A number-theoretic conjecture and its implication for set theory.
A one axiom set theory based on higher order predicate calculus.
A partial comparison of two conditions on the intersections of regressive sets.
A partial order where all monotone maps are definable
It is consistent that there is a partial order (P,≤) of size such that every monotone function f:P → P is first order definable in (P,≤).
A partition theorem for α-large sets
A polarized partition relation and failure of GCH at singular strong limit
The main result is that for λ strong limit singular failing the continuum hypothesis (i.e. ), a polarized partition theorem holds.
A poset hierarchy
This article extends a paper of Abraham and Bonnet which generalised the famous Hausdorff characterisation of the class of scattered linear orders. They gave an inductively defined hierarchy that characterised the class of scattered posets which do not have infinite incomparability antichains (i.e. have the FAC). We define a larger inductive hierarchy κℌ* which characterises the closure of the class of all κ-well-founded linear orders under inversions, lexicographic sums and FAC weakenings. This...
A possibilistic view on set and multiset comparison
A primrose path from Krull to Zorn
Given a set of “indeterminates” and a field , an ideal in the polynomial ring is called conservative if it contains with any polynomial all of its monomials. The map yields an isomorphism between the power set and the complete lattice of all conservative prime ideals of . Moreover, the members of any system of finite character are in one-to-one correspondence with the conservative prime ideals contained in , and the maximal members of correspond to the maximal ideals contained in...
A principal topology obtained from uninorms
We obtain a principal topology and some related results. We also give some hints of possible applications. Some mathematical systems are both lattice and topological space. We show that a topology defined on the any bounded lattice is definable in terms of uninorms. Also, we see that these topologies satisfy the condition of the principal topology. These topologies can not be metrizable except for the discrete metric case. We show an equivalence relation on the class of uninorms on a bounded lattice...
A problem on series of ordinals
A proof (involving Martin's axiom) of a partition relation
A proof of Rasiowa-Sikorski theorem via complete sequences
A proof of the independence of the Axiom of Choice from the Boolean Prime Ideal Theorem
We present a proof of the Boolean Prime Ideal Theorem in a transitive model of ZF in which the Axiom of Choice does not hold. We omit the argument based on the full Halpern-Läuchli partition theorem and instead we reduce the proof to its elementary case.
A proper class of pairwise incomparable cardinals
A propos des distances entre ensembles de parties
Soit un ensemble fini. Nous nous proposons ici d’établir des indices de distance entre les divers ensembles de parties de . Nous généraliserons pour cela, à , la notion de fonction caractéristique bien connue dans .
A propos des quasi-ordres - Note
A Reduction in Consistency Strength for Universal Indestructibility
We show how to reduce the assumptions in consistency strength used to prove several theorems on universal indestructibility.
A reflection on what is a membership function.
This paper is just a first approach to the idea that the membership function μP of a fuzzy set labelled P is, basically, a measure on the set of linguistic expressions x is P for each x in the corresponding universe of discourse X. Estimating that the meaning of P (relatively to X) is nothing else than the use of P on X, these measures seem to be reached by generalizing to a preordered set the concept of Fuzzy Measure, introduced by M. Sugeno, when the preorder translates the primary use of the...
A reflection phenomenon in descriptive set theory