This paper investigates the question of how we manage to single out the natural number structure as the intended interpretation of our arithmetical language. Horsten submits that the reference of our arithmetical vocabulary is determined by our knowledge of some principles of arithmetic on the one hand, and by our computational abilities on the other. We argue against such a view and we submit an alternative answer. We single out the structure of natural numbers through our intuition of the (...) absolute notion of finiteness. (shrink)

The collapse models of arithmetic are inconsistent, nontrivial models obtained from ℕ and set out in the Logic of Paradox (LP). They are given a general treatment by Priest (Priest, 2000). Finite collapse models are decidable, and thus axiomatizable, because finite. LP, however, is ill-suited to normal axiomatic reasoning, as it invalidates Modus Ponens, and almost all other usual conditional inferences. I set out a logic, A3, first given by Avron (Avron, 1991), and give a first order axiom system (...) for the finite collapse models. I present some standard arithmetical axioms in addition to a cyclic axiom and prove that these axioms are sound and complete for the cyclic models, reporting a similar result for the heap models. The state of the situation for the each of the kinds of infinite collapse model is, however, left an open question. (shrink)

I make an attempt at the description of the delicate role of the standard model of arithmetic for the syntax of formal systems. I try to assess whether the possible instability in the notion of finiteness deriving from the nonstandard interpretability of arithmetic affects the very notions of syntactic metatheory and of formal system. I maintain that the crucial point of the whole question lies in the evaluation of the phenomenon of formalization. The ideas of Skolem, (...) Zermelo, Beth and Carnap (among others) on the problem are discussed. ‘A tries to explain to B the meaning of negation. Finally A gives up, saying: “You don’t understand what I mean, and I am not going to explain any longer,” to which B replies: “Yes, I see what you mean, and I am glad you are willing to continue your explanations”’. G. Mannoury, reported by E. W. Beth (Beth, 1963, 489). (shrink)

The paper is concerned with the old conjecture that there are infinitely many twin primes. In the paper we show that this conjecture is true, that is, it is true in the standard model of arithmetic. The proof is based on Rasiowa–Sikorski Lemma. The key role are played by the derived notion of a Rasiowa–Sikorski set and the method of forcing adjusted to arbitrary first–order languages. This approach was developed in the papers Czelakowski [ 4, 5 ]. (...) The central idea consists in constructing an appropriate countable model \(\textbf{A}\) of Peano arithmetic by means of a Rasiowa–Sikorski set. This model validates the twin prime conjecture. Since \(\textbf{A}\) is elementarily equivalent to the standard model, the conjecture follows. Thus the standard model validates the twin primes conjecture. More generally, it is shown that de Polignac’s conjecture has a positive solution. The paper employs methods borrowed from the contemporary mathematical logic. Such a ’logical’ approach may be viewed as a useful addition to the dominant methodology in number theory based on mathematical analysis. (shrink)

We investigate theories of initial segments of the standard models for arithmetics. It is easy to see that if the ordering relation is definable in the standard model then the decidability results can be transferred from the infinite model into the finite models. On the contrary we show that the Σ₂—theory of multiplication is undecidable in finite models. We show that this result is optimal by proving that the Σ₁—theory of multiplication and order is decidable in (...) finite models as well as in the standard model. We show also that the exponentiation function is definable in finite models by a formula of arithmetic with multiplication and that one can define in finite models the arithmetic of addition and multiplication with the concatenation operation. We consider also the spectrum problem. We show that the spectrum of arithmetic with multiplication and arithmetic with exponentiation is strictly contained in the spectrum of arithmetic with addition and multiplication. (shrink)

The paper proves a predicate version of Solovay's well-known theorem on provability interpretations of modal logic: If a closed modal predicate-logical formula R is not valid in some finite Kripke model, then there exists an arithmetical interpretation f such that $PA \nvdash fR$ . This result implies the arithmetical completeness of arithmetically correct modal predicate logics with the finite model property (including the one-variable fragments of QGL and QS). The proof was obtained by adding "the predicate part" as (...) a specific addition to the standard Solovay construction. (shrink)

In this paper, I pursue such a logical foundation for arithmetic in a variant of Zermelo set theory that has axioms of subset separation only for quantifier-free formulae, and according to which all sets are Dedekind finite. In section 2, I describe this variant theory, which I call ZFin0. And in section 3, I sketch foundations for arithmetic in ZFin0 and prove that certain foundational propositions that are theorems of the standard Zermelian foundation for arithmetic are (...) independent of ZFin0.<br><br>An equivalent theory of sets and an equivalent foundation for arithmetic was introduced by John Mayberry and developed by the current author in his doctoral thesis. In that thesis, the independence results mentioned above are proved using proof-theoretic methods. In this paper, I offer model-theoretic proofs of the central independence results using the technique of cumulation models, which was introduced by Steve Popham, a doctoral student of Mayberry<br>from the early 1980s. (shrink)

The principal result of this paper answers a long-standing question in the model theory of arithmetic [R. Kossak, J. Schmerl, The Structure of Models of Peano Arithmetic, Oxford University Press, 2006, Question 7] by showing that there exists an uncountable arithmetically closed family of subsets of the set ω of natural numbers such that the expansion of the standard model of Peano arithmetic has no conservative elementary extension, i.e., for any elementary extension of , (...) there is a subset of ω* that is parametrically definable in but whose intersection with ω is not a member of . We also establish other results that highlight the role of countability in the model theory of arithmetic. Inspired by a recent question of Gitman and Hamkins, we furthermore show that the aforementioned family can be arranged to further satisfy the curious property that forcing with the quotient Boolean algebra collapses 1 when viewed as a notion of forcing. (shrink)

The paper concerns interpretations of the paraconsistent logic LP which model theories properly containing all the sentences of first order arithmetic. The paper demonstrates the existence of such models and provides a complete taxonomy of the finite ones.

By a theorem of R. Kaye, J. Paris and C. Dimitracopoulos, the class of the Πn+1-sentences true in the standard model is the only consistent Πn+1-theory which extends the scheme of induction for parameter free Πn+1-formulas. Motivated by this result, we present a systematic study of extensions of bounded quantifier complexity of fragments of first-order Peano Arithmetic. Here, we improve that result and show that this property describes a general phenomenon valid for parameter free schemes. As a (...) consequence, we obtain results on the quantifier complexity, finite axiomatizability and relative strength of schemes for Δn+1-formulas. (shrink)

We prove that in IST, Nelson's internal set theory, the Uniqueness and Collection principles, hold for all (including external) formulas. A corollary of the Collection theorem shows that in IST there are no definable mappings of a set X onto a set Y of greater (not equal) cardinality unless both sets are finite and #(Y) ≤ n #(X) for some standard n. Proofs are based on a rather general technique which may be applied to other nonstandard structures. In particular (...) we prove that in a nonstandard model of PA, Peano arithmetic, every hyperinteger uniquely definable by a formula of the PA language extended by the predicate of standardness, can be defined also by a pure PA formula. (shrink)

We show that for any, the structure of sets that are hereditarily of size at most k is decidable. We provide a transparent complete axiomatization of its theory, a quantifier elimination result, and tight bounds on its computational complexity. This stands in stark contrast to the structure of hereditarily finite sets, which is well known to be bi‐interpretable with the standard model of arithmetic.

In response to Bhupinder Singh Anand''s article CAN WE REALLY FALSIFY TRUTH BY DICTAT? in THE REASONER II, 1, January 2008,that denies the existence of nonstandard models of Peano Arithmetic, we prove from Compactness the existence of such models.

Computationalism holds that our grasp of notions like ‘computable function’ can be used to account for our putative ability to refer to the standard model of arithmetic. Tennenbaum's Theorem has been repeatedly invoked in service of this claim. I will argue that not only do the relevant class of arguments fail, but that the result itself is most naturally understood as having the opposite of a reference-fixing effect — i.e., rather than securing the determinacy of number-theoretic reference, (...) Tennenbaum's Theorem points towards a sense in which portions of our computational vocabulary should be regarded as model-relative. (shrink)

We carry out a study of definability issues in the standard models of Presburger and Skolem arithmetics (henceforth referred to simply as Presburger and Skolem arithmetics, for short, because we only deal with these models, not the theories, thus there is no risk of confusion) supplied with free unary predicates—which are strongly related to definability in the monadic SOA (second-order arithmetic) without × or + , respectively. As a consequence, we obtain a very direct proof for ${\Pi^1_1}$ -completeness (...) of Presburger, and also Skolem, arithmetic with a free unary predicate, generalize it to all ${\Pi^1_n}$ -levels, and give an alternative description of the analytical hierarchy without × or + . Here ‘direct’ means that one explicitly m-reduces the truth of ${\Pi^1_1}$ -formulae in SOA to the truth in the extended structures. Notice that for the case of Presburger arithmetic, the ${\Pi^1_1}$ -completeness was already known, but the proof was indirect and exploited some special ${\Pi^1_1}$ -completeness results on so-called recurrent nondeterministic Turing machines—for these reasons, it was hardly able to shed any light on definability issues or possible generalizations. (shrink)

In this paper I begin by extending two results of Solovay; the first characterizes the possible Turing degrees of models of True Arithmetic (TA), the complete first-order theory of the standard model of PA, while the second characterizes the possible Turing degrees of arbitrary completions of P. I extend these two results to characterize the possible Turing degrees of m-diagrams of models of TA and of arbitrary complete extensions of PA. I next give a construction showing that (...) the conditions Solovay identified for his characterization of degrees of models of arbitrary completions of PA cannot be dropped (I showed that these conditions cannot be simplified in the paper. (shrink)

We observe that the classification problem for countable models of arithmetic is Borel complete. On the other hand, the classification problems for finitely generated models of arithmetic and for recursively saturated models of arithmetic are Borel; we investigate the precise complexity of each of these. Finally, we show that the classification problem for pairs of recursively saturated models and for automorphisms of a fixed recursively saturated model are Borel complete.

Understanding arithmetic word problems involves a complex interaction of text comprehension and mathematical processes. This article presents a computer simulation designed to capture the working memory demands required in “bottomup” comprehension of arithmetic word problems. The simulation's sentence‐level parser and text integration component reflect the importance of processing the problem from its original natural language presentation. Children's probability of solution was analyzed in exploratory regression analyses as a function of the simulation's sentence‐level and text integration processes. Working memory (...) variables measuring the combined effects of concepts to remember and text integration inferences account for a significant proportion of variance in children's solution probabilities across the first four grade levels (K‐3). Consistent with previous results from others, which highlighted the significance of small changes in problem wording, the simulation offers a process‐oriented perspective as to why natural language presentation constrains the comprehension of mathematical relationships. (shrink)

In [4] the authors studied the residue field of a model M of IΔ0 + Ω1 for the principal ideal generated by a prime p. One of the main results is that M/ has a unique extension of each finite degree. In this paper we are interested in understanding the structure of any quotient field of M, i.e. we will study the quotient M/I for I a maximal ideal of M. We prove that any quotient field of M satisfies (...) the property of having a unique extension of each finite degree. We will use some of Cherlin's ideas from [3], where he studies the ideal theory of non standard algebraic integers. (shrink)

In most accounts of realization of computational processes by physical mechanisms, it is presupposed that there is one-to-one correspondence between the causally active states of the physical process and the states of the computation. Yet such proposals either stipulate that only one model of computation is implemented, or they do not reflect upon the variety of models that could be implemented physically. In this paper, I claim that mechanistic accounts of computation should allow for a broad variation of models (...) of computation. In particular, some non-standard models should not be excluded a priori. The relationship between mathematical models of computation and mechanistically adequate models is studied in more detail. (shrink)

Given a classical theory T, a Kripke model K for the language L of T is called T-normal or locally PA just in case the classical L-structure attached to each node of K is a classical model of T. Van Dalen, Mulder, Krabbe, and Visser showed that Kripke models of Heyting Arithmetic (HA) over finite frames are locally PA, and that Kripke models of HA over frames ordered like the natural numbers contain infinitely many PA-nodes. We show (...) that Kripke models of the latter sort are in fact PA-normal. This result is extended to a somewhat larger class of frames. (shrink)

Arithmetical definability has been extensively studied over the natural numbers. In this paper, we take up the study of arithmetical definability over finite structures, motivated by the correspondence between uniform AC0 and FO. We prove finite analogs of three classic results in arithmetical definability, namely that < and TIMES can first-order define PLUS, that < and DIVIDES can first-order define TIMES, and that < and COPRIME can first-order define TIMES. The first result sharpens the equivalence FO =FO to FO = (...) FO, answering a question raised by Barrington et al. about the Crane Beach Conjecture. Together with previous results on the Crane Beach Conjecture, our results imply that FO is strictly less expressive than FO = FO = FO. In more colorful language, one could say that, for parallel computation, multiplication is harder than addition. (shrink)

The paper presents an outline of the general theory of countable arithmetically saturated models of PA and some of its applications. We consider questions concerning the automorphism group of a countable recursively saturated model of PA. We prove new results concerning fixed point sets, open subgroups, and the cofinality of the automorphism group. We also prove that the standard system of a countable arithmetically saturated model of PA is determined by the lattice of its elementary substructures.

Let P 0 be the subsystem of Peano arithmetic obtained by restricting induction to bounded quantifier formulas. Let M be a countable, nonstandard model of P 0 whose domain we suppose to be the standard integers. Let T be a recursively enumerable extension of Peano arithmetic all of whose existential consequences are satisfied in the standard model. Then there is an initial segment M ' of M which is a model of T such (...) that the complete diagram of M ' is Turing reducible to the atomic diagram of M. Moreover, neither the addition nor the multiplication of M is recursive. (shrink)

We study those models of ZFCwhich are embeddable, as the class of all standard sets, in a model of internal set theory >ISTor models of some other nonstandard set theories.

In this paper we will show that for every cutIof any countable nonstandard model$\mathcal {M}$of$\mathrm {I}\Sigma _{1}$, eachI-small$\Sigma _{1}$-elementary submodel of$\mathcal {M}$is of the form of the set of fixed points of some proper initial self-embedding of$\mathcal {M}$iffIis a strong cut of$\mathcal {M}$. Especially, this feature will provide us with some equivalent conditions with the strongness of the standard cut in a given countable model$\mathcal {M}$of$ \mathrm {I}\Sigma _{1} $. In addition, we will find some criteria for (...) extendability of initial self-embeddings of countable nonstandard models of$ \mathrm {I}\Sigma _{1} $to larger models. (shrink)

We calculate the possible Scott ranks of countable models of Peano arithmetic. We show that no non-standard model can have Scott rank less than $\omega $ and that non-standard models of true arithmetic must have Scott rank greater than $\omega $. Other than that there are no restrictions. By giving a reduction via $\Delta ^{\mathrm {in}}_{1}$ bi-interpretability from the class of linear orderings to the canonical structural $\omega $ -jump of models of an arbitrary completion (...) T of $\mathrm {PA}$ we show that every countable ordinal $\alpha>\omega $ is realized as the Scott rank of a model of T. (shrink)

Global properties of canonical derivability predicates in Peano Arithmetic) are studied here by means of a suitable propositional modal logic GL. A whole book [1] has appeared on GL and we refer to it for more information and a bibliography on GL. Here we propose a sequent calculus for GL and, by exhibiting a good proof procedure, prove that such calculus admits the elimination of cuts. Most of standard results on GL are then easy consequences: completeness, decidability, finite (...) model property, interpolation and the fixed point theorem. (shrink)

We study the following open question in computable model theory: does there exist a structure of computable dimension two which is the prime model of its first-order theory? We construct an example of such a structure by coding a certain family of c.e. sets with exactly two one-to-one computable enumerations into a directed graph. We also show that there are examples of such structures in the classes of undirected graphs, partial orders, lattices, and integral domains.

In most accounts of realization of computational processes by physical mechanisms, it is presupposed that there is one-to-one correspondence between the causally active states of the physical process and the states of the computation. Yet such proposals either stipulate that only one model of computation is implemented, or they do not reflect upon the variety of models that could be implemented physically. -/- In this paper, I claim that mechanistic accounts of computation should allow for a broad variation of (...) models of computation. In particular, some non-standard models should not be excluded a priori. The relationship between mathematical models of computation and mechanistically adequate models is studied in more detail. (shrink)

Model theory is concerned mainly, although not exclusively, with infinite structures. In recent years, finite structures have risen to greater prominence, both within the context of mainstream model theory, e.g., in work of Lachlan, Cherlin, Hrushovski, and others, and with the advent of finite model theory, which incorporates elements of classical model theory, combinatorics, and complexity theory. The purpose of this survey is to provide an overview of what might be called the model theory of (...) finite structures. Some topics in finite model theory have strong connections to theoretical computer science, especially descriptive complexity theory. In fact, it has been suggested that finite model theory really is, or should be, logic for computer science. These connections with computer science will, however, not be treated here.It is well-known that many classical results of ‘infinite model theory’ fail over the class of finite structures, including the compactness and completeness theorems, as well as many preservation and interpolation theorems. The failure of compactness in the finite, in particular, means that the standard proofs of many theorems are no longer valid in this context. At present, there is no known example of a classical theorem that remains true over finite structures, yet must be proved by substantially different methods. It is generally concluded that first-order logic is ‘badly behaved’ over finite structures.From the perspective of expressive power, first-order logic also behaves badly: it is both too weak and too strong. Too weak because many natural properties, such as the size of a structure being even or a graph being connected, cannot be defined by a single sentence. Too strong, because every class of finite structures with a finite signature can be defined by an infinite set of sentences. Even worse, every finite structure is defined up to isomorphism by a single sentence. In fact, it is perhaps because of this last point more than anything else that model theorists have not been very interested in finite structures. Modern model theory is concerned largely with complete first-order theories, which are completely trivial here. (shrink)

A practical viewpoint links reality, representation, and language to calculation by the concept of Turing (1936) machine being the mathematical model of our computers. After the Gödel incompleteness theorems (1931) or the insolvability of the so-called halting problem (Turing 1936; Church 1936) as to a classical machine of Turing, one of the simplest hypotheses is completeness to be suggested for two ones. That is consistent with the provability of completeness by means of two independent Peano arithmetics discussed in Section (...) I. Many modifications of Turing machines cum quantum ones are researched in Section II for the Halting problem and completeness, and the model of two independent Turing machines seems to generalize them. Then, that pair can be postulated as the formal definition of reality therefore being complete unlike any of them standalone, remaining incomplete without its complementary counterpart. Representation is formal defined as a one-to-one mapping between the two Turing machines, and the set of all those mappings can be considered as “language” therefore including metaphors as mappings different than representation. Section III investigates that formal relation of “reality”, “representation”, and “language” modeled by (at least two) Turing machines. The independence of (two) Turing machines is interpreted by means of game theory and especially of the Nash equilibrium in Section IV. Choice and information as the quantity of choices are involved. That approach seems to be equivalent to that based on set theory and the concept of actual infinity in mathematics and allowing of practical implementations. (shrink)

We give a constructive analysis of learning as it arises in various computational interpretations of classical Peano Arithmetic, such as Aschieri and Berardi learning based realizability, Avigad’s update procedures and epsilon substitution method. In particular, we show how to compute in Gödel’s system T upper bounds on the length of learning processes, which are themselves represented in T through learning based realizability. The result is achieved by the introduction of a new non standard model of Gödel’s T, (...) whose new basic objects are pairs of non standard natural numbers and moduli of convergence, where the latter are objects giving constructive information about the former. As a foundational corollary, we obtain that that learning based realizability is a constructive interpretation of Heyting Arithmetic plus excluded middle over Σ10 formulas and of all Peano Arithmetic when combined with Gödel’s double negation translation. As a byproduct of our approach, we also obtain a new proof of Avigad’s theorem for update procedures and thus of the termination of the epsilon substitution method for PA. (shrink)

A new technique for proving realisability results is presented, and is illustrated in detail for the simple case of arithmetic minus induction. CL is a Gentzen formulation of classical logic. CPQ is CL minus the Cut Rule. The basic proof theory and model theory of CPQ and CL is developed. For the semantics presented CPQ is a paraconsistent logic, i.e. there are non-trivial CPQ models in which some sentences are both true and false. Two systems of arithmetic (...) minus induction are introduced, CL-A and CPQ-A based on CL and CPQ, respectively. The realisability theorem for CPQ-A is proved: It is shown constructively that to each theorem A of CPQ-A there is a formula A *, a so-called “realised disjunctive form of A ”, such that variables bound by essentially existential quantifiers in A * can be written as recursive functions of free variables and variables bound by essentially universal quantifiers. Realisability is then applied to prove the consistency of CL-A, making use of certain finite non-trivial inconsistent models of CPQ-A. (shrink)

We study the expressive power in the finite of the logic Fixed-Point+Counting, the extension of first-order logic which is obtained through adding both the fixed-point constructor and the ability to count. To this end an isomorphism preserving (`generic') model of computation is introduced whose PTime restriction exactly corresponds to this level of expressive power, while its PSpace restriction corresponds to While+Counting. From this model we obtain a normal form which shows a rather clear separation of the relational vs. (...) the arithmetical side of the algorithms involved. In parallel, we study the relations of Fixed-Point+Counting with the infinitary logics L r ∞ω (∃ ≥ m ) m∈ω and the corresponding pebble games. The main result, however, involves the concept of an arithmetical invariant. By this we mean a functor taking every finite relational structure to an expansion of (an initial segment of) the standard arithmetical structure. In particular its values are linearly ordered structures. We establish the existence of a family of arithmetical invariants (J r ) r≥ 1 with the following properties: $\bullet$ The invariants themselves can be evaluated in polynomial time. $\bullet$ A class of finite relational structures is definable in Fixed-Point+Counting if and only if membership can be decided in polynomial time on the basis of the values of one of the invariants. $\bullet$ The invariant J r classifies all finite relational structures exactly up to equivalence with respect to the logic L r ∞ω (∃ ≥ m ) m∈ω . We also give a characterization of Fixed-Point+Counting in terms of sequences of formulae in the L r ωω (∃ ≥ m ) m∈ω : It corresponds exactly to the polynomial time computable families (φ n ) n∈ω in these logics. Towards a positive assessment of the expressive power of Fixed-Point+Counting, it is shown that the natural extension of fixed-point logic by Lindström quantifiers, which capture all the PTime computable properties of cardinalities of definable predicates, is strictly weaker than what we get here. This implies in particular that every extension of fixed-point logic by means of monadic Lindstrom quantifiers, which stays within PTime, must be strictly contained in Fixed-Point+Counting. (shrink)

The paper is concerned with the old conjecture that there are infinitely many Mersenne primes. It is shown in the work that this conjecture is true in the standard model of arithmetic. The proof refers to the general approach to first–order logic based on Rasiowa-Sikorski Lemma and the derived notion of a Rasiowa–Sikorski set. This approach was developed in the papers [ 2 – 4 ]. This work is a companion piece to [ 4 ].

A survey of the isomorphic submodels of Vω, the set of hereditarily finite sets. In the usual language of set theory, Vω has 2ℵ0 isomorphic submodels. But other set-theoretic languages give different systems of submodels. For example, the language of adjunction allows only countably many isomorphic submodels of Vω.

The complete characterisation of order types of non-standard models of Peano arithmetic and its extensions is a famous open problem. In this paper, we consider subtheories of Peano arithmetic (both with and without induction), in particular, theories formulated in proper fragments of the full language of arithmetic. We study the order types of their non-standard models and separate all considered theories via their possible order types. We compare the theories with and without induction and observe (...) that the theories without induction tend to have an algebraic character that allows model constructions by closing a model under the relevant algebraic operations. (shrink)

The complete characterisation of order types of non-standard models of Peano arithmetic and its extensions is a famous open problem. In this paper, we consider subtheories of Peano arithmetic (both with and without induction), in particular, theories formulated in proper fragments of the full language of arithmetic. We study the order types of their non-standard models and separate all considered theories via their possible order types. We compare the theories with and without induction and observe (...) that the theories without induction tend to have an algebraic character that allows model constructions by closing a model under the relevant algebraic operations. (shrink)

Let us recall that Raphael Robinson's Arithmetic Q is an axiom system that differs from Peano Arithmetic essentially by containing no Induction axioms [13], [18]. We will generalize the semantic-tableaux version of the Second Incompleteness Theorem almost to the level of System Q. We will prove that there exists a single rather long Π 1 sentence, valid in the standard model of the Natural Numbers and denoted as V, such that if α is any finite consistent (...) extension of Q + V then α will be unable to prove its Semantic Tableaux consistency. The same result will also apply to axiom systems α with infinite cardinality when these infinite-sized axiom systems satisfy a minor additional constraint, called the Conventional Encoding Property. Our formalism will also imply that the semantic-tableaux version of the Second Incompleteness Theorem generalizes for the axiom system IΣ 0 , as well as for all its natural extensions. (This answers an open question raised twenty years ago by Paris and Wilkie [15].). (shrink)

We show that every model of IΔ0 has an end extension to a model of a theory where log-space computable function are formalizable. We also show the existence of an isomorphism between models of IΔ0 and models of linear arithmetic LA.

It is well known that the relevant arithmetic R# admits finite models whose domains are the integers modulo n rather than the expected natural numbers. Less well appreciated is the fact that the logic of these models is much more subtle than that of the three-valued structure in which they are usually presented. In this paper we consider the DeMorgan monoids in which R# can be modelled, deriving a fairly complete account of those modelling the stronger arithmetic RM# (...) modulo n and a partial account for the case of R# modulo a prime number. The more general case in which the modulus is arbitrary is shown to lead to infinite propositional structures even with the additional constraint that '0=1' implies everything. (shrink)