Related papers: Aritm\'etica
We present a version of arithmetic in all finite types which allows for a definition of equality at higher types for which all congruence are derivable, for which the soundness of the Dialectica interpretation is provable inside the system…
We exposit two previously unpublished theorems of Leo Harrington. The first theorem says that there exist arithmetical singletons which are arithmetically incomparable. The second theorem says that there exists a ranked point which is not…
I develop in depth the machinery of $(\mathcal L, n)$-models originally introduced by Shelah and, independently in a slightly different form by Kripke. This machinery allows fairly routine constructions of true but unprovable sentences in…
We consider implicit definability of the standard part {0,1,...} in nonstandard models of Peano arithmetic (PA), and we ask whether there is a model of PA in which the standard part is implicitly definable. In section 1, we define a certain…
The purpose of the present work is to provide short and supple teaching notes for a $30$ hours introductory course on elementary \textit{Enumerative Algebraic Combinatorics}. We fully adopt the \textit{Rota way}. The themes are organized…
We state the defining characteristic of mathematics as a type of symmetry where one can change the connotation of a mathematical statement in a certain way when the statement's truth value remains the same. This view of mathematics as…
We give another proof of ordinal analysis of $I\Sigma_{k}$-fragments of Peano Arithmetic which is free from cut-elimination of $\omega$-logic. Our main tool is a direct witnessing argument utilizing game notion, motivated from the realm of…
In a forthcoming book, professional computer scientist and physicist Paul Budnik presents an exposition of classical mathematical theory as the backdrop to an elegant thesis: we can interpret any model of a formal system of Peano Arithmetic…
The general completeness problem of Hoare logic relative to the standard model $N$ of Peano arithmetic has been studied by Cook, and it allows for the use of arbitrary arithmetical formulas as assertions. In practice, the assertions would…
The prime number problem falls within the realm of number theory, specifically elementary number theory. Current research approaches have unnecessarily complicated this matter. In contrast to more advanced mathematical tools, the methods of…
A major question in philosophy of science involves the unreasonable effectiveness of mathematics in physics. Why should mathematics, created or discovered, with nothing empirical in mind be so perfectly suited to describe the laws of the…
This article analyzes the emerging ethical turn in mathematics education, arguing that it is a nuanced extension of the sociopolitical turn. While sociopolitical studies of mathematics have highlighted systemic issues and group concerns…
Peano Arithmetic is known to be provably equivalent to reflection over Elementary Arithmetic. We prove a characterization of Predicative Analysis in the guise of ATR0 in terms of stronger reflection principles.
We unify in a large class of additive functions the results obtained in the first part of this work. The proof rests on series involving the Riemann zeta function and certain sums of primes which may have their own interest.
Mathematical reasoning---a core ability within human intelligence---presents some unique challenges as a domain: we do not come to understand and solve mathematical problems primarily on the back of experience and evidence, but on the basis…
This paper establishes grounds for deeper exploration into the question of dual nature of mathematics as an abstract discipline and as a concrete science. It is argued, as one of the consequences of the discussion, that the division into…
Linear arithmetics are extensions of Presburger arithmetic (Pr) by one or more unary functions, each intended as multiplication by a fixed element (scalar), and containing the full induction schemes for their respective languages. In this…
The aim of these notes is to study some of the structural aspects of the ring of arithmetical functions. We prove that this ring is neither Noetherian nor Artinian. Furthermore, we construct various types of prime ideals. We show arithmetic…
The Sigma formulas of the language of arithmetic express semidecidable relations on the natural numbers. More generally, whenever a totality of objects is regarded as incomplete, the Sigma formulas express relations that are witnessed in a…
The book is devoted to investigation of arithmetic of the matrix rings over certain classes of commutative finitely generated principal ideals domains. We mainly concentrate on constructing of the matrix factorization theory. We reveal a…