§1. Introduction. By and large, definitions of a differentiable structure on a set involve two ingredients, topology and algebra. However, in some cases, partial information on one or both of these is sufficient. A very simple example is that of the field ℝ where algebra alone determines the ordering and hence the topology of the field:In the case of the field ℂ, the algebraic structure is insufficient to determine the Euclidean topology; another topology, Zariski, is associated with the ield but (...) this will be too coarse to give a diferentiable structure.A celebrated example of how partial algebraic and topological data determines a differentiable structure is Hilbert's 5th problem and its solution by Montgomery-Zippin and Gleason.The main result which we discuss here is of a similar flavor: we recover an algebraic and later differentiable structure from a topological data. We begin with a linearly ordered set ⟨M, <⟩, equipped with the order topology, and its cartesian products with the product topologies. We then consider the collection of definable subsets of Mn, n = 1, 2, …, in some first order expansion ℳ of ⟨M, <⟩. (shrink)

Suppose $D \subset M$ is a strongly minimal set definable in M with parameters from C. We say D is locally modular if for all $X, Y \subset D$ , with $X = \operatorname{acl}(X \cup C) \cap D, Y = \operatorname{acl}(Y \cup C) \cap D$ and $X \cap Y \neq \varnothing$ , dim(X ∪ Y) + dim(X ∩ Y) = dim(X) + dim(Y). We prove the following theorems. Theorem 1. Suppose M is stable and $D \subset M$ is strongly minimal. (...) If D is not locally modular then in M eq there is a definable pseudoplane. (For a discussion of M eq see [M, § A].) This is the main part of Theorem 1 of [Z2] and the trichotomy theorem of [Z3]. Theorem 2. Suppose M is stable and $D, D' \subset M$ are strongly minimal and nonorthogonal. Then D is locally modular if and only if D' is locally modular. (shrink)

We show that is a quasi-minimal torsion-free divisible abelian group. After discussing the axiomatization of the theory of this structure, we present its ω-saturated quasi-minimal model.

The Cherlin–Zil'ber Conjecture states that all simple groups of finite Morley rank are algebraic. We prove that any minimal counterexample to this conjecture has a unique conjugacy class of Carter subgroups, which are analogous to Cartan subgroups in algebraic groups.

In this note we treat maximal and minimal normal subgroups of a superstable group and prove that these groups are definable under certain conditions. Main tool is a superstable version of Zil'ber's indecomposability theorem.

We know quite a lot about the general structure of ω-stable solvable centerless groups of finite Morley rank. Abelian groups of finite Morley rank are also well-understood. By comparison, nonabelian nilpotent groups are a mystery except for the following general results:• An ω1-categorical torsion-free nonabelian nilpotent group is an algebraic group over an algebraically closed field of characteristic 0 [Z3].• A nilpotent group of finite Morley rank is the central product of a definable subgroup of finite exponent and of a (...) definable divisible subgroup [N3].• A divisible nilpotent group of finite Morley rank is the direct product of its torsion part (which is central) and of a torsion-free subgroup [N3].However, we do not understand nilpotent groups of bounded exponent. It seems that the classification of nilpotent (but nonabelian)p-groups of finite Morley rank is impossible. Even the nilpotent groups of Morley rank 2 contain insurmountable difficulties [C], [T]. At first glance, this may seem to be an obstacle to proving the Cherlin-Zil'ber conjecture (“simple groups of finite Morley rank are algebraic groups”). Our purpose in this article is to show that if such a group is a definable subgroup of a nonnilpotent group, then it is possible to obtain a classification within the boundaries of our present knowledge. In this respect, our article may be considered as a relief to those who are trying to classify simple groups of finite Morley rank.Before explicitly stating our result, we need the following definition. (shrink)

In this paper the following theorem is proved regarding groups of finite Morley rank which are perfect central extensions of quasisimple algebraic groups.Theorem1.Let G be a perfect group of finite Morley rank and let C0be a definable central subgroup of G such that G/C0is a universal linear algebraic group over an algebraically closed field; that is G is a perfect central extension of finite Morley rank of a universal linear algebraic group. Then C0= 1.Contrary to an impression which exists in (...) some circles, the center of the universal extension of a simple algebraic group, as an abstract group, is not finite in general. Thus the finite Morley rank assumption cannot be omitted.Corollary1.Let G be a perfect group of finite Morley rank such that G/Z(G) is a quasisimple algebraic group. Then G is an algebraic group. In particular, Z(G) is finite([4],Section27.5).An understanding of central extensions of quasisimple linear algebraic groups which are groups of finite Morley rank is necessary for the classification oftamesimpleK*-groupsof finite Morley rank, which constitutes an approach to the Cherlin-Zil’ber conjecture. For this reason the theorem above and its corollary were proven in [1] (Theorems 4.1 and 4.2) under the assumption oftameness, which simplifies the argument considerably. The result of the present paper shows that this assumption can be dropped. The main line of argument is parallel to that in [1]; the absence of the tameness assumption will be countered by a model-theoretic result and results fromK-theory. The model-theoretic result places limitations on definability in stable fields, and may possibly be relevant to eliminating certain other uses of tameness. (shrink)

The received view on the development of the correspondence rules in Carnap’s philosophy of science is that at first, Carnap assumed the explicit definability of all theoretical terms in observational terms and later weakened this assumption. In the end, he conjectured that all observational terms can be explicitly defined in in theoretical terms, but not vice versa. I argue that from the very beginning, Carnap implicitly held this last view, albeit at times in contradiction to his professed position. To establish (...) this point I argue that, first, Carnap’s ‘Über die Aufgabe der Physik’ is a contribution to the philosophy of science of logical empiricism, contrary to Thomas Mormann and in agreement with Herbert Feigl. Second, Michael Friedman misunderstands the ‘Aufgabe’ with his claim that it describes a method for arriving at explicit definitions for theoretical terms. Another received view on Carnap’s philosophy of science is that it assumed a formalization of physical theories that was too detached from actual physics and thus justly disavowed in favor of the semantic view as, for example, developed by van Fraassen. But the ‘Aufgabe’ and related works including the Aufbau show that from the very beginning to his last works, Carnap suggested formalizing physical theories as restrictions in mathematical spaces, as in the state-space conception of scientific theories favored by van Fraassen. (shrink)

A structure ${\mathbb Y}$ of a relational language L is called almost chainable iff there are a finite set $F \subset Y$ and a linear order $\,<$ on the set $Y\setminus F$ such that for each partial automorphism $\varphi $ of the linear order $\langle Y\setminus F, <\rangle $ the mapping $\mathop {\mathrm {id}}\nolimits _F \cup \varphi $ is a partial automorphism of ${\mathbb Y}$. By theorems of Fraïssé and Pouzet, an infinite structure ${\mathbb Y}$ is almost chainable iff the (...) profile of ${\mathbb Y}$ is bounded; namely, iff there is a positive integer m such that ${\mathbb Y}$ has $\leq m$ non-isomorphic substructures of size n, for each positive integer n. A complete first order L-theory ${\mathcal T}$ having infinite models is called almost chainable iff all models of ${\mathcal T}$ are almost chainable and it is shown that the last condition is equivalent to the existence of one countable almost chainable model of ${\mathcal T}$. In addition, it is proved that an almost chainable theory has either one or continuum many non-isomorphic countable models and, thus, the Vaught conjecture is confirmed for almost chainable theories. (shrink)

We investigate how weak square principles are denied by Chang’s Conjecture and its generalizations. Among other things we prove that Chang’s Conjecture does not imply the failure of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\square_{\omega_1, 2}}$$\end{document}, i.e. Chang’s Conjecture is consistent with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\square_{\omega_1, 2}}$$\end{document}.

Rado’s Conjecture is a compactness/reflection principle that says any nonspecial tree of height ω1 has a nonspecial subtree of size ℵ1. Though incompatible with Martin’s Axiom, Rado’s Conjecture turns out to have many interesting consequences that are also implied by certain forcing axioms. In this paper, we obtain consistency results concerning Rado’s Conjecture and its Baire version. In particular, we show that a fragment of PFA, which is the forcing axiom for Baire Indestructibly Proper forcings, is compatible (...) with the Baire Rado’s Conjecture. As a corollary, the Baire Rado’s Conjecture does not imply Rado’s Conjecture. Then we discuss the strength and limitations of the Baire Rado’s Conjecture regarding its interaction with stationary reflection principles and some families of weak square principles. Finally, we investigate the influence of Rado’s Conjecture on some polarized partition relations. (shrink)

Kueker's conjecture is proved for stable theories, for theories that interpret a linear ordering, and for theories with Skolem functions. The proof of the stable case involves certain results on coordinatization that are of independent interest.

We introduce a natural generalization of Borel's Conjecture. For each infinite cardinal number $\kappa$, let ${\sf BC}_{\kappa}$ denote this generalization. Then ${\sf BC}_{\aleph_0}$ is equivalent to the classical Borel conjecture. Assuming the classical Borel conjecture, $\neg{\sf BC}_{\aleph_1}$ is equivalent to the existence of a Kurepa tree of height $\aleph_1$. Using the connection of ${\sf BC}_{\kappa}$ with a generalization of Kurepa's Hypothesis, we obtain the following consistency results: 1. If it is consistent that there is a 1-inaccessible cardinal (...) then it is consistent that ${\sf BC}_{\aleph_1}$.2. If it is consistent that ${\sf BC}_{\aleph_1}$, then it is consistent that there is an inaccessible cardinal.3. If it is consistent that there is a 1-inaccessible cardinal with $\omega$ inaccessible cardinals above it, then $\neg{\sf BC}_{\aleph_{\omega}} + (\forall n < \omega){\sf BC}_{\aleph_n}$ is consistent.4. If it is consistent that there is a 2-huge cardinal, then it is consistent that ${\sf BC}_{\aleph_{\omega}}$.5. If it is consistent that there is a 3-huge cardinal, then it is consistent that ${\sf BC}_{\kappa}$ for a proper class of cardinals $\kappa$ of countable cofinality. (shrink)

Goldbach’s conjecture, if not read in number theory, but in a precise foundation theory of mathematics, that refers to the metaphysical ‘theory of the participation’ of Thomas Aquinas, poses a surprising analogy between the category of the quantity, within which the same arithmetic conjecture is formulated, and the transcendental/formal dimension. It says: every even number is ‘like’ a two, that is: it has the form-of-two. And that means: it is the composition of two units; not two equal arithmetic (...) units, but two different formal-transcendental units, which are, in arithmetic, two prime numbers. (shrink)

We prove that Rado's Conjecture implies that the nonstationary ideal on ω 1 is presaturated.

In this paper, we explore some of the consequences of Martin’s Conjecture on degree invariant Borel maps. These include the strongest conceivable ergodicity result for the Turing equivalence relation with respect to the filter on the degrees generated by the cones, as well as the statement that the complexity of a weakly universal countable Borel equivalence relation always concentrates on a null set.

Suppose that $\sigma\in{\mathcal{L}}_{\omega _{1},\omega }$ is such that all equations occurring in $\sigma$ are positive, have the same set of variables on each side of the equality symbol, and have at least one function symbol on each side of the equality symbol. We show that $\sigma$ satisfies Vaught’s conjecture. In particular, this proves Vaught’s conjecture for sentences of $ {\mathcal{L}}_{\omega _{1},\omega }$ without equality.

Quine often argued for a simple, untyped system of logic rather than the typed systems that were championed by Russell and Carnap, among others. He claimed that nothing important would be lost by eliminating sorts, and the result would be additional simplicity and elegance. In support of this claim, Quine conjectured that every many-sorted theory is equivalent to a single-sorted theory. We make this conjecture precise, and prove that it is true, at least according to one reasonable notion of (...) theoretical equivalence. Our clarification of Quine’s conjecture, however, exposes the shortcomings of his argument against many-sorted logic. (shrink)

Todorčević showed that Rado's Conjecture implies CC*, a strengthening of Chang's Conjecture. We generalize this by showing that also CC**, a global version of CC*, follows from RC. As a corollary we obtain that RC implies Semistationary Reflection and, i.e. the statement that all forcings that preserve the stationarity of subsets of ω1 are semiproper.

We answer a question of Slaman and Steel by showing that a version of Martin’s conjecture holds for all regressive functions on the hyperarithmetic degrees. A key step in our proof, which may have applications to other cases of Martin’s conjecture, consists of showing that we can always reduce to the case of a continuous function.

Alexander George; Discussions: ‘Goldbach's Conjecture Can Be Decided in One Minute’: On an Alleged Problem for Intuitionism, Proceedings of the Aristotelian Soc.

We prove that Kreisel's Conjecture is true, if Peano arithmetic is axiomatised using minimality principle and axioms of identity (theory $PA_M $ )-The result is independent on the choice of language of $PA_M $ . We also show that if infinitely many instances of A(x) are provable in a bounded number of steps in $PA_M $ then there existe k ∈ ω s. t. $PA_M $ ┤ ∀x > k̄ A(x). The results imply that $PA_M $ does not prove (...) scheme of induction or identity schemes in a bounded number of steps. (shrink)

We prove Fraïssé’s conjecture within the system of Π11-comprehension. Furthermore, we prove that Fraïssé’s conjecture follows from the Δ20-bqo-ness of 3 over the system of Arithmetic Transfinite Recursion, and that the Δ20-bqo-ness of 3 is a Π21-statement strictly weaker than Π11-comprehension.

This is a continuation of our paper where we show that Rado's Conjecture can trivialize ‐sequences in some cases when ϑ is not necessarily a successor cardinal.

It is proved that Vaught's conjecture is true for modules over an arbitrary countable serial ring. It follows from the structural result that every module with few models over a (countable) serial ring is ω-stable.

We formulate an analogue of Zilber's conjecture for o-minimal structures in general, and then prove it for a class of o-minimal structures over the reals. We conclude in particular that if is an ordered reduct of ,<,+,·,ex whose theory T does not have the CF property then, given any model of T, a real closed field is definable on a subinterval of.

In an article published in 1974, Menas conjectured that any stationary subset of can be split in many pairwise disjoint stationary subsets. Even though the conjecture was shown long ago by Baumgartner and Taylor to be consistently false, it is still haunting papers on. In which situations does it hold? How much of it can be proven in ZFC? We start with an abridged history of the conjecture, then we formulate a new version of it, and finally we (...) keep weakening this new assertion until, building on the work of Usuba, we hit something we can prove. (shrink)

The introduction of contraceptive technologies hasresulted in the separation of sex and procreation. Theintroduction of new reproductive technologies (mainlyIVF and embryo transfer) has led not only to theseparation of procreation and sex, but also to there-definition of the terms mother and family.For the purpose of this essay, I will distinguishbetween:1. the genetic mother – the donor of the egg;2. the gestational mother – she who bears and gives birth to the baby;3. the social mother – the woman who raises the (...) child.This essay will deal only with the form of gestationalsurrogacy in which the genetic parents intend to bethe social parents, and the surrogate mother has nogenetic relationship to the child she bears anddelivers. I will raise questions regarding medicalethical aspects of surrogacy and the obligation(s) ofthe physician(s) to the parties involved. I will arguethat the gestational surrogate is a womb to rent,that there is great similarity between gestationalcommercial surrogacy and organ transplant marketing.Furthermore, despite claims to freedom of choice andfree marketing, I will claim that gestationalsurrogacy is a form of prostitution and slavery,exploitation of the poor and needy by those who arebetter off. The right to be a parent, although notconstitutional, is intuitive and deeply rooted.However, the issue remains whether this rightoverrules all other rights, and at what price to theparties involved. I will finally raise the followingprovocative question to society: In the interim periodbetween today''s limited technology and tomorrow''sextra-corporeal gestation technology (ectogenesis),should utilizing females in PVS (persistent vehetativestate) for gestational surrogacy be sociallyacceptable/permissible – provided they have leftpermission in writing? (shrink)

We develop an approach to the longstanding conjecture of Kierstead concerning the character of strongly nontrivial automorphisms of computable linear orderings. Our main result is that for any η-like computable linear ordering, such that has no interval of order type η, and such that the order type of is determined by a -limitwise monotonic maximal block function, there exists computable such that has no nontrivial automorphism.

We show that the complete first order theory of an MV algebra has equation image countable models unless the MV algebra is finitely valued. So, Vaught's Conjecture holds for all MV algebras except, possibly, for finitely valued ones. Additionally, we show that the complete theories of finitely valued MV algebras are equation image and that all ω-categorical complete theories of MV algebras are finitely axiomatizable and decidable. As a final result we prove that the free algebra on countably many (...) generators of any locally finite variety of MV algebras is ω-categorical. (shrink)

Kreisel’s conjecture is the statement: if, for all$n\in \mathbb {N}$,$\mathop {\text {PA}} \nolimits \vdash _{k \text { steps}} \varphi (\overline {n})$, then$\mathop {\text {PA}} \nolimits \vdash \forall x.\varphi (x)$. For a theory of arithmeticT, given a recursive functionh,$T \vdash _{\leq h} \varphi $holds if there is a proof of$\varphi $inTwhose code is at most$h(\#\varphi )$. This notion depends on the underlying coding.${P}^h_T(x)$is a predicate for$\vdash _{\leq h}$inT. It is shown that there exist a sentence$\varphi $and a total recursive functionhsuch (...) that$T\vdash _{\leq h}\mathop {\text {Pr}} \nolimits _T(\ulcorner \mathop {\text {Pr}} \nolimits _T(\ulcorner \varphi \urcorner )\rightarrow \varphi \urcorner )$, but, where$\mathop {\text {Pr}} \nolimits _T$stands for the standard provability predicate inT. This statement is related to a conjecture by Montagna. Also variants and weakenings of Kreisel’s conjecture are studied. By the use of reflexion principles, one can obtain a theory$T^h_\Gamma $that extendsTsuch that a version of Kreisel’s conjecture holds: given a recursive functionhand$\varphi (x)$a$\Gamma $-formula (where$\Gamma $is an arbitrarily fixed class of formulas) such that, for all$n\in \mathbb {N}$,$T\vdash _{\leq h} \varphi (\overline {n})$, then$T^h_\Gamma \vdash \forall x.\varphi (x)$. Derivability conditions are studied for a theory to satisfy the following implication: if, then$T\vdash \forall x.\varphi (x)$. This corresponds to an arithmetization of Kreisel’s conjecture. It is shown that, for certain theories, there exists a functionhsuch that$\vdash _{k \text { steps}}\ \subseteq\ \vdash _{\leq h}$. (shrink)

Managing aggression in mental health hospitals is an important and challenging task for clinical nursing staff. A majority of studies focus on the perspective of clinicians, and research mainly depicts aggression by referring to patient-related factors. This qualitative study investigates how aggression is communicated in forensic mental health nursing records. The aim of the study was to gain insight into the discursive practices used by forensic mental health nursing staff when they record observed aggressive incidents. Textual accounts were extracted from (...) the Staff Observation Aggression Scale-Revised (SOAS-R), and Fairclough’s Critical Discourse Analysis was used to identify short narrative entries depicting patients and staffs in typical ways. The narratives contained descriptions of complex interactions between patient and staff that were linked to specific circumstances surrounding the patient. These antecedents, combined with the aggression incident itself, created stereotyping representations of forensic psychiatric patients as deviant, unpredictable and dangerous. Patient and staff identities were continually (re)produced by an automatic response from the staff that was solely focused on the patient’s behavior. Such response might impede implementation of new strategies for managing aggression. (shrink)

Answering a question of Sakai :29–45, 2013), we show that the existence of an \-Erdős cardinal suffices to obtain the consistency of Chang’s Conjecture with \. By a result of Donder, volume 872 of lecture notes in mathematics. Springer, Berlin, pp 55–97, 1981) this is best possible. We also give an answer to another question of Sakai relating to the incompatibility of \ and \ \twoheadrightarrow \) for uncountable \.