# István NémetiAlfréd Rényi Institute of Mathematics · Research Division of Algebraic Logic

István Németi

Doctor of sciences with the academy

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234

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Introduction

**Skills and Expertise**

## Publications

Publications (234)

Interdefinability of timelike, lightlike and spacelike relatedness of Minkowski spacetime is investigated in detail in the paper, with the aim of finding the simplest definitions. Based on ideas scattered in the literature, definitions are given between any two of these binary relations that use 4 variables, i.e., they use only 2 auxiliary variable...

Interdefinability of timelike, lightlike and spacelike relatedness of Minkowski spacetime is investigated in detail in the paper, with the aim of finding the simplest definitions. Based on ideas scattered in the literature, definitions are given between any two of these relations that use only 4 variables. All these definitions work over arbitrary...

We prove that the two-variable fragment of first-order logic has the weak Beth definability property. This makes the two-variable fragment a natural logic separating the weak and the strong Beth properties since it does not have the strong Beth definability property.

We prove that the two-variable fragment of first-order logic has the weak Beth definability property. This makes the two-variable fragment a natural logic separating the weak and the strong Beth properties since it does not have the strong Beth definability property. The proof relies on the special property of 2-variable logic that each model is 2-...

A. Tarski proved that the m-generated free algebra of CAα, the class of cylindric algebras of dimension α, contains exactly 2m zero-dimensional atoms, when m≥1 is a finite cardinal and α is an arbitrary ordinal. He conjectured that, when α is infinite, there are no more atoms other than the zero-dimensional atoms. This conjecture has not been confi...

We prove that persistently finite algebras are not created by completions of algebras, in any ordered discriminator variety. A persistently finite algebra is one without infinite simple extensions. We prove that finite measurable relation algebras are all persistently finite. An application of these theorems is that the variety generated by the com...

Nonrepresentable relation algebras from groups - H. Andréka, S. Givant, I. Németi

We explore the first-order logic conceptual structure of special relativistic spacetime. Namely, we describe the algebra of explicitly definable relations of Minkowski-spacetime, and we draw conclusions such as “the concept of lightlike-separability can be defined from that of timelike-separability by using 4 variables but not by using three variab...

This text is intended to be readable without looking at the slides. However, looking at the slides, especially at the drawings in slides 17 – 19, may be illuminating. The numbering of the items below coincides with that of the slides.

A. Tarski proved that the m-generated free algebra of $\mathrm{CA}_{\alpha}$, the class of cylindric algebras of dimension $\alpha$, contains exactly $2^m$ zero-dimensional atoms, when $m\ge 1$ is a finite cardinal and $\alpha$ is an arbitrary ordinal. He conjectured that, when $\alpha$ is infinite, there are no more atoms. This conjecture has not...

The marriage of groups and Boolean algebras.
In memory of Steven Givant
Talk by H. Andréka and I. Németi
Renyi Institute, lecture room on the 3rd floor
December 21, 2018. 15.00—17.00
Put together a group and a Boolean algebra and you get a relation algebra. Typically, the complexes (i.e., subsets) of a group form a relation algebra. The representa...

This is an abstract for a talk about a long-term joint work with Steven R. Givant

Abstract for
The marriage of groups and Boolean algebras.
In memory of Steven Givant
Talk by H. Andréka and I. Németi
Renyi Institute, lecture room on the 3rd floor
December 21, 2018. 15.00—17.00

Put together a group and a Boolean algebra and you get a relation algebra. Typically, the complexes (i.e., subsets) of a group form a relation algebra. The representation theory of relation algebras is hard, due to the fact that the whole of axiomatic mathematics can be expressed in the equational theory of relation algebras ([TG]). The talk is abo...

We prove that persistently finite algebras are not created by completions of algebras, in any ordered discriminator variety. A persistently finite algebra is one without infinite simple extensions. We prove that finite measurable relation algebras are all persistently finite. An application of these theorems is that the variety generated by the com...

A series of nonrepresentable relation algebras is constructed from groups. We use them to prove that there are continuum many subvarieties between the variety of representable relation algebras and the variety of coset relation algebras. We present our main construction in terms of polygroupoids.

We exhibit two relation algebra atom structures such that they are elementarily equivalent but their term algebras are not. This answers Problem 14.19 in the book Hirsch, R. and Hodkinson, I., "Relation Algebras by Games", North-Holland, 2002.

Adventures in the network of theories, feedback to cylindric algebras
Talk given at Salzburg Conference for Young Analytic Philosophy on Sept 7, 2016 and at the Renyi Institute of Mathematics in Budapest on Sept 25, 2016.
Network of theories is an efficient way of organizing scientific knowledge. In this talk, we are concerned with theories formul...

It is known that nontrivial ultraproducts of complete partially ordered sets (posets) are almost never complete. We show that complete additivity of functions is preserved in ultraproducts of posets. Since failure of this property is clearly preserved by ultraproducts, this implies that complete additivity of functions is an elementary property.

Talk given at Salzburg Conference for Young Analytic Philosophy on Sept 7, 2016 and at the Renyi Institute of Mathematics in Budapest on Sept 25, 2016. This is the presentation. The text of the presentation is available separately. Abstract: Network of theories is an efficient way of organizing scientific knowledge. In this talk, we are concerned w...

It is shown that Tarski's set of ten axioms for the calculus of relations is independent in the sense that no axiom can be derived from the remaining axioms. It is also shown that by modifying one of Tarski's axioms slightly, and in fact by replacing the right-hand distributive law for relative multiplication with its left-hand version, we arrive a...

We prove that all definable pre-orders are atomic, in a finitely generated free algebra of a discriminator variety of finite similarity type which is generated by its finite members.

This is an entry in Supplement III of Encyclopaedia of Mathematics. Ed: Hazewinkel, Kluwer, 2002. pp.31-34. Abstract: Algebraic logic can be divided into two major parts: (i) abstract (or universal) algebraic logic and (ii) " concrete " algebraic logic (or algebras of relations of various ranks). (1) Abstract algebraic logic is built around a duali...

Cylindric algebras, or concept algebras in another name, form an interface
between algebra, geometry and logic; they were invented by Alfred Tarski around
1947. We prove that there are 2 to the alpha many varieties of geometric (i.e.,
representable) alpha-dimensional cylindric algebras, this means that 2 to the
alpha properties of definable relatio...

It is known that nontrivial ultraproducts of complete partially ordered
sets (posets) are almost never complete. We show that complete additivity
of functions is preserved in ultraproducts of posets.

The generalized models for higher-order logics introduced by Leon Henkin, and their multiple offspring over the years, have become a standard tool in many areas of logic. Even so, discussion has persisted about their technical status, and perhaps even their conceptual legitimacy. This paper gives a systematic view of generalized model techniques, d...

The talk is about an interaction between mathematical logic and relativity theory. In our story there is a world without time and space, hence no motion no change, and then time and space emerge as a result of a cognitive process. We define time and space (over timeless objects) as derived notions. Technically, we define two first-order logic theor...

Logicians at the Rényi Mathematical Institute in Budapest have spent several years developing versions of relativity theory (special, general, and other variants) based wholly on first-order logic, and have argued in favour of the physical decidability, via exploitation of cosmological phenomena, of formally unsolvable questions such as the Halting...

We prove that n-variable logics do not have the weak Beth definability
property, for all n greater than 2. This was known for n=3 (Ildik\'o Sain and
Andr\'as Simon), and for n greater than 4 (Ian Hodkinson). Neither of the
previous proofs works for n=4. In this paper we settle the case of n=4, and we
give a uniform, simpler proof for all n greater...

There are several first-order logic (FOL) axiomatizations of special
relativity theory in the literature, all looking essentially different but
claiming to axiomatize the same physical theory. In this paper, we elaborate a
comparison, in the framework of mathematical logic, between these FOL theories
for special relativity. For this comparison, we...

Algebraic logic conference, Budapest, 1988 - Volume 54 Issue 2 - H. Andréka, M. Ferenczi, I. Németi, Gy. Serény

Seeing the many examples in the literature of causality violations based on faster than light (FTL) signals one naturally thinks that FTL motion leads inevitably to the possibility of time travel. We show that this logical inference is invalid by demonstrating a model, based on (3+1)-dimensional Minkowski spacetime, in which FTL motion is permitted...

Alfred Tarski in 1953 formalized set theory in the equational theory of relation algebras [Tar,53a, Tar,53b]. Why did he do so? Because the equational theory of relation algebras (RA) corresponds to a logic without individual variables, in other words, to a propositional logic. This is why the title of the book [Tar-Giv,87] is “Formalizing set theo...

This is Part I of the presentation of a talk given on September 27, 2013 at the Logic and Philosophy of Mathematics seminar of ELTE University, Budapest.
We present a concrete mathematical realization for the Leibnizian relationist concept of time and space, via a logical analysis of exploring time and space experimentally. We start out from ideas...

We present a concrete mathematical realization for the Leibnizian relationist concept of time and space, via a logical analysis of exploring time and space experimentally. We start out from ideas in James Ax’s 1978 paper entitled “The elementary foundations of spacetime”.
This is Part II of the presentation of a talk given on September 27, 2013 at...

We introduce several axiom systems for general relativity and show that they
are complete with respect to the standard models of general relativity, i.e.,
to Lorentzian manifolds having the corresponding smoothness properties.

We show that the transformations J. M. Hill and B. J. Cox introduce between
inertial observers moving faster than light with respect to each other are
consistent with Einstein's principle of relativity only if the spacetime is 2
dimensional.

We construct an infinite dimensional quasi-polyadic equality algebra \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bb...

Algebraic logic is a subject in the interface between logic, algebra and geometry, it has strong connections with category theory and combinatorics. Tarski’s quest for finding structure in logic leads to cylindric-like algebras as studied in this book, they are among the main players in Tarskian algebraic logic. Cylindric algebra theory can be view...

Logicians at the R\'enyi Mathematical Institute in Budapest have spent
several years developing versions of relativity theory (special, general, and
other variants) based wholly on first order logic, and have argued in favour of
the physical decidability, via exploitation of cosmological phenomena, of
formally undecidable questions such as the Halt...

In this paper we investigate the logical decidability and undecidability properties of relativity theories. If we include into our theory the whole theory of the reals, then relativity theory still can be decidable. However, if we actually assume the structure of the quantities in our models to be the reals, or at least to be Archimedean, then we g...

We present a streamlined axiom system of special relativity in first-order
logic. From this axiom system we "derive" an axiom system of general relativity
in two natural steps. We will also see how the axioms of special relativity
transform into those of general relativity. This way we hope to make general
relativity more accessible for the non-spe...

In this paper, we investigate the possibility of using closed timelike curves
(CTCs) in relativistic hypercomputation. We introduce a wormhole based
hypercomputation scenario which is free from the common worries, such as the
blueshift problem. We also discuss the physical reasonability of our scenario,
and why we cannot simply ignore the possibili...

Within an axiomatic framework, we investigate the possible structures of
numbers (as physical quantities) in different theories of relativity.

We show that there is no finitely axiomatizable class of algebras that would serve as an analogue to Kozen's class of Kleene algebras if we include the residuals of composition in the similarity type of relation algebras.

Languages and families of binary relations are standard interpretations of Kleene algebras. It is known that the equational theories of these interpretations coincide and that the free Kleene algebra is representable both as a relation and as a language algebra. We investigate the identities valid in these interpretations when we expand the signatu...

We show that first-order logic can be translated into a very simple and weak
logic, and thus set theory can be formalized in this weak logic. This weak
logical system is equivalent to the equational theory of Boolean algebras with
three commuting complemented closure operators, i.e., that of diagonal-free
3-dimensional cylindric algebras (Df_3's)....

The aim of this paper is to give an introduction to our axiomatic logical analysis of relativity theories.

In this paper we present some of our school's results in the area of building
up relativity theory (RT) as a hierarchy of theories in the sense of logic. We
use plain first-order logic (FOL) as in the foundation of mathematics (FOM) and
we build on experience gained in FOM.
The main aims of our school are the following: We want to base the theory o...

This paper consists mostly of pictures visualizing ideas leading to Godel's rotating cosmological model. The pictures are constructed according to concrete metric tensor fields. Information about these are in the last chapters. The main aim is to visualise ideas.

The aim of this paper is to give an introduction to our axiomatic logical
analysis of relativity theories.

The main result gives a sufficient condition for a class K of finite dimensional cylindric algebras to have the property that not every epimorphism in K is surjective. In particular, not all epimorphisms are surjective in the classes CA
n
of n-dimensional cylindric algebras and the class of representable algebras in CA
n
for finite n > 1, solving P...

Looking at very recent developments in spacetime theory, we can wonder whether these results exhibit features of hypercomputation that traditionally seemed impossible or absurd. Namely, we describe a physical device in relativistic spacetime which can compute a non-Turing computable task, e.g. which can decide the halting problem of Turing ma- chin...

We show that the variety of n-dimensional weakly higher order cylindric algebras, introduced in Németi [9], [8], is finitely axiomatizable when n > 2. Our result implies that in certain non-well-founded set theories the finitization problem of algebraic logic admits
a positive solution; and it shows that this variety is a good candidate for being t...

A part of relativistic dynamics is axiomatized by simple and purely geometrical axioms formulated within first-order logic.
A geometrical proof of the formula connecting relativistic and rest masses of bodies is presented, leading up to a geometric
explanation of Einstein’s famous E= mc
2. The connection of our geometrical axioms and the usual axio...

We give a novel application of algebraic logic to first order logic. A new, flexible construction is presented for representable but not completely rep- resentable atomic relation and cylindric algebras of dimension n (for finite n > 2) with the additional property that they are one-generated and the set of all n by n atomic matrices forms a cylind...

Several versions of the Gravitational Time Dilation effect of General Relativity are formulated by the use of Einstein's Equivalence Principle. It is shown that all of them are logical consequence of a first-order axiom system of Special Relativity extended to accelerated observers.

We investigate Kerr-Newman black holes in which a rotating charged
ring-shaped singularity induces a region which contains closed timelike curves
(CTCs). Contrary to popular belief, it turns out that the time orientation of
the CTC is opposite to the direction in which the singularity or the ergosphere
rotates. In this sense, CTCs "counter-rotate"...

Motivation and perspective for an exciting new research direction interconnecting logic, spacetime theory, relativity--including such revolutionary areas as black hole physics, relativistic computers, new cosmology--are presented in this paper. We would like to invite the logician reader to take part in this grand enterprise of the new century. Bes...

We examine the current status of the physical version of the Church-Turing Thesis (PhCT for short) in view of latest developments in spacetime theory. This also amounts to investigating the status of hypercomputation in view of latest results on spacetime. We agree with [D. Deutsch, A. Ekert, R. Lupacchini, Machines, logic and quantum physics, Bull...

Can general relativistic computers break the Turing bar- rier? - Are there final limits to human knowledge? - Limitative results versus human creativity (paradigm shifts). - Godel's logical results in comparison/combination with Godel's relativistic results. - Can Hilbert's programme be carried through after all? 1 Aims, perspective The Physical Ch...

We study the foundation of space-time theory in the framework of first-order
logic (FOL). Since the foundation of mathematics has been successfully carried
through (via set theory) in FOL, it is not entirely impossible to do the same
for space-time theory (or relativity). First we recall a simple and streamlined
FOL-axiomatization SpecRel of specia...

Does new physics give us a chance for designing computers, at least in principle, which could compute beyond the Turing barrier?
By the latter we mean computers which could compute some functions which are not Turing computable. Part of what we call “new
physics” is the surge of results in black hole physics in the last 15 years, which certainly ch...

We study relativity theory as a theory in the sense of mathematical logic. We use first-order logic (FOL) as a framework to
do so. We aim at an “analysis of the logical structure of relativity theories”. First we build up (the kinematics of) special
relativity in FOL, then analyze it, and then we experiment with generalizations in the direction of...

We give two theories, Th1 and Th2, which are explicitly definable over each other (i.e. the relation symbols of one theory are explic- itly definable in the other, and vice versa), but are not definitionally equivalent. The languages of the two theories are disjoint.

Here the emphasis is on the main pillars of Tarskian structuralist approach to logic: relation algebras, cylindric algebras, polyadic algebras, and Boolean algebras with operators. We also tried to highlight the recent renaissance of these areas and their fusion with new trends related to logic, like the guarded fragment or dynamic logic. Tarskian...

In this paper we develop definability theory in such a way that we allow to define new elements also, not only new relations on already existing elements. This is in harmony with our everyday mathematical practice, for example we define new entities when we define a geometry over a field. We will see that, in many respects, defining new elements is...

An n -ary operation (n ∈ ℕ0) on a set A is a map ω: A n → A, where A 0 := { ∅ }. The number n is called the arity of ω. A universal algebra is a pair 𝒜 = (A, Ω) consisting of a non-empty set A and a set Ω of operations on A. The set A and the members of Ω are called the universe and the fundamental operations of the algebra 𝒜, respectively. In prac...

We characterize the finite-dimensional elements of a free cylindric algebra. This solves Problem 2.10 in [Henkin, Monk, Tarski: Cylindric Algebras, North-Holland, 1971 and 1985]. We generalize the characterization to quasi-varieties of Boolean algebras with operators in place of cylindric algebras.

This is the third supplementary volume to Kluwer's highly acclaimed twelve-volume Encyclopaedia of Mathematics. This additional volume contains nearly 500 new entries written by experts and covers developments and topics not included in the previous volumes. These entries are arranged alphabetically throughout and a detailed index is included. This...

SCa, CAa, QAa and QEAa stand for the classes of Pinter's substitution algebras, Tarski's cylindric algebras, Halmos' quasipolyadic algebras, and quasipolyadic equality algebras of dimension a, respectively. Generalizing a result of Németi on cylindric algebras, we show that for K ? {SC, CA, QA, QEA} and ordinals a NraKß of a-dimensional neat reduct...

We investigate the Church-Kalm\'ar-Kreisel-Turing Theses concerning theoretical (necessary) limitations of future computers and of deductive sciences, in view of recent results of classical general relativity theory. We argue that (i) there are several distinguished Church-Turing-type Theses (not only one) and (ii) validity of some of these theses...

Algebraic logic can be divided into two main parts. Part I studies algebras which are relevant to logic(s), e.g. algebras which were obtained from logics (one way or another). Since Part I studies algebras, its methods are, basically, algebraic. One could say that Part I belongs to ‘Algebra Country’. Continuing this metaphor, Part II deals with stu...

We consider classes of relation algebras expanded with new operations based on the formation of ordered pairs. Examples for such algebras are pairing (or projection) algebras of algebraic logic and fork algebras of computer science. It is proved by Sain and Németi [38] that there is no 'strong' representation theorem for all abstract pairing algebr...

Among others we will prove that the equational theory of ω dimensional representable polyadic equality algebras (RPEA
ω
's) is not schema axiomatizable. This result is in interesting contrast with the Daigneault-Monk representation theorem, which states that the class of representable polyadic algebras is finite schema-axiomatizable (and hence the...