We prove that, building upon the universal-existential orthogonality-based axiom system for metric planes presented in [28], one can provide universal-existential axiom systems – expressed solely in terms of the ternary predicate I, with I(abc) standing for ‘ab is congruent to ac’, which Pieri has introduced 100 years ago – for metric planes, for absolute geometry with the circle axiom, for Euc...

We present a streamlined axiom system of special relativity in first-order logic. From this axiom system we “derive” an axiom system of general relativity in two natural steps. We will also see how the axioms of special relativity transform into those of general relativity. This way we hope to make general relativity more accessible for the non-specialist.

In multiwinner approval voting, the goal is to select k-member committees based on voters' ballots. A well-studied concept of proportionality in this context justified representation (JR) axiom, which demands that no large cohesive group voters remains unrepresented. However, JR axiom may conflict with other desiderata, such as coverage (maximizing number who approve at least one committee memb...

We study a standard model of exchange economies with individual endowments. It is well known that no rule is individually rational, e cient, and strategyproof. In order to quantify the extent of this impossibility, we parametrize axioms on allocation rules. Given an axiom A, a parametrization of A is a continuum of axioms {δ-A}δ∈[0,1] such that (i) δ-A is equivalent to A only if δ = 1; (ii) δ-A...

We propose an axiomatization of Sergeyev’s theory of Grossone, trying to comply with his methodological principles. We find that a simplified form of his Divisibility axiom is sufficient. We use for easier readability a second order language and a predicative second order logic. Our theory is not finitely axiomatizable and is a conservative extension of Peano’s arithmetic.

Under Martin’s Axiom a c.c.c. nonseparable compact space is constructed which maps continuously into [0, 1] with linear fibers. Such a space can not, for instance, map onto [0, 1]א1 .

We show that several logics of common belief and common knowledge are not only complete, but also strongly complete, hence compact. These logics involve a weakened monotonicity axiom, and no other restriction on individual belief. The semantics is of the ordinary fixed-point type.

We show that bounded forcing axioms (for instance, the Bounded Proper Forcing Axiom and the Bounded Semiproper Forcing Axiom) are consistent with the existence of (!2; !2)-gaps and thus do not imply the Open Coloring Axiom. They are also consistent with Jensen’s combinatorial principles for L at the level !2, and therefore with the existence of an !2-Suslin tree. We also show that the axiom we ...