову и двукратного вручения регалий Ордена Св. князя Александра Невского светлейшему князю Александру Александровичу Меншикову. The article lists and analyzes a few details of the Imperial Order of Saint Alexander Nevsky that are available from the Statute of Russian Orders of 1797. This juridical artifact owes its advent to the failed attempt of Paul I of Russia to establish the only Knights Hospitaller in Russia which was aimed at providing the sovereign status of Saint Alexander Nevsky. Since the Statute of 1797 was not intended to establish the absent orders of St. Andrew the First-Called (Andrew the Apostle) and Alexander Nevsky, laconic and fragmentary mentions of both phaleronyms in the document left many gaps in the description of the order organization. Particularly, the Order of Saint Alexander Nevsky conceived by Peter the Great as the first unified themed award of merit of such a high rank did not receive clearly stated recipient specifications. Following the failure of Paul I's project of Russian Knights Hospitaller, the order institution of Saint Alexander Nevsky retained its sovereignty until the last days of historical Russia; however, a full-fledged statute was not developed. During the Era of Palace Coups, the social disturbance resulted in a series of defeats in the rights of the elite political class forcing some representatives to repeatedly re-enter the same order. In the article, the phenomenon of Russian phaleristics is studied on the example of Prince Vasily Vladimirovich Dolgorukov triply awarded insignia of the Danish Order of the Elephant and Prince Alexander Alexandrovich Menshikov awarded the regalia of the Order of Saint Alexander Nevsky twice.

HYPERCLASS FORCING IN MORSE-KELLEY CLASS THEORY

In this article we introduce and study hyperclass-forcing (where the conditions of the forcing notion are themselves classes) in the context of an extension of Morse-Kelley class theory, called MK**. We define this forcing by using a symmetry between MK** models and models of ZFC− plus there exists a strongly inaccessible cardinal (called SetMK**). We develop a coding between β-models ${\cal M}$ of MK** and transitive models M+ of SetMK** which will allow us to go from ${\cal M}$ to M+ and vice versa. So instead of forcing with a hyperclass in MK** we can force over the corresponding SetMK** model with a class of conditions. For class-forcing to work in the context of ZFC− we show that the SetMK** model M+ can be forced to look like LK*[X], where κ* is the height of M+, κ strongly inaccessible in M+ and $X \subseteq \kappa$. Over such a model we can apply definable class forcing and we arrive at an extension of M+ from which we can go back to the corresponding β-model of MK**, which will in turn be an extension of the original ${\cal M}$. Our main result combines hyperclass forcing with coding methods of [3] and [4] to show that every β-model of MK** can be extended to a minimal such model of MK** with the same ordinals. A simpler version of the proof also provides a new and analogous minimality result for models of second-order arithmetic.

CLASS FORCING, THE FORCING THEOREM AND BOOLEAN COMPLETIONS

The forcing theorem is the most fundamental result about set forcing, stating that the forcing relation for any set forcing is definable and that the truth lemma holds, that is everything that holds in a generic extension is forced by a condition in the relevant generic filter. We show that both the definability (and, in fact, even the amenability) of the forcing relation and the truth lemma can fail for class forcing.In addition to these negative results, we show that the forcing theorem is equivalent to the existence of a (certain kind of) Boolean completion, and we introduce a weak combinatorial property (approachability by projections) that implies the forcing theorem to hold. Finally, we show that unlike for set forcing, Boolean completions need not be unique for class forcing.

LARGE CARDINALS AND LIGHTFACE DEFINABLE WELL-ORDERS, WITHOUT THE GCH

This paper deals with the question whether the assumption that for every inaccessible cardinal κ there is a well-order of H(κ+) definable over the structure $\langle {\rm{H}}({\kappa ^ + }), \in \rangle$ by a formula without parameters is consistent with the existence of (large) large cardinals and failures of the GCH. We work under the assumption that the SCH holds at every singular fixed point of the ℶ-function and construct a class forcing that adds such a well-order at every inaccessible cardinal and preserves ZFC, all cofinalities, the continuum function, and all supercompact cardinals. Even in the absence of a proper class of inaccessible cardinals, this forcing produces a model of “V = HOD” and can therefore be used to force this axiom while preserving large cardinals and failures of the GCH. As another application, we show that we can start with a model containing an ω-superstrong cardinal κ and use this forcing to build a model in which κ is still ω-superstrong, the GCH fails at κ and there is a well-order of H(κ+) that is definable over H(κ+) without parameters. Finally, we can apply the forcing to answer a question about the definable failure of the GCH at a measurable cardinal.

Pointwise definable models of set theory

A pointwise definable model is one in which every object is definable without parameters. In a model of set theory, this property strengthens V = HOD, but is not first-order expressible. Nevertheless, if ZFC is consistent, then there are continuum many pointwise definable models of ZFC. If there is a transitive model of ZFC, then there are continuum many pointwise definable transitive models of ZFC. What is more, every countable model of ZFC has a class forcing extension that is pointwise definable. Indeed, for the main contribution of this article, every countable model of Gödel-Bernays set theory has a pointwise definable extension, in which every set and class is first-order definable without parameters.