The Gödel Hierarchy and Reverse Mathematics

Reverse Mathematics

In math we typically assume a set of axioms to prove a theorem. In reverse mathematics, the premise is reversed: we start with a theorem and try to determine the minimal axiomatic system required to prove the theorem (over a weak base system). This produces interesting results, as it can be shown that theorems from different fields of math such as group theory and analysis are in fact equivalen...

Mass problems and measure-theoretic regularity

A well known fact is that every Lebesgue measurable set is regular, i.e., it includes an Fσ set of the same measure. We analyze this fact from a metamathematical or foundational standpoint. We study a family of Muchnik degrees corresponding to measure-theoretic regularity at all levels of the effective Borel hierarchy. We prove some new results concerning Nies’s notion of LR-reducibility. We bu...

Fuzzy Linear Programming Method for Deriving Priorities in the Fuzzy Analytic Hierarchy Process

There are various methods for obtaining the preference vector of pair-wise comparison matrix factors. These methods can be employed when the elements of pair-wise comparison matrix are crisp while they are inefficient for fuzzy elements of pair-wise comparison matrix. In this paper, a method is proposed by which the preference vector of pair-wise comparison matrix elements can be obtained even ...

Beautifying Gödel

The incompleteness theorems of Kurt Gödel [1931] are considered to be among the most important results of mathematics. They are regarded as “deep”, and they strongly influenced the course of modern mathematics. They are popularly thought to prove the limitations (or even futility!) of mathematical formalism. At any rate, they deserve to be presented as simply, as elegantly, as beautifully as po...

Combined Data Envelopment Analysis and Analytical Hierarchy Process Methods to obtain the Favorable Weights from Pairwise Comparison Matrix