Nonstandard Arithmetic Research Articles

Introduction to Hyperspaces

The development of mathematics brought mathematicians to infinite structures. This process started with transcendent real numbers and infinite sequences going through infinite series to transfinite numbers to nonstandard numbers to hypernumbers. From mathematics, infinity came to physics where physicists have been trying to get rid of infinity inventing a variety of techniques for doing this. In contrast to this, mathematicians as well as some physicists suggested ways to work with infinity introducing new mathematical structures such distributions and extrafunctions. The goal of this paper is to extend mathematical tools for treating infinity by considering hyperspaces and developing their theory.

Short Proofs for Slow Consistency

Let $\operatorname{Con}(\mathbf T)\!\restriction\!x$ denote the finite consistency statement "there are no proofs of contradiction in $\mathbf T$ with $\leq x$ symbols". For a large class of natural theories $\mathbf T$, Pudl\'ak has shown that the lengths of the shortest proofs of $\operatorname{Con}(\mathbf T)\!\restriction\!n$ in the theory $\mathbf T$ itself are bounded by a polynomial in $n$. At the same time he conjectures that $\mathbf T$ does not have polynomial proofs of the finite consistency statements $\operatorname{Con}(\mathbf T+\operatorname{Con}(\mathbf T))\!\restriction\!n$. In contrast we show that Peano arithmetic ($\mathbf{PA}$) has polynomial proofs of $\operatorname{Con}(\mathbf{PA}+\operatorname{Con}^*(\mathbf{PA}))\!\restriction\!n$, where $\operatorname{Con}^*(\mathbf{PA})$ is the slow consistency statement for Peano arithmetic, introduced by S.-D. Friedman, Rathjen and Weiermann. We also obtain a new proof of the result that the usual consistency statement $\operatorname{Con}(\mathbf{PA})$ is equivalent to $\varepsilon_0$ iterations of slow consistency. Our argument is proof-theoretic, while previous investigations of slow consistency relied on non-standard models of arithmetic.

Dense enough non-standard reals

Non-standard analysis is a branch of Mathematics introduced by Abraham Robinson in 1966[1]. In 1977, Edward Nelson gave an axiomatic approach to Non-standard analysis [2]. One of the main features of these approaches is the reduction of infiniteness to finiteness. In this paper we have come up with functions in the nonstandard setting, especially in the extended Real number system consisting of infinitesimals and infinite numbers, that are analogous with functions in classical analysis and show that non-standard numbers are dense enough to facilitate continuity of functions

Vector-like quarks with non-renormalizable interactions

We study the impact of the leading non-renormalizable terms in the effective field theory that describes general extensions of the Standard Model with vector-like quarks that can decay into Standard Model particles. Dropping the usual assumption of renormalizability has several phenomenological consequences for the production and decay of the heavy quarks and also for Higgs physics. The most dramatic effects, including those associated with a long lifetime, occur for vector-like quarks with non-standard quantum numbers.

Supersymmetric Calogero and Calogero-Sutherland models from gauging

We describe how the new kinds of N = 2 and N = 4 supersymmetric extensions of the rational and hyperbolic Calogero models can be derived by gauging U(n) symmetry of the appropriate superfield matrix models. These systems feature non-standard numbers N n2 of physical fermionic variables as compared with N n in the standard case. An essential ingredient of N = 4 models is the necessary presence of semi-dynamical spin variables described by d = 1 Wess-Zumino terms. The bosonic cores of N = 4 models are U(2) spin Calogero and Calogero-Sutherland models. In the hyperbolic case two non-equivalent N = 4 extensions exist, with and without the interacting center-of-mass coordinate in the bosonic sector. The talk is based on joint works with Sergey Fedoruk and Olaf Lechtenfeld.

The Monotonic Sequence Theorem and Measurement of Lengths and Areas in Axiomatic Non-Standard Hyperrational Analysis

This paper lies in the framework of axiomatic non-standard analysis based on the non-standard arithmetic axiomatic theory. This arithmetic includes actual infinite numbers. Unlike the non-standard model of arithmetic, this approach does not take models into account but uses an axiomatic research method. In the axiomatic theory of non-standard arithmetic, hyperrational numbers are defined as triplets of hypernatural numbers. Since the theory of hyperrational numbers and axiomatic non-standard analysis is mainly published in Russian, in this article we give a brief review of its basic concepts and required results. Elementary hyperrational analysis includes defining and evaluating such notions as continuity, differentiability and integral calculus. We prove that a bounded monotonic sequence is a Cauchy sequence. Also, we solve the task of line segment measurement using hyperrational numbers. In fact, this allows us to approximate real numbers using hyperrational numbers, and shows a way to model real numbers and real functions using hyperrational numbers and functions.

Fully encrypted high-speed microprocessor architecture: the secret computer in simulation

Copyright © 2019 Inderscience Enterprises Ltd. The architecture of an encrypted high-performance microprocessor designed on the principle that a nonstandard arithmetic generates encrypted processor states is described here. Data in registers, in memory and on buses exists in encrypted form. Any block encryption is feasible, in principle. The processor is (initially) intended for cloud-based remote computation. An encrypted version of the standard OpenRISC instruction set is understood by the processor. It is proved here, for programs written in a minimal subset of instructions, that the platform is secure against ‘Iago’ attacks by the privileged operator or a subverted operating system, which cannot decrypt the program output, nor change the program’s output to a particular value of their choosing. Performance measures from cycle-accurate behavioural simulation of the platform are given for 64-bit RC2 (symmetric, keyed) and 72-bit Paillier (asymmetric, additively homomorphic, no key in-processor) encryptions. Measurements are centred on a nominal 1 GHz clock with 3 ns cache and 15 ns memory latency, which is conservative with respect to available technology.

A note on non-classical nonstandard arithmetic

Recently, a number of formal systems for Nonstandard Analysis restricted to the language of finite types, i.e. nonstandard arithmetic, have been proposed. We single out one particular system by Dinis–Gaspar, which is categorised by the authors as being part of intuitionistic nonstandard arithmetic. Their system is indeed inconsistent with the Transfer axiom of Nonstandard Analysis, and the latter axiom is classical in nature as it implies (higher-order) comprehension. Inspired by this observation, the main aim of this paper is to provide answers to the following questions:(Q1)In the spirit of Reverse Mathematics, what is the minimal fragment of Transfer that is inconsistent with the Dinis–Gaspar system?(Q2)What other axioms are inconsistent with the Dinis–Gaspar system? Our answer to the first question suggests that the aforementioned inconsistency actually derives from the axiom of extensionality relative to the standard world, and that other (much stronger) consequences of Transfer are actually harmless. Perhaps surprisingly, our answer to the second question shows that the Dinis–Gaspar system is inconsistent with a number of (non-classical) continuity theorems which one would – in our opinion – categorise as intuitionistic in the sense of Brouwer. Finally, we show that the Dinis–Gaspar system involves a standard part map, suggesting this system also pushes the boundary of what still counts as ‘Nonstandard Analysis’ or ‘internal set theory’.

Independence of the Grossone-Based Infinity Methodology from Non-standard Analysis and Comments upon Logical Fallacies in Some Texts Asserting the Opposite

This commentary considers non-standard analysis and a recently introduced computational methodology based on the notion of \G1 (this symbol is called \emph{grossone}). The latter approach was developed with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework and in all the situations requiring these notions. Non-standard analysis is a classical purely symbolic technique that works with ultrafilters, external and internal sets, standard and non-standard numbers, etc. In its turn, the \G1-based methodology does not use any of these notions and proposes a more physical treatment of mathematical objects separating the objects from tools used to study them. It both offers a possibility to create new numerical methods using infinities and infinitesimals in floating-point computations and allows one to study certain mathematical objects dealing with infinity more accurately than it is done traditionally. In these notes, we explain that even though both methodologies deal with infinities and infinitesimals, they are independent and represent two different philosophies of Mathematics that are not in a conflict. It is proved that texts \cite{Flunks, Gutman_Kutateladze_2008, Kutateladze_2011} asserting that the \G1-based methodology is a part of non-standard analysis unfortunately contain several logical fallacies. Their attempt to prove that the \G1-based methodology is a part of non-standard analysis is similar to trying to show that constructivism can be reduced to the traditional mathematics.

The Logical Complexity of Finitely Generated Commutative Rings

AbstractWe characterize those finitely generated commutative rings which are (parametrically) bi-interpretable with arithmetic: a finitely generated commutative ring A is bi-interpretable with $(\mathbb{N},{+},{\times })$ if and only if the space of non-maximal prime ideals of A is nonempty and connected in the Zariski topology and the nilradical of A has a nontrivial annihilator in $\mathbb{Z}$. Notably, by constructing a nontrivial derivation on a nonstandard model of arithmetic we show that the ring of dual numbers over $\mathbb{Z}$ is not bi-interpretable with $\mathbb{N}$.

The Gandy–Hyland functional and a computational aspect of Nonstandard Analysis

In this paper, we highlight a new computational aspect of Nonstandard Analysis relating to higher-order computability theory. In particular, we prove that the Gandy-Hyland functional equals a primitive recursive functional involving nonstandard numbers inside Nelson's internal set theory. From this classical and ineffective proof in Nonstandard Analysis, a term from Godel's system T can be extracted which computes the Gandy-Hyland functional in terms of a modulus-of-continuity functional and the special fan functional. We obtain several similar relative computability results not involving Nonstandard Analysis from their associated nonstandard theorems. By way of reversal, we show that certain relative computability results, called Herbrandisations, also imply the nonstandard theorem from whence they were obtained. Thus, we establish a direct two-way connection between the field Computability (in particular theoretical computer science) and the field Nonstandard Analysis.

Intuitionistic nonstandard bounded modified realisability and functional interpretation

We present a bounded modified realisability and a bounded functional interpretation of intuitionistic nonstandard arithmetic with nonstandard principles.The functional interpretation is the intuitionistic counterpart of Ferreira and Gaspar's functional interpretation and has similarities with Van den Berg, Briseid and Safarik's functional interpretation but replacing finiteness by majorisability.We give a threefold contribution: constructive content and proof-theoretical properties of nonstandard arithmetic; filling a gap in the literature; being in line with nonstandard methods to analyse compactness arguments.

Cauchy, infinitesimals and ghosts of departed quantifiers

Procedures relying on infinitesimals in Leibniz, Euler and Cauchy have been interpreted in both a Weierstrassian and Robinson's frameworks. The latter provides closer proxies for the procedures of the classical masters. Thus, Leibniz's distinction between assignable and inassignable numbers finds a proxy in the distinction between standard and nonstandard numbers in Robinson's framework, while Leibniz's law of homogeneity with the implied notion of equality up to negligible terms finds a mathematical formalisation in terms of standard part. It is hard to provide parallel formalisations in a Weierstrassian framework but scholars since Ishiguro have engaged in a quest for ghosts of departed quantifiers to provide a Weierstrassian account for Leibniz's infinitesimals. Euler similarly had notions of equality up to negligible terms, of which he distinguished two types: geometric and arithmetic. Euler routinely used product decompositions into a specific infinite number of factors, and used the binomial formula with an infinite exponent. Such procedures have immediate hyperfinite analogues in Robinson's framework, while in a Weierstrassian framework they can only be reinterpreted by means of paraphrases departing significantly from Euler's own presentation. Cauchy gives lucid definitions of continuity in terms of infinitesimals that find ready formalisations in Robinson's framework but scholars working in a Weierstrassian framework bend over backwards either to claim that Cauchy was vague or to engage in a quest for ghosts of departed quantifiers in his work. Cauchy's procedures in the context of his 1853 sum theorem (for series of continuous functions) are more readily understood from the viewpoint of Robinson's framework, where one can exploit tools such as the pointwise definition of the concept of uniform convergence. Keywords: historiography; infinitesimal; Latin model; butterfly model

Nonstandard Functional Interpretations and Categorical Models

Recently, the second author, Briseid, and Safarik introduced nonstandard Dialectica, a functional interpretation capable of eliminating instances of familiar principles of nonstandard arithmetic—including overspill, underspill, and generalizations to higher types—from proofs. We show that the properties of this interpretation are mirrored by first-order logic in a constructive sheaf model of nonstandard arithmetic due to Moerdijk, later developed by Palmgren, and draw some new connections between nonstandard principles and principles that are rejected by strict constructivism. Furthermore, we introduce a variant of the Diller–Nahm interpretation with two different kinds of quantifiers, similar to Hernest’s light Dialectica interpretation, and show that one can obtain nonstandard Dialectica by weakening the computational content of the existential quantifiers—a process called herbrandization. We also define a constructive sheaf model mirroring this new functional interpretation, and show that the process of herbrandization has a clear meaning in terms of these sheaf models.

Interpreting weak Kőnig's lemma in theories of nonstandard arithmetic

We show how to interpret weak Kőnig's lemma in some recently defined theories of nonstandard arithmetic in all finite types. Two types of interpretations are described, with very different verifications. The celebrated conservation result of Friedman's about weak Kőnig's lemma can be proved using these interpretations. We also address some issues concerning the collecting of witnesses in herbrandized functional interpretations.

$$\mathcal {F}$$ F -finite embeddabilities of sets and ultrafilters

Let S be a semigroup, let $$n\in \mathbb {N}$$nźN be a positive natural number, let $$A,B\subseteq S$$A,B⊆S, let $$\mathcal {U},\mathcal {V}\in \beta S$$U,VźβS and let let $$\mathcal {F}\subseteq \{f:S^{n}\rightarrow S\}$$F⊆{f:SnźS}. We say that A is $$\mathcal {F}$$F-finitely embeddable in B if for every finite set $$F\subseteq A$$F⊆A there is a function $$f\in \mathcal {F}$$fźF such that $$f\left( A^{n}\right) \subseteq B$$fAn⊆B, and we say that $$\mathcal {U}$$U is $$\mathcal {F}$$F-finitely embeddable in $$\mathcal {V}$$V if for every set $$B\in \mathcal {V}$$BźV there is a set $$A\in \mathcal {U}$$AźU such that A is $$\mathcal {F}$$F-finitely embeddable in B. We show that $$\mathcal {F}$$F-finite embeddabilities can be used to study certain combinatorial properties of sets and ultrafilters related with finite structures. We introduce the notions of set and of ultrafilter maximal for $$\mathcal {F}$$F-finite embeddability, whose existence is proved under very mild assumptions. Different choices of $$\mathcal {F}$$F can be used to characterize many combinatorially interesting sets/ultrafilters as maximal sets/ultrafilters, for example thick sets, AP-rich sets, $$\overline{K(\beta S)}$$K(βS)¯ and so on. The set of maximal ultrafilters for $$\mathcal {F}$$F-finite embeddability can be characterized algebraically in terms of $$\mathcal {F}$$F. This property can be used to give an algebraic characterization of certain interesting sets of ultrafilters, such as the ultrafilters whose elements contain, respectively, arbitrarily long arithmetic, geoarithmetic or polynomial progressions. As a consequence of the connection between sets and ultrafilters maximal for $$\mathcal {F}$$F-finite embeddability we are able to prove a general result that entails, for example, that given a finite partition of a set that contains arbitrarily long geoarithmetic (resp. polynomial) progressions, one cell must contain arbitrarily long geoarithmetic (resp. polynomial) progressions. Finally we apply $$\mathcal {F}$$F-finite embeddabilities to study a few properties of homogeneous partition regular diophantine equations. Some of our results are based on connections between ultrafilters and nonstandard models of arithmetic.

Finite Embeddability of Sets and Ultrafilters

A set A of natural numbers is finitely embeddable in another such set B if every finite subset of A has a rightward translate that is a subset of B. This notion of finite embeddability arose in combinatorial number theory, but in this paper we study it in its own right. We also study a related notion of finite embeddability of ultrafilters on the natural numbers. Among other results, we obtain connections between finite embeddability and the algebraic and topological structure of the Stone-Cech compactification of the discrete space of natural numbers. We also obtain connections with nonstandard models of arithmetic.

Nonstandardness and the bounded functional interpretation

The bounded functional interpretation of arithmetic in all finite types is able to interpret principles like weak König's lemma without the need of any form of bar recursion. This interpretation requires the use of intensional (rule-governed) majorizability relations. This is a somewhat unusual feature. The main purpose of this paper is to show that if the base domain of the natural numbers is extended with nonstandard elements, then the bounded functional interpretation can be seen as falling out from a functional interpretation of nonstandard number theory without intensional notions. The original bounded functional interpretation can be seen as the trace left behind by the new interpretation when one sees it restricted to the standard number theoretical setting.We also answer an open question regarding the conservativity of the transfer principle vis-à-vis functional interpretations of nonstandard arithmetic.

Totally non‐immune sets

Let be a countable first‐order language and be an ‐structure. “Definable set” means a subset of M which is ‐definable in with parameters. A set is said to be immune if it is infinite and does not contain any infinite definable subset. X is said to be partially immune if for some definable A, is immune. X is said to be totally non‐immune if for every definable A, and are not immune. Clearly every definable set is totally non‐immune. Here we ask whether the converse is true and prove that it is false for every countable structure whose class of definable sets satisfies a mild condition. We investigate further the possibility of an alternative construction of totally non‐immune non‐definable sets with the help of a subclass of immune sets, the class of cohesive sets, as well as with the help of a generalization of definable sets, the semi‐definable ones (the latter being naturally defined in models of arithmetic). Finally connections are found between totally non‐immune sets and generic classes in nonstandard models of arithmetic.

Constant Regions in Models of Arithmetic

This paper introduces a new theory of constant regions, which generalizes that of interstices, in nonstandard models of arithmetic. In particular, we show that two homogeneity notions introduced by Richard Kaye and the author, namely, constantness and pregenericity, are equivalent. This led to some new characterizations of generic cuts in terms of existential closedness.