In this article we introduce and study the notion of operational closure: a transitive set d is called operationally closed iff it contains all constants of OST and any operation f ∈ d applied to an element a ∈ d yields an element fa ∈ d, provided that f applied to a has a value at all. We will show that there is a direct relationship between operational closure and stability in the sense that ...

We prove that a random choice rule satisfies Luce's Choice Axiom if and only its support is correspondence the Weak of Revealed Preference, thus it consists alternatives are optimal according to some preference, then occurs tie-breaking mechanism among such Renyi's Conditioning Axiom. Our result shows is, in precise formal sense, probabilistic version It supports view his own axiom as “canon ra...

The project of dialetheic set theories is based on a simple motivation: we want to see how much mathematics we can do by keeping the following claim of Graham Priest in his celebrated ‘In Contradiction’: I wish to claim that (Abst) and (Ext) are true, and in fact that they analytically characterise the notion of set. ([4, p.30]) Here, (Abst) and (Ext) are ∃y∀x(x∈y↔β) and ∀x(x∈z↔x∈y)→z=y respect...

We present equiconsistency results at the level of subcompact cardinals. Assuming SBHδ, a special case of the Strategic Branches Hypothesis, we prove that if δ is a Woodin cardinal and both 2(δ) and 2δ fail, then δ is subcompact in a class inner model. If in addition 2(δ) fails, we prove that δ is Π21 subcompact in a class inner model. These results are optimal, and lead to equiconsistencies. A...

Basu and Van (1998) present a fundamental framework of child labor with two important axioms: the luxury axiom and the substitution axiom. A number of empirical studies, however, reveal a “wealth paradox”. The current paper has two aims. First, it develops a model that provides an explanation for “the wealth paradox” in light of the luxury axiom and the substitution axiom. Second, it helps unde...

Based on the standard axioms of quantum theory, we provide a counterexample which invalidates the full compatibility between consciousness and quantum theory. In particular, we present an example of a natural phenomenon in which an observer’s the mental state can be fully described in mathematical terms analogous to the state vector that is being observed. This mathematical description of the o...

We logically model uncertainty by expanding language without changing logical reasoning rules. We expand the language of set theory by adding new predicate symbols, uncertain membership relations ∈+ and ∈−. We define the set theory ZF± as an extension of ZF with new symbols in classical logic. In this system we can represent uncertainty which is naturally represented in the model of 3-valued lo...

It is possible, but difficult, to reason in Hoare logic about programs which address and modify data structures defined by pointers. The challenge is to approach the simplicity of Hoare logic’s treatment of variable assignment, where substitution affects only relevant assertion formulæ. The axiom of assignment to object components treats each component name as a pointerindexed array. This permi...

A convexity structure satisfies the separation property 5'4 if any two disjoint convex sets extend to complementary half-spaces. This property is investigated for alignment spaces, n-ary convexities, and graphs. In particular, it is proven that a) an n-ary convexity is $4 iff every pair of disjoint polytopes with at most n vertices can be separated by complementary half spaces, and b) an interv...

In recent publications Alain Connes [1] and John Barrett [2] proposed to change the KO-dimension of the internal space of the standard model in its noncommutative representation [3] from zero to six. This apparently minor modification allowed to resolve the fermion doubling problem [4], and the introduction of Majorana mass terms for the right-handed neutrino. The price which had to be paid was...