LetKbe an algebraically closed field and letLbe itscanonical language; that is,Lconsists of all relations onKwhich are definable from addition, multiplication, and parameters fromK. Two sublanguagesL1andL2ofLaredefinably equivalentif each relation inL1can be defined by anL2-formula with parameters inK, and vice versa. The equivalence classes of sublanguages ofLform a quotient lattice of the power set ofLabout which very little is known. We will not distinguish between a sublanguage and its equivalence class.LetLmdenote the language of multiplication alone, and letLadenote the language (...) of addition alone. Letf∈K[X, Y] and consider thealgebraic functiondefined byf(x, y) = 0 forx, y∈K. LetLfdenote the language consisting of the relation defined byf. The possibilities forLm∨Lfare examined in §2, and the possibilities forLa∨Lfare examined in §3. In fact the only comprehensive results known are under the additional hypothesis thatfactually defines a rational function (i.e., whenfis linear in one of the variables), and in positive characteristic, only expansions of addition by polynomials (i.e., whenfis linear and monic in one of the variables) are understood. It is hoped that these hypotheses will turn out to be unnecessary, so that reasonable generalizations of the theorems described below to algebraic functions will be true. The conjecture is thatLcoversLmand that the only languages betweenLaandLare expansions ofLaby scalar multiplications. (shrink)

Fix an algebraically closed field of characteristic zero and let G be its geometry of transcendence degree one extensions. Let X be a set of points of G. We show that X extends to a projective subgeometry of G exactly if the partial derivatives of the polynomials inducing dependence on its elements satisfy certain separability conditions. This analysis produces a concrete representation of the coordinatizing fields of maximal projective subgeometries of G.

The main scope of this short article is to provide a modification of the axioms given by Messmer and Wood for the theory of separably closed fields of positive characteristic and finite imperfectness degree. As their original axioms failed to meet natural expectations, a new axiomatization was given (i.e., Ziegler’s one), but the new axioms do not follow Messmer and Wood’s initial idea. Therefore, we aim to give a correct axiomatization that is more similar to the original (...) one and that, as with the original axioms, involves only one Hasse–Schmidt derivation, this time based on the iterativity conditions corresponding to the Witt group. (shrink)

We construct and study structures imitating the field of complex numbers with exponentiation. We give a natural, albeit non first-order, axiomatisation for the corresponding class of structures and prove that the class has a unique model in every uncountable cardinality. This gives grounds to conjecture that the unique model of cardinality continuum is isomorphic to the field of complex numbers with exponentiation.

One of the main theorems of the paper states the following. Let R-K-M be finite extensions of a rational one variable function field R over a finite field of constants. Let S be a finite set of valuations of K. Then the ring of elements of K having no poles outside S has a Diophantine definition over its integral closure in M.

In this paper we extend of the notion of algebraically closed given in the case of groups and skew fields to an arbitrary h-inductive theory. The main subject of this paper is the study of the notion of positive algebraic closedness and its relationship with the notion of positive closedness and the amalgamation property.

We study structures on the fields of characteristic zero obtained by introducing operations of raising to power. Using Hrushovski–Fraisse construction we single out among the structures exponentially-algebraically closed once and prove, under certain Diophantine conjecture, that the first order theory of such structures is model complete and every its completion is superstable.

This thesis is concerned with the expansions of algebraic structures and their fit in Shelah’s classification landscape.The first part deals with the expansion of a theory by a random predicate for a substructure model of a reduct of the theory. Let T be a theory in a language $\mathcal {L}$. Let $T_0$ be a reduct of T. Let $\mathcal {L}_S = \mathcal {L}\cup \{S\}$, for S a new unary predicate symbol, and $T_S$ be the $\mathcal {L}_S$ -theory that axiomatises the (...) following structures: $$ consist of a model $\mathscr {M}$ of T and S is a predicate for a model $\mathscr {M}_0$ of $T_0$ which is a substructure of $\mathscr {M}$. We present a setting for the existence of a model-companion $TS$ of $T_S$. As a consequence, we obtain the existence of the model-companion of the following theories, for $p>0$ a prime number: • $\mathrm {ACF}_p$, $\mathrm {SCF}_{e,p}$, $\mathrm {Psf}_p$, $\mathrm {ACFA}_p$, $\mathrm {ACVF}_{p,p}$ in appropriate languages expanded by arbitrarily many predicates for additive subgroups;• $\mathrm {ACF}_p$, $\mathrm {ACF}_0$ in the language of rings expanded by a single predicate for a multiplicative subgroup;• $\mathrm {PAC}_p$ -fields, in an appropriate language expanded by arbitrarily many predicates for additive subgroups.From an independence relation in T, we define independence relations in $TS$ and identify which properties of are transferred to those new independence relations in $TS$, and under which conditions. This allows us to exhibit hypotheses under which the expansion from T to $TS$ preserves $\mathrm {NSOP}_{1}$, simplicity, or stability. In particular, under some technical hypothesis on T, we may draw the following picture : Configuration $T_0\subseteq T$ Generic expansion $TS$ $T_0 = T$ Preserves stability $T_0\subseteq T$ Preserves $\mathrm {NSOP}_{1}$ $T_0 = \emptyset $ Preserves simplicityIn particular, this construction produces new examples of $\mathrm {NSOP}_{1}$ not simple theories, and we study in depth a particular example: the expansion of an algebraically closed field of positive characteristic by a generic additive subgroup. We give a full description of imaginaries, forking, and Kim-forking in this example.The second part studies expansions of the group of integers by p-adic valuations. We prove quantifier elimination in a natural language and compute the dp-rank of these expansions: it equals the number of independent p-adic valuations considered. Thus, the expansion of the integers by one p-adic valuation is a new dp-minimal expansion of the group of integers. Finally, we prove that the latter expansion does not admit intermediate structures: any definable set in the expansion is either definable in the group structure or is able to “reconstruct” the valuation using only the group operation.prepared by Christian d’Elbée.E-mail: delbee@math.univ-lyon1.frURL: https://choum.net/~chris/page_perso. (shrink)

Journal of Mathematical Logic, Ahead of Print. We present numerous natural algebraic examples without the so-called Canonical Base Property (CBP). We prove that every commutative unitary ring of finite Morley rank without finite-index proper ideals satisfies the CBP if and only if it is a field, a ring of positive characteristic or a finite direct product of these. In addition, we construct a CM-trivial commutative local ring with a finite residue field without the CBP. Furthermore, we (...) also show that finite-dimensional non-associative algebras over an algebraically closed field of characteristic [math] give rise to triangular rings without the CBP. This also applies to Baudisch’s [math]-step nilpotent Lie algebras, which yields the existence of a [math]-step nilpotent group of finite Morley rank whose theory, in the pure language of groups, is CM-trivial and does not satisfy the CBP. (shrink)

We study the generic theory of algebraically closed fields of fixed positive characteristic with a predicate for an additive subgroup, called $\mathrm {ACFG}$. This theory was introduced in [16] as a new example of $\mathrm {NSOP}_{1}$ nonsimple theory. In this paper we describe more features of $\mathrm {ACFG}$, such as imaginaries. We also study various independence relations in $\mathrm {ACFG}$, such as Kim-independence or forking independence, and describe interactions between them.

The generalization properties of algebraically closed fields $ACF_p$ of characteristic $p > 0$ and $ACF_0$ of characteristic 0 are investigated in the sequent calculus with blocks of quantifiers. It is shown that $ACF_p$ admits finite term bases, and $ACF_0$ admits term bases with primality constraints. From these results the analogs of Kreisel's Conjecture for these theories follow: If for some $k$ , $A(1 + \cdots + 1)$ ( $n$ 1's) is provable in $k$ steps, then $(\forall (...) x)A(x)$ is provable. (shrink)

We present two of the three major steps in the construction of motivic integration, that is, a homomorphism between Grothendieck semigroups that are associated with a first-order theory of algebraically closed valued fields, in the fundamental work of Hrushovski and Kazhdan [8]. We limit our attention to a simple major subclass of V-minimal theories of the form ACV FS, that is, the theory of algebraically closed valued fields of pure characteristic 0 expanded by a -generated (...) substructure S in the language . The main advantage of this subclass is the presence of syntax. It enables us to simplify the arguments with many different technical details while following the major steps of the Hrushovski–Kazhdan theory. (shrink)

We present numerous natural algebraic examples without the so-called Canonical Base Property (CBP). We prove that every commutative unitary ring of finite Morley rank without finite-index proper ideals satisfies the CBP if and only if it is a field, a ring of positive characteristic or a finite direct product of these. In addition, we construct a CM-trivial commutative local ring with a finite residue field without the CBP. Furthermore, we also show that finite-dimensional non-associative algebras over (...) an algebraically closed field of characteristic [Formula: see text] give rise to triangular rings without the CBP. This also applies to Baudisch’s [Formula: see text]-step nilpotent Lie algebras, which yields the existence of a [Formula: see text]-step nilpotent group of finite Morley rank whose theory, in the pure language of groups, is CM-trivial and does not satisfy the CBP. (shrink)

According to Belegradek, a first order structure is weakly small if there are countably many $1$-types over any of its finite subset. We show the following results. A field extension of finite degree of an infinite weakly small field has no Artin-Schreier extension. A weakly small field of characteristic $2$ is finite or algebraically closed. A weakly small division ring of positive characteristic is locally finite dimensional over its centre. A weakly small (...) division ring of characteristic $2$ is a field. (shrink)

The first two steps of the construction of motivic integration in the fundamental work of Hrushovski and Kazhdan [8] have been presented in Yin [12]. In this paper we present the final third step. As in Yin [12], we limit our attention to the theory of algebraically closed valued fields of pure characteristic 0 expanded by a -generated substructure S in the language . A canonical description of the kernel of the homomorphism is obtained.

In this paper we deal with the model theory of differentially closed fields of characteristic zero with finitely many commuting derivations. First we observe that the only known lower bound for the Lascar rank of types in differentially closed fields, announced in a paper of McGrail, is false. This gives us a new class of regular types which are orthogonal to fields. Then we classify the subgroups of the additive group of Lascar rank omega with differential-type 1 (...) which are nonorthogonal to fields. The last parts consist of an analaysis of the quotients of the heat variety. We show that the generic type of such a quotient is locally modular. Finally, we answer a question of Phylliss Cassidy about the existence of certain Jordan-Holder type series in the negative. (shrink)

Theorem: For every k, there is an expansion of the theory of algebraically closed fields (of any fixed characteristic) which is almost strongly minimal with Morley rank k.

Let ℛ be an o-minimal expansion of a real closed field R. We continue here the investigation we began in [11] of differentiability with respect to the algebraically closed field [Formula: see text]. We develop the basic theory of such K-differentiability for definable functions of several variables, proving theorems on removable singularities as well as analogues of the Weierstrass preparation and division theorems for definable functions. We consider also definably meromorphic functions and prove that every (...) definable function which is meromorphic on Kn is necessarily a rational function. We finally discuss definable analogues of complex analytic manifolds, with possible connections to the model theoretic work on compact complex manifolds, and present two examples of "nonstandard manifolds" in our setting. (shrink)

This paper studies unbounded pseudo-algebraically closed fields and shows an amalgamation result for types over algebraically closed sets. It discusses various applications, for instance that omega-free PAC fields have the property NSOP3. It also contains a description of imaginaries in PAC fields.

The aim of this paper is to develop the theory of groups definable in the p-adic field ${{\mathbb {Q}}_p}$, with “definable f-generics” in the sense of an ambient saturated elementary extension of ${{\mathbb {Q}}_p}$. We call such groups definable f-generic groups.So, by a “definable f-generic” or $dfg$ group we mean a definable group in a saturated model with a global f-generic type which is definable over a small model. In the present context the group is definable over ${{\mathbb {Q}}_p}$, (...) and the small model will be ${{\mathbb {Q}}_p}$ itself. The notion of a $\mathrm {dfg}$ group is dual, or rather opposite to that of an $\operatorname {\mathrm {fsg}}$ group (group with “finitely satisfiable generics”) and is a useful tool to describe the analogue of torsion-free o-minimal groups in the p-adic context.In the current paper our group will be definable over ${{\mathbb {Q}}_p}$ in an ambient saturated elementary extension $\mathbb {K}$ of ${{\mathbb {Q}}_p}$, so as to make sense of the notions of f-generic type, etc. In this paper we will show that every definable f-generic group definable in ${{\mathbb {Q}}_p}$ is virtually isomorphic to a finite index subgroup of a trigonalizable algebraic group over ${{\mathbb {Q}}_p}$. This is analogous to the o-minimal context, where every connected torsion-free group definable in $\mathbb {R}$ is isomorphic to a trigonalizable algebraic group [5, Lemma 3.4]. We will also show that every open definable f-generic subgroup of a definable f-generic group has finite index, and every f-generic type of a definable f-generic group is almost periodic, which gives a positive answer to the problem raised in [28] of whether f-generic types coincide with almost periodic types in the p-adic case. (shrink)

We study model theory of actions of finite groups on substructures of a stable structure. We give an abstract description of existentially closed actions as above in terms of invariants and PAC structures. We show that if the corresponding PAC property is first order, then the theory of such actions has a model companion. Then, we analyze some particular theories of interest (mostly various theories of fields of positive characteristic) and show that in all the cases considered (...) the PAC property is first order. (shrink)

Let R be a real closed field, and X⊆Rm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${X\subseteq R^m}$$\end{document} semi-algebraic and 1-dimensional. We consider complete first-order theories of modules over the ring of continuous semi-algebraic functions X→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${X\to R}$$\end{document} definable with parameters in R. As a tool we introduce -piecewise vector bundles on X and show that the category of piecewise vector bundles on X is equivalent to the category of (...) syzygies of finitely generated submodules of free modules. We give an explicit method to determine the Baur–Monk invariants of free modules in terms of pre-piecewise vector bundles. When R is a recursive real closed field this yields the decidability of the theory of free modules. Where it makes sense, we address the same questions for continuous definable functions in o-minimal expansions of a real closed field. From the free module case we are able to deduce generalisations of some results to arbitrary modules over the ring. We present a geometrically motivated quantifier elimination result down to the level of positive primitive formulae with a certain block decomposition of the matrix of coefficients. (shrink)

Let A be an abelian variety in an algebraically closed field of characteristic 0. We prove that the expansion of A by a generic divisible subgroup of A with the same torsion exists provided A has few algebraic endomorphisms, namely. The resulting theory is NSOP1 and not simple. Note that there exist abelian varieties A with of any genus.

Let n ≥ 3. The following theorems are proved. Theorem. The theory of the class of strictly upper triangular n × n matrix rings over fields is finitely axiomatizable. Theorem. If R is a strictly upper triangular n × n matrix ring over a field K, then there is a recursive map σ from sentences in the language of rings with constants for K into sentences in the language of rings with constants for R such that $K \vDash \varphi$ (...) if and only if $R \vDash \sigma$. Theorem. The theory of a strictly upper triangular n × n matrix ring over an algebraically closed field is ℵ 1 -categorical. (shrink)

We construct an existentially undecidable complete discretely valued field of mixed characteristic with existentially decidable residue field and decidable algebraic part, answering a question by Anscombe–Fehm in a strong way. Along the way, we construct an existentially decidable field of positive characteristic with an existentially undecidable finite extension, modifying a construction due to Kesavan Thanagopal.

Vector spaces over unspecified fields can be axiomatized as one-sorted structures, namely, abelian groups with the relation of parallelism. Parallelism is binary linear dependence. When equipped with the n-ary relation of linear dependence for some positive integer n, a vector-space is existentially closed if and only if it is n-dimensional over an algebraically closed field. In the signature with an n-ary predicate for linear dependence for each positive integer n, the theory of infinite-dimensional vector (...) spaces over algebraically closed fields is the model-completion of the theory of vector spaces. (shrink)

This paper introduces a natural extension of Kolchin's differential Galois theory to positive characteristic iterative differential fields, generalizing to the non-linear case the iterative Picard—Vessiot theory recently developed by Matzat and van der Put. We use the methods and framework provided by the model theory of iterative differential fields. We offer a definition of strongly normal extension of iterative differential fields, and then prove that these extensions have good Galois theory and that a G-primitive element theorem holds. In (...) addition, making use of the basic theory of arc spaces of algebraic groups, we define iterative logarithmic equations, finally proving that our strongly normal extensions are Galois extensions for these equations. (shrink)

Sophists’ apologia. -/- Sophists were the first paid teachers ever. These ancient Greek enlighteners taught wisdom. Protagoras, Antiphon, Prodicus, Hippias, Lykophron are most famous ones. Sophists views and concerns made a unified encyclopedic system aimed at teaching common wisdom, virtue, management and public speaking. Of the contemporary “enlighters”, Deil Carnegy’s educational work seems to be the most similar to sophism. Sophists were the first intellectuals – their trade was to sell knowledge. They introduced a new type of teacher-student relationship – (...) the mutually beneficial communication on equal terms. They taught pupils how to think independently and how to persuade others, which was inseparably connected with the rise of democracy in the most advanced Greek polices. Sophists were the first to proclaim that all people are naturally equal. They put forward the idea of natural rights and social contract, working out the fundamentals of the present-day law. In addition, they developed the basics of philology, psychology, logic, and gave a scientific explanation to the origins of religions. They embodied definite positive ideals of their epoch. Sophistry flourished in the second half of the Vth through he first third of the IVth centuries B. C. The period coincides with the heyday of the entire Greek history – the so called Greek Miracle epoch. Sophists expressed some present-day concepts of Ancient Greek ideology and mentality, characteristic of its Golden Age times. The Ancient Athens of the times of Periclus were similar to the Florence of Renaissance as to the type of social structure. That is why (?) the art of eloquence (public speaking) was highly respected in Florence. The two cities had a lot of features of the present-day capitalist society, which was due to their role as the main centres of industry of their “worlds-economies”. During the subsequent Hellenic period, the Old Order was restored, based on the feudal rent recipients’ rule. The degradation of society resulted in the decline of sophistry. Sophistry represents the highest point in the evolution of Greek philosophy. It is also the most prominent school as to the novelty of its ideas. That was owing to the common propensity for innovation typical of Periclus’ Golden Age, which in its turn resulted from the establishment of the individualistic and rationalistic thinking similar to present-day mentality. Among other aspects, close attention was paid to ethnic issues; some humanistic ideas such as Master and Servant ethics and Enlightment as the primary condition of man’s freedom, received scientific grounding. The genre of philosophical dialogue was developed, as well as the rationalistic scientific gnoceology, aimed at consistently opposing any religion. Dialectics, the most outstanding achievement of the Antiquity, was also worked out by the sophists. Zeno of Helleas was one of the sophists. Democrites’ ideas were, too, quite close to dialectics. Socrates’ contemporaries regarded him ironically, as shown, for example, in the Aristophanes’ comedy “Clouds”. Nevertheless, this scholar can still be referred to as a sophist, although of somewhat weird personality. Socrates’ popularity, which came later, should be attributed to the fact that he was the first philosopher ever to have been sentenced and executed by law, rather than to any valuable scientific contributions. Sophistry as such, rather than Socrates, marks the line between pre-Socratic and post- Socratic schools. Plato, with the exception of his latest works, is considered to belong to the same philosophical school. Specifically, there are valid reasons to believe that he was paid fees by his disciples. In Plato’s Dialogues, all points of view will be proven and then disproved, so that an integral philosophical system is hard to observe. Plato’s “Idealism” is but one of the many possible theories. His ideal philosophy is a high-minded argument of wise men. As late as towards the end of his life, Plato betrayed the ideals of sophistry, turning into the world’s first proponent of totalitarianism. Sophistry is Greek classical literature. Later on, philosophy, as well as the entire Greek civilization, started to fall into decline. Over 500 subsequent years, Hellenistic-Roman scholars never came up with anything novel. The entire process of antique philosophical evolution can be subdivided into the following six stages: Phoenician (IX–VIIth centuries B. C.), archaic (VIth – the first half of IVth century B. C., classic (the epoch of sophistry: the second half of the Vth through the first decades of the IIIrd centuries B. C.), late antique (middle of the IIIrd – first decades of the VI th centuries B.C.) The fact that early in the VIth century B. C. sophistry was rejected in Athens, has a lot to do with the consequences of their defeat in Peloponnesian war, a social catastrophe. Ever since, the prevalent part of the criticism of sophistry is a criticism of sophists’ immoral ways of life and personalities, rather than the essentials of their philosophy. For instance, Aristotle’s criticism primarily concerns their pragmatism and tendency to make a profit out of teaching. However, Aristotle himself taught his disciples along similar lines, showing them how to “be able to prove both opposites”. It was Aristotle who summed up sophists’ century-long elaborations. The traditional modern interpretation of sophists as shown in Plato’s dialogues appears to be groundless. The truth of the matter is that Plato regards sophists as more serious philosophers than Socrates was. It becomes especially clear in the dialogue titled “Protagoras”, which shows the latter as gaining an undoubted victory over Socrates. Generally speaking, Plato’s works never regard sophists as weak or worthless opponents. A common opinion as to the essence of sophists’, in particular, Protagoras’s philosophy, has hitherto not been worked out. Many researchers renounce the very existence of any specific philosophical position in sophistry. The term “relativism” itself now has a bad name. However, it should be borne in mind that Protagoras doctrine emerged as a result of an attempt to get out of the ontological dead-end to which the entire Greek natural philosophy had come. To this purpose, he employed Anaxagoras’s idea about everything comprising everything, whereupon he built his physical-ontological theory, and Zeno of Helleas’s concept that all objects are such and different at the same time in the same respect. According to Protagoras’s philosophy, every object does not only seem, but is different from every other object, including human beings. Therefore, every person is his or her own measure of all other entities. The objective reality is the Ego’s relations with the outer world. Every object’s essence is defined through its interaction with ourselves. Nothing remains stagnant – every existence already contains non-existence, and vice-versa. However, the relativity of being is not to be regarded as absolute. That means, that relatively stable and therefore relatively cognizable entities do exist, among them the words we say. Doubt is not considered absolute by relativists either, which differentiates them from skeptics and agnostics. They dare to believe. Uncertainty is Godly – here, Protagoras becomes very close to the concept of apathetic theology. Gorgius shared Protagoras’s views and tried to prove them applying the same method as Zeno applied when upholding Parmenidis’s philosophy. He argued that if one accepts the opposite point of view and agrees that nothing can exist and not exist at the same time, one is bound to come to an the absurd conclusion that nothing is cognizable and nothing exists. To prove that there is a universal contradiction within the Existence itself, Protagoras and his followers in their turn put forward a number of special logical arguments, called rules of contraries, or sophisms. The most well-known of them is the Euathle’s sophism, which proves that an object can be such and different at the same time in the same respect. Generally speaking, all relationships of any entity are extraneous. The idea that all entities, including concepts, exist and do not exist at the same time, was also accepted by Plato. It should be stressed, that Plato’s philosophy as a whole is relativists’ greatest achievement. In his dispute against relativism, Aristotle put forward the main concepts of his ontology: the existence falls into the existence in possibility and the existence in reality, or substance and accidence – the ever changing “sublunary” world and the stable world of divine heavenly bodies. This division made his system self-contradictory to the point of absurd. Aristotle quite realized, that the direct disproof of relativism is hardly possible, which fact makes his criticism a mere tautology. It is very indicative in this respect that materialists, idealists and even skeptical agnostics applied the same arguments do disprove relativism. That demonstrates that the inner contradiction between any forms of “absolutism” and relativism is the most fundamental philosophical problem. Relativist ideas were repeatedly renewed throughout history, provided the social conditions were favourable. In variations, the concepts were advanced by Tzuan Tsi and ancient Chinese sophists, apostle Paul, Nicolas of Cusa, Galileo, Rene Descartes, G. Berkley, D. Hume, Ch. S. Pierce, E. Mach, A. Bogdanov, F. de Saussure, N. Bore, P. Feuerabend, U. Quain and others. However, no consistent relativist ontological system was developed after Protagoras. Meanwhile, a valid relativist theory could provide a clue to a fuller and more adequate general physical concept of the world, and to a deeper understanding of quantum mechanics. -/- Relativism as an ontological system of beliefs. -/- The relativist principle of relativity of all that exists is fundamental for the entire system of philosophy. Absolutely everything exists, but, at the same time, no existence is absolute. Anything is possible, but those entities only exist for us with which we interact this way or another, i. e., reality is interaction, “I interact – hence, I exist”. However, an overly close relationship between entities leads to non-existence – in other words, there is a certain existential optimum. For all of us, information, or perceptible heterogeneities, is real. Each of us is the center of a definite system of interactions with the world. If God is an endless Possibility, then the World is but the man’s dialogue with God. However, the ties that bind us with the world are of some definite kind, which predetermines our reality. The principle of relativity constrains itself, as any relativity is relative. On the one hand, that calls forth the Evil – caused by our own weakness. On the other hand, it makes our world relatively stable and cognizable. Generally speaking, the world is what we are. Everybody has an own Universe, but since people are fairly similar, our universes can be regarded as one Universe with common laws and regulations. It is possible for us to change the degree of objects’ existence in relation to ourselves and one another. It is equally important, though, to remain ourselves (keep following our “warrior’s path”). Since everything exists and no existence is absolute, the process of cognition should involve a criticism of everything but never a complete rebuttal of anything. The criterion of verity for any theory then is in its capacity to build itself upon a bigger number of diverse statements than in any other theory. Theoretically, all philosophies should ideally be complementary. The question of what in fact happened is meaningless unless we specify the “we”, the “where” and the “when”. The past can only exist as part of the present. When we change, our past undergoes corresponding qualitative objective changes, all of which are irreversible. The contemporary physics regards all objects of the Universe as existing for one another, which results in absurd conclusions. In fact, only what we interact with, exists. The so called “speed of light in vacuum” c is the maximum speed of direct interaction and direct exchange of signals. Specifically, at the oncoming speed c with a significant blue parallax the gamma-quantum is unable to interact with the matter. When this speed is reached, the gravitational mass of objects in relation to one another becomes equal to zero (imaginary), whereas their inertial mass equals infinity. Further acceleration will bring us into an entirely different Universe. That means that the Universe is an open system, which fact destroys all present-day cosmological models. Anything is possible in this world. Our reality is magic. Every bit of space contains a potentially limitless number of bits of other, however virtual, universes, as well as an infinite number of particles, which are still virtual for us (as postulated by quantum mechanics). In other words, there exists an infinity of different realities. With all the above in mind, the different levels of entities’ existence, in particular, of Life and Reason, should be taken into consideration. For example, to which extent a live system exists in relation to the inanimate forms of being is very indefinite, as what life is has to be determined by living creatures. An even greater compass of levels of existence can be brought about through interaction – that is the core idea of Godel’s theorem. As a result, the range of forms of existence tends towards infinity – the evolution of matter through super-intelligence will bring us up to God’s heights. A more generalized mathematical model of the physical world, including a universal operator of the co-existence of objects, can be developed based on the above considerations. The bonds between objects become weaker owing to 1) a high speed, 2) acceleration, 3) rotation, 4) strong gravity, 5) strong fields of other kinds, 6) the difference of levels of existence , 7) phase shifts. Yet, there always is an existential optimum. Such mathematical model is to operate certain non-transitive algebras. Relativist ontology assumes that the notion of “temperature” is relative, thus making it possible to ignore the Second Law of Thermodynamics. In its turn, that will allow for phase shifts. A perpetual motion machine of the second type will become a reality. Rotation could presumably be used to increase inertial and decrease gravitational mass, slowing down the time. An experimental research could shed light on the mentioned effects. (shrink)

We show that for any uncountable cardinal λ, the category of sets of cardinality at least λ and monomorphisms between them cannot appear as the category of points of a topos, in particular is not the category of models of a ${L_{\infty,\omega }}$-theory. More generally we show that for any regular cardinal $\kappa < \lambda$ it is neither the category of κ-points of a κ-topos, in particular, nor the category of models of a ${L_{\infty,\kappa }}$-theory.The proof relies on the construction (...) of a categorified version of the Scott topology, which constitute a left adjoint to the functor sending any topos to its category of points and the computation of this left adjoint evaluated on the category of sets of cardinality at least λ and monomorphisms between them. The same techniques also apply to a few other categories. At least to the category of vector spaces of with bounded below dimension and the category of algebraic closed fields of fixed characteristic with bounded below transcendence degree. (shrink)

We consider the theory P of pairs Falgebraically closed fields of a given characteristic p. We exhibit a collection of additional sorts in which this theory has geometric elimination of imaginaries. The sorts are essentially of the form aBVa/Ga, where G,V,B are varieties over the prime field, G is a group scheme over B and V is a scheme over B , G acts algebraically on V over B, and for generic bB the (...) action of Gb on Vb is generically free. (shrink)

We prove that there exists no sentence F of the language of rings with an extra binary predicat I2 satisfying the following property: for every definable set X ⊆ ℂ2, X is connected if and only if ⊧ F, where I2 is interpreted by X. We conjecture that the same result holds for closed subset of ℂ2. We prove some results motivated by this conjecture.

We prove that if a strongly minimal nonlocally modular reduct of an algebraically closed valued field of characteristic 0 contains +, then this reduct is bi-interpretable with the underlying field.

A first-order theory is Noetherian with respect to the collection of formulae [Formula: see text] if every definable set is a Boolean combination of instances of formulae in [Formula: see text] and the topology whose subbasis of closed sets is the collection of instances of arbitrary formulae in [Formula: see text] is Noetherian. We show the Noetherianity of the theory of proper pairs of algebraically closed fields in any characteristic with respect to the family of tame (...) formulae as introduced in [A. Martin-Pizarro and M. Ziegler, Equational theories of fields, J. Symbolic Logic 85 (2020) 828–851, https://arxiv.org/abs/1702.05735 ], thus answering a question which was left open there. (shrink)

Here we prove that if T and T′ are strongly minimal theories, where T′ satisfies a certain property related to triviality and T does not, and T′ is model complete, then there is no computable embedding of Mod(T) into Mod(T′). Using this, we answer a question from [4], showing that there is no computable embedding of VS into ZS, where VS is the class of infinite vector spaces over Q, and ZS is the class of models of Th(Z, S). Similarly, (...) we show that there is no computable embedding of ACF into ZS, where ACF is the class of algebraically closed fields of characteristic 0. (shrink)

We study the question of $\mathcal {L}_{\mathrm {ring}}$ -definability of non-trivial henselian valuation rings. Building on previous work of Jahnke and Koenigsmann, we provide a characterization of henselian fields that admit a non-trivial definable henselian valuation. In particular, we treat the cases where the canonical henselian valuation has positive residue characteristic, using techniques from the model theory and algebra of tame fields.

This work is inspired by the correspondence of Malcev between rings and groups. Let A be a domain with unit, and S a multiplicative group of invertible elements. We define A S as the structure obtained from A by restraining the multiplication to A × S, and σ(A S ) as the group of functions from A to A of the form $x \longrightarrow xa + b$ , where (a, b) belongs to S × A. We show that A S (...) and σ(A S ) are interpretable in each other, and then, that we can transfer some properties between classes (or theories) of "reduced" domains and corresponding groups, such as being elementary, axiomatisability (for classes), decidability, completeness, or, in some cases, existence of a model-completion (for theories). We study the extensions of the additive group of A by the group S, acting by right multiplication, and show that sometimes σ(A S ) is the unique extension of this type. We also give conditions allowing us to eliminate parameters appearing in interpretations. We emphasize the case where the domain is a division ring K and S is its multiplicative group K × . Here, the interpretations can always be done without parameters. If the centre of K contains more than two elements, then σ(K) is the only extension of the additive group of K by its multiplicative group acting by right multiplication, and the class of all such σ(K)'s is elementary and finitely axiomatisable. We give, in particular, an axiomatisation for this class and for the class of σ(K)'s where K is an algebraically closed field of characteristic 0. From these results it follows that some classical model-companion results about theories of fields can be translated and restated as results about theories of solvable groups of class 2. (shrink)

The fields K) of generalized power series with coefficients in a field K and exponents in an additive abelian ordered group G play an important role in the study of real closed fields. The subrings K) consisting of series with non-positive exponents find applications in the study of models of weak axioms for arithmetic. Berarducci showed that the ideal JK) generated by the monomials with negative exponents is prime when is the additive group of the reals, and (...) asked whether the same holds for any G. We prove that this is the case and that in the quotient ring K)/J, each element admits at least one factorization into irreducibles. (shrink)

We show that if R is a nonconstant regular (semi-)local subring of a rational function field over an algebraically closed field of characteristic zero, Hilbert's Tenth Problem for this ring R has a negative answer; that is, there is no algorithm to decide whether an arbitrary Diophantine equation over R has solutions over R or not. This result can be seen as evidence for the fact that the corresponding problem for the full rational field (...) is also unsolvable. (shrink)

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)