The consistency of arithmetic

This chapter opens the part of the book that deals with ordinal proof theory. Here the systems of interest are not purely logical ones, but rather formalized versions of mathematical theories, and in particular the first-order version of classical arithmetic built on top of the sequent calculus. Classical arithmetic goes beyond pure logic in that it contains a number of specific axioms for, among other symbols, 0 and the successor function. In particular, it contains the rule of induction, which is the essential rule characterizing the natural numbers. Proving a cut-elimination theorem for this system is hopeless, but something analogous to the cut-elimination theorem can be obtained. Indeed, one can show that every proof of a sequent containing only atomic formulas can be transformed into a proof that only applies the cut rule to atomic formulas. Such proofs, which do not make use of the induction rule and which only concern sequents consisting of atomic formulas, are called simple. It is shown that simple proofs cannot be proofs of the empty sequent, i.e., of a contradiction. The process of transforming the original proof into a simple proof is quite involved and requires the successive elimination, among other things, of “complex” cuts and applications of the rules of induction. The chapter describes in some detail how this transformation works, working through a number of illustrative examples. However, the transformation on its own does not guarantee that the process will eventually terminate in a simple proof.

Panaitopol-Bandila-Lascu Type Inequalities for Generalized Hyperbolic Functions

In this paper, we establish Panaitopol-Bandila-Lascu type inequalities for some generalized hyperbolic functions. The established results extend and generalize some earlier results due Barbu and Piscoran. The procedure makes use of the classical arithmetic-geometric mean inequality and the Cauchy-Schwarz inequality.

Zeta-regularization of arithmetic sequences

Is it possible to give a reasonable value to the infinite product 1 × 2 × 3 × · · · × n × · · · ? In other words, can we define some sort of convergence of the finite product 1 × 2 × 3 × · · · × n when n goes to infinity? One way is to relate this product to the R√iemann zeta function and to its analytic continuation. This approach leads to: 1 × 2 × 3 × · · · × n × · · · = 2π. More generally the “zeta-regularization” of an infinite product consists of introducing a related Dirichlet series and its analytic continuation at 0 (if it exists). We will survey some properties of this generalized product and allude to applications. Then we will give two families of possibly new examples: one unifies and generalizes known results for the zeta-regularization of the products of Fibonacci, balanced and Lucas-balanced numbers; the other studies the zeta-regularized products of values of classical arithmetic functions. Finally we ask for a possible zeta-regularity notion of complexity.

Application of Fuzzy Numbers to Assessment Processes

A fuzzy number (FN) is a special kind of FS on the set R of real numbers. The four classical arithmetic operations can be defined on FNs, which play an important role in fuzzy mathematics analogous to the role played by the ordinary numbers in crisp mathematics. The simplest form of FNs is the triangular FNs (TFNs), while the trapezoidal FNs (TpFNs) are straightforward generalizations of the TFNs. In the chapter, a combination of the COG defuzzification technique and of the TFNs (or TpFNs) is used as an assessment tool. Examples of assessing student problem-solving abilities and basketball player skills are also presented illustrating in practice the results obtained. This new fuzzy assessment method is validated by comparing its outcomes in the above examples with the corresponding outcomes of two commonly used assessment methods of the traditional logic, the calculation of the mean values, and of the grade point average (GPA) index. Finally, the perspectives of future research on the subject are discussed.

Application of Fuzzy Numbers to Assessment Processes

A Fuzzy Number (FN) is a special kind of FS on the set R of real numbers. The four classical arithmetic operations can be defined on FNs, which play an important role in fuzzy mathematics analogous to the role played by the ordinary numbers in crisp mathematics (Kaufmann & Gupta, 1991). The simplest form of FNs is the Triangular FNs (TFNs), while the Trapezoidal FNs (TpFNs) are straightforward generalizations of the TFNs. In the present work a combination of the COG defuzzification technique and of the TFNs (or TpFNs) is used as an assessment tool. Examples of assessing student problem-solving abilities and basket-ball player skills are also presented illustrating in practice the results obtained. This new fuzzy assessment method is validated by comparing its outcomes in the above examples with the corresponding outcomes of two commonly used assessment methods of the traditional logic, the calculation of the mean values and of the Grade Point Average (GPA) index. Finally, the perspectives of future research on the subject are discussed.

Homaloidal nets and ideals of fat points II: Subhomaloidal nets

This work is a natural sequel to a previous paper by the authors in that it tackles problems of the same nature. Here, one aims at the ideal theoretic and homological properties of an ideal of general plane fat points for which the corresponding second symbolic power has virtual multiplicities of a proper homaloidal type. For this purpose, one carries a detailed examination of the underlying linear system at the initial degree, where a good deal of the results depends on the method of the classical arithmetic quadratic transformations of Hudson–Nagata. A subsidiary guide to understand these ideals through their initial linear systems has been supplied by questions of birationality with source [Formula: see text] and target higher dimensional spaces. This leads, in particular, to the retrieval of birational maps studied by Geramita–Gimigliano–Pitteloud, including a few of the celebrated Bordiga–White parameterizations.

The Problem of Consistency

This chapter talks about how the translations from classical to intuistic arithmetic that Gödel and Gentzen found in 1932–33 showed that the consistency of classical arithmetic reduces to that of intuitionistic arithmetic. The main aim of Hilbert's program in the 1920s had been to show that the infinitistic component of arithmetic, namely the use of quantificational inferences in indirect proofs, does not lead to contradiction. It followed from Gödel's and Gentzen's result that these classical steps are harmless. Therefore, the consistency problem of arithmetic has an intuitionistic sense and, consequently, possibly also an intuitionistic solution. Bernays was quick to point out that intuitionism transcends the boundaries of strictly finitary reasoning.