Order Number Theory Research Articles

The Ordinal Interpretation of the Integers and Its Use in Number Theory

The author recently published a paper which claimed that an ordinal interpretation of numbers had limited applicability for cryptography. A further examination of this subject, in particular to what extent an ordinal interpretation is useful for recurrence sequences, is needed. Hilbert favored an interpretation of the natural numbers that placed their ordinal properties prior to their cardinal properties [1] [2]. The author examines ordinal uses of the integers in number theory in order to discuss the possibilities and limitations of this approach. The author hopes this paper will be useful in clarifying or even correcting some matters that were discussed in his paper of January of 2018. I was trained informally in philosophical realism, and while I think idealism too has a place, at this time in my life I believe that the weight of evidence and usefulness is more on the side of philosophical materialism. I hope this discussion will help supplement for my readers the material in Number in Mathematical Cryptography. I still maintain that a lack of clarity on these matters has hindered progress in cryptography; and it has taken time for me to better understand these things. I hope others who have interest and ability will assist in making these matters clearer. My intention was to work in pure mathematics, and the transition to an applied mindset was difficult for me. As a result, I feel most comfortable in a more middle-of-the road attitude, but have had to slowly move to a more precise analysis of the physical quantities involved. I hope my readers will be patient with my terminology, which is still evolving, and my discussion of things which are more indirectly related, and which are necessary for my expression. These are important things for the mathematical community to understand, and I hope smarter and more knowledgeable people will address my errors, and improve upon the things I might have correct. I am discussing sequences which are sometimes a use of both ordinal and cardinal numbers.

Pure patterns of order 2

We provide mutual elementary recursive order isomorphisms between classical ordinal notations, based on Skolem hulling, and notations from pure elementary patterns of resemblance of order 2, showing that the latter characterize the proof-theoretic ordinal 1∞ of the fragment Π11–CA0 of second order number theory, or equivalently the set theory KPℓ0. As a corollary, we prove that Carlson's result on the well-quasi orderedness of respecting forests of order 2 implies transfinite induction up to the ordinal 1∞. We expect that our approach will facilitate analysis of more powerful systems of patterns.

Triangular intuitionistic fuzzy number theory for driver-pedestrians interactions and risk exposure modeling

Abstract: We travel every day as pedestrians on the road network, if only for a short time, in order to reach another means of transport. But this activity is not as risk-free as it should be. These accidents represent the product of an overall system combining the individual behaviors of pedestrians and drivers, transport activity, and the environmental system, under conditions that make these malfunctions both predictable and avoidable. This predictive skill is an essential step in ensuring the safety of pedestrians in road traffic. In this paper, we use intuitionistic fuzzy number theory in order to model the exposure of pedestrians to the accidents risks by new indicators. This approach aligns the behavioral psychology of pedestrians and drivers with intuitive methods based on the theory of fuzzy sets. A software solution is developed for this instance by reusing the models of pedestrian simulation developed in our previous works. The attempt in this article considers a novelty in partial aspect of risk modeling. The fuzzy intuitionistic theory has the flexibility to investigate hesitation and indecision. Some other behavioral and environmental factors must be taken into account in order to explore deeply the issue.

Full and hat inductive definitions are equivalent in NBG

A new research project has, quite recently, been launched to clarify how different, from systems in second order number theory extending ACA 0, those in second order set theory extending NBG (as well as those in n + 3-th order number theory extending the so-called Bernays−Gödel expansion of full n + 2-order number theory etc.) are. In this article, we establish the equivalence between \({\Delta^1_0\mbox{\bf-LFP}}\) and \({\Delta^1_0\mbox{\bf-FP}}\), which assert the existence of a least and of a (not necessarily least) fixed point, respectively, for positive elementary operators (or between \({\Delta^{n+2}_0\mbox{\bf-LFP}}\) and \({\Delta^{n+2}_0\mbox{\bf-FP}}\)). Our proof also shows the equivalence between ID 1 and \({\widehat{\it ID}_1}\), both of which are defined in the standard way but with the starting theory PA replaced by ZFC (or full n + 2-th order number theory with global well-ordering).

RELATIVE PREDICATIVITY AND DEPENDENT RECURSION IN SECOND-ORDER SET THEORY AND HIGHER-ORDER THEORIES

AbstractThis article reports that some robustness of the notions of predicativity and of autonomous progression is broken down if as the given infinite total entity we choose some mathematical entities other than the traditionalω. Namely, the equivalence between normal transfinite recursion scheme and newdependent transfinite recursionscheme, which does hold in the context of subsystems of second order number theory, does not hold in the context of subsystems of second order set theory where the universeVof sets is treated as the given totality (nor in the contexts of those ofn+3-th order number or set theories, where the class of alln+2-th order objects is treated as the given totality).

Evaluation on Coal Miner’s Emergency Response Capacity Based on the Catastrophe Theory and Triangular Fuzzy Number

The production environment of coal mine is complex and tough, so once dangerous situation appears it requires coal miners take accurate and effective emergency measures. Actually, the occurrence and further expansion of a large number of mining accidents are associated with coal miners’ mishandling of emergency. Therefore, evaluating coal miners’ emergency response capacity is the significant way to improve the coal miners' emergency response capacity level and pre-control coal mine accidents. Considering the drawbacks of traditional evaluation methods, this paper introduced the principle of catastrophe theory in evaluation, and proposed the evaluation model of coal miner’s emergency response capacity by introducing the triangular fuzzy number theory in order to reduce the uncertainties of underlying index grades made by experts. In this method, the importance and quantitative process of each assessing indices were determined by the intrinsic mechanism of the normalized formula in catastrophe model, thus the problems of estimating the assessing index weights were avoided and the effect of subjective factors was mitigated. Finally, the verification of evaluation was implemented on specific example, and the results show that this evaluation method is reasonable, feasible, exact, and stable.

Asymptotics of counts of small components in random structures and models of coagulation-fragmentation

We establish necessary and sufficient conditions for the convergence (in the sense of finite dimensional distributions) of multiplicative measures on the set of partitions. The multiplicative measures depict distributions of component spectra of random structures and also the equilibria of classic models of statistical mechanics and stochastic processes of coagulation-fragmentation. We show that the convergence of multiplicative measures is equivalent to the asymptotic independence of counts of components of fixed sizes in random structures. We then apply Schur’s tauberian lemma and some results from additive number theory and enumerative combinatorics in order to derive plausible sufficient conditions of convergence. Our results demonstrate that the common belief, that counts of components of fixed sizes in random structures become independent as the number of particles goes to infinity, is not true in general.

Behavior of lacunary series at the natural boundary

We develop a local theory of lacunary Dirichlet series of the form ∑ k = 1 ∞ c k exp ( − z g ( k ) ) , Re ( z ) > 0 as z approaches the boundary i R , under the assumption g ′ → ∞ and further assumptions on c k . These series occur in many applications in Fourier analysis, infinite order differential operators, number theory and holomorphic dynamics among others. For relatively general series with c k = 1 , the case we primarily focus on, we obtain blow up rates in measure along the imaginary line and asymptotic information at z = 0 . When sufficient analyticity information on g exists, we obtain Borel summable expansions at points on the boundary, giving exact local description. Borel summability of the expansions provides property-preserving extensions beyond the barrier. The singular behavior has remarkable universality and self-similarity features. If g ( k ) = k b , c k = 1 , b = n or b = ( n + 1 ) / n , n ∈ N , behavior near the boundary is roughly of the standard form Re ( z ) − b ′ Q ( x ) where Q ( x ) = 1 / q if x = p / q ∈ Q and zero otherwise. The Bötcher map at infinity of polynomial iterations of the form x n + 1 = λ P ( x n ) , | λ | < λ 0 ( P ) , turns out to have uniformly convergent Fourier expansions in terms of simple lacunary series. For the quadratic map P ( x ) = x − x 2 , λ 0 = 1 , and the Julia set is the graph of this Fourier expansion in the main cardioid of the Mandelbrot set.

More undecidable lattices of Steinitz exchange systems

AbstractWe show that the first order theory of the lattice &lt;ω(S) of finite dimensional closed subsets of any nontrivial infinite dimensional Steinitz Exhange System S has logical complexity at least that of first order number theory and that the first order theory of the lattice (S∞) of computably enumerable closed subsets of any nontrivial infinite dimensional computable Steinitz Exchange System S∞ has logical complexity exactly that of first order number theory. Thus, for example, the lattice of finite dimensional subspaces of a standard copy of ⊕ωQ interprets first order arithmetic and is therefore as complicated as possible. In particular, our results show that the first order theories of the lattice (V∞) of c.e. subspaces of a fully effective ℵ0-dimensional vector space V∞ and the lattice of c.e. algebraically closed subfields of a fully effective algebraically closed field F∞ of countably infinite transcendence degree each have logical complexity that of first order number theory.

A reflection principle and its applications to nonstandard models

Some methods of constructing nonstandard models work only for particular theories, such as ZFC, or CA + AC (which is second order number theory with the choice scheme). The examples of this which motivated the results of this paper occur in the main theorems of [5], which state that if T is any consistent extension of either ZFC0 (which is ZFC but with only countable replacement) or CA + AC and if κ and λ are suitably chosen cardinals, then T has a model which is κ-saturated and has the λ-Bolzano-Weierstrass property. (Compare with Theorem 3.5.) Another example is a result from [12] which states that if T is any consistent extension of CA + AC and cf (λ) &gt; ℵ0, then T has a natural λ-Archimedean model. (Compare with Theorem 3.1 and the comments following it.) Still another example is a result in [6] in which it is shown that if a model of Peano arithmetic is expandable to a model of ZF or of CA, then so is any cofinal extension of . (Compare with Theorem 3.10.) Related types of constructions can also be found in [10] and [11].A reflection principle will be proved here, allowing these constructions to be extended to models of many other theories, among which are some exceedingly weak theories and also all of their completions.

Fisher ideal output, input, and productivity indexes revisited

A productivity index for a firm is generally defined as an output index divided by an input index. The first part of the paper uses the test or axiomatic approach to index number theory in order to determine the appropriate functional form for the output and input indexes. It is found that the Fisher ideal index satisfies 21 reasonable tests and is uniquely characterized by a subset of these tests. In the remainder of the paper, the economic approach to productivity indexes introduced by Caves, Christensen, and Diewert is adopted and, again, a strong justification for the Fisher productivity index is provided.

About the proof-theoretic ordinals of weak fixed point theories

AbstractThis paper presents several proof-theoretic results concerning weak fixed point theories over second order number theory with arithmetic comprehension and full or restricted induction on the natural numbers. It is also shown that there are natural second order theories which are proof-theoretically equivalent but have different proof-theoretic ordinals.

Arithmetic transfinite induction and recursive well-orderings

A uniform, algebraic proof that every number-theoretic assertion provable in any of the intuitionistic theories T listed below has a well-founded recursive proof tree (demonstraby in T) is given. Thus every such assertion is provable by transfinite induction over some primitive recursive well-ordering. T can be higher order number theory, set theory, or its extensions equiconsistent with large cardinals. It is shown that there is a number-theoretic assertion B( n) (independent of T) with a parameter n such that any primitive recursive linear ordering R on ω for which transfinite induction on R for B( n) is provable in T is in fact a well-ordering.

Sue Toledo. Tableau systems for first order number theory and certain higher order theories. Lecture notes in mathematics, vol. 447. Springer-Verlag, Berlin, Heidelberg, and New York, 1975, iii + 339 pp.

Sue Toledo. Tableau systems for first order number theory and certain higher order theories. Lecture notes in mathematics, vol. 447. Springer-Verlag, Berlin, Heidelberg, and New York, 1975, iii + 339 pp.

An Elementary Analysis of the Factorization of Integers and the Determination of Primes by the Use of Integral Binary Quadratic Forms

In several previous issues of the MATHEMATICS MAGAZINE, we have disclissed the factorization of integers by the use of integral binary quadratic forms. In this paper, we analyze our previous work along with some of the modern advances in number theory in order that we may develop an elementary approach, not only for the factorization of integers, but also as an aid in the determination of primes. Before continuing, we introduce the following transformation: Let M= r1r2, where ri and r2 are any pair of integral factors of M, be an integer properly represented in form,