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An Introduction to Hydrogels and Some Recent Applications

Authors:

Morteza Bahram, Naimeh Mohseni, Mehdi MoghtaderPublished: Aug 2016

- #Hydrogels #Applications #AllAboutHydrogels

Logical Loop

Published: Jul 2020

We discuss the dubbed term "logical loop" and it's implication regarding provability in undecidable theorems.

The logic of orthomodular posets of finite height

Authors:

Ivan Chajda, Helmut LängerPublished: Mar 2020

Orthomodular posets form an algebraic formalization of the logic of quantummechanics. The question is how to introduce the connective implication in sucha logic. We show that this is possible when the orthomodular poset in questionis of finite height. The main point is that the corresponding algebra, calledimplication orthomodular poset, i.e. a poset equipped with a binary operator ofimplication, corresponds to the original orthomodular poset and this operatoris everywhere defined. We present here the complete list of axioms forimplication orthomodular posets. This enables us to derive an axiomatization inGentzen style for the algebraizable logic of orthomodular posets of finiteheight.

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Residuation in finite posets

Authors:

Ivan Chajda, Helmut LängerPublished: Oct 2019

When an algebraic logic based on a poset instead of a lattice is investigatedthen there is a natural problem how to introduce the connective implication tobe everywhere defined and satisfying (left) adjointness with the connectiveconjunction. We have already studied this problem for the logic of quantummechanics which is based on an orthomodular poset or the logic of quantumeffects based on a so-called effect algebra which is only partial and need notbe lattice-ordered. For this, we introduced the so-called operator residuationwhere the values of implication and conjunction need not be elements of theunderlying poset, but only certain subsets of it. However, this approach can begeneralized for posets satisfying more general conditions. If these posets areeven finite, we can focus on maximal or minimal elements of the correspondingsubsets and the formulas for the mentioned operators can be essentiallysimplified. This is shown in the present paper where all theorems are explainedby corresponding examples.

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A model with everything except for a well-ordering of the reals

Authors:

Jörg Brendle, Fabiana Castiblanco, Ralf Schindler, Liuzhen Wu, Liang YuPublished: Sep 2018

We construct a model of $\mathsf{ZF} + \mathsf{DC}$ containing a Luzin set, aSierpi\'{n}ski set, as well as a Burstin basis but in which there is no a wellordering of the continuum.

Retrieved from arxiv

Residuated operators in complemented posets

Authors:

Ivan Chajda, Helmut LängerPublished: Sep 2018

Using the operators of taking upper and lower cones in a poset with a unaryoperation, we define operators M(x,y) and R(x,y) in the sense of multiplicationand residuation, respectively, and we show that by using these operators, ageneral modification of residuation can be introduced. A relativelypseudocomplemented poset can be considered as a prototype of such an operatorresiduated poset. As main results we prove that every Boolean poset as well asevery pseudo-orthomodular poset can be organized into a (left) operatorresiduated structure. Some results on pseudo-orthomodular posets are presentedwhich show the analogy to orthomodular lattices and orthomodular posets.

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Infinitary propositional relevant languages with absurdity

Author:

Guillermo BadiaPublished: Sep 2018

Analogues of Scott's isomorphism theorem, Karp's theorem as well as resultson lack of compactness and strong completeness are established for infinitarypropositional relevant logics. An "interpolation theorem" (of a particular sortintroduced by Barwise and van Benthem) for the infinitary quantificationalboolean logic $L_{\infty \omega}$ holds. This yields a preservation resultcharacterizing the expressive power of infinitary relevant languages withabsurdity using the model-theoretic relation of relevant directed bisimulationas well as a Beth definability property.

Retrieved from arxiv

Naming the largest number: Exploring the boundary between mathematics
and the philosophy of mathematics

Author:

David SimmonsPublished: Sep 2018

What is the largest number accessible to the human imagination? The questionis neither entirely mathematical nor entirely philosophical. Mathematicalformulations of the problem fall into two classes: those that fail to fullycapture the spirit of the problem, and those that turn it back into aphilosophical problem.

Retrieved from arxiv

Fraisse classes of graded relational structures

Authors:

Guillermo Badia, Carles NogueraPublished: Sep 2018

We study classes of graded structures satisfying the properties ofamalgamation, joint embedding and hereditariness. Given appropriate conditions,we can build a graded analogue of the Fraisse limit. Some examples such as theclass of all finite weighted graphs or the class of all finite fuzzy orders(evaluated on a particular countable algebra) will be examined.

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