we introduce some new concepts of topological spaces which say α − separable topologicalspace and o-topological group, α − first axiom, α − second axiom, and we find some relations betweenthem with some applications in normed spaces.

Let us recall that Raphael Robinson’s Arithmetic Q is an axiom system that differs from Peano Arithmetic essentially by containing no Induction axioms [13, 18]. We will generalize the semantictableaux version of the Second Incompleteness Theorem almost to the level of System Q. We will prove that there exists a single rather long Π1 sentence, valid in the standard model of the Natural Numbers a...

In the present paper, we introduce the fuzzy Nehring axiom, fuzzy Sen axiom and weaker form of the weak fuzzycongruence axiom. We establish interrelations between these axioms and their relation with fuzzy Chernoff axiom. Weexpress full rationality of a fuzzy choice function using these axioms along with the fuzzy Chernoff axiom.

For α, β, n ordinals α → (β, n) means that any graph on the ordinal α either contains an independent set of order-type β or a complete subgraph of order-type n. We only consider cases where n is a natural number, hence the naming. The partition relation α → (β, n) for α, β finite multiples of κ where κ = ω or κ is weakly compact is characterized by a finitary problem as is the same for α, β fin...

We prove that the Covering Property Axiom CPAprism, which holds in the iterated perfect set model, implies the following facts. • If G is an intersection of ω1-many open sets of a Polish space and G has cardinality continuum, then G contains a perfect set. • There exists a subset G of the Cantor set which is an intersection of ω1-many open sets but is not a union of ω1-many closed sets. The exa...

We introduce infinitary propositional theories over a set and their models which are subsets of the set, and define a generalized geometric theory as an infinitary propositional theory of a special form. The main result is that the class of models of a generalized geometric theory is set-generated. Here a class X of subsets of a set is set-generated if there exists a subset G of X such that for...

The Tsallis entropy given for a positive parameter α can be considered as a generalization of the classical Shannon entropy. For the latter, corresponding to α = 1, there exist many axiomatic characterizations. One of them based on the well-known Khinchin-Shannon axioms has been simplified several times and adapted to Tsallis entropy, where the axiom of (generalized) Shannon additivity is playi...

We develop a set of five axioms for fairness measures in resource allocation: the axiom of continuity, of homogeneity, of saturation, of partition, and of starvation. We prove that there is a unique family of fairness measures satisfying the axioms, which is constructed and then shown to include Atkinson’s index, αfairness, Jain’s index, entropy, and other “decomposable” global measures of fair...

Rays in a vector space are simply one-dimensional subspaces. The state space axiom therefore says that states of a quantum systems are identified with an entire one-dimensional subspace tα|ψy : α P Cu, where |ψy is some unit vector. When performing calculations the convention is to represent physical states by unit vectors, and so henceforth we will assume that all kets are normalized unless ot...