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There are well known problems which are not decidable: halting problem, provability in PA, being a first order tautology, and others. Since all these problem deal with notions like computability and provability, they are beyond the scope of “usual” mathematics - mathematical analysis, for instance. Here, we will show a bunch of examples of simple undecidable statements of such mathematics.
It is shown that a chain of type of w + 1 of modal logics: Tr = Grz + B1 ⊃ Grz + B2 ⊃ Grz + B3 ⊃...⊃ Grz (all of them are extensions of the Grzegorczyk logic Grz), contains all and only such modal logics which can be obtained as sets of formulae that are valid in the Stone spaces of countable superatomic Boolean algebras. Some topological conditions which correspond to the Grzegorczyk logic are presented.
This article proposes to bring some information about possible ways t develop the capacity for inductive thinking that are exemplified in the series of mathematics textbooks for children from 6th to 9th grades published in the Czech Republic.
Artykuł ten dotyczy logiki zdaniowej S* przedstawionej w pracy [5]. Rozważana logika zawiera funktory: ∨,∧,⇒,⟺,¬,*, z których cztery pierwsze są dwuargumentowe, dwa ostatnie - jednoargumentowe. Logika S* posiada adekwatną matrycę trójelementową, której uniwersum jest zbiór {l, 0,-l}, wartościami wyróżnionymi są elementy 1 i -1, zaś działania odpowiadające funktorom określone są następującymi tabelkami:...
The sum operation, as introduced by Andrzej Wroński, is used to decompose any finite distributive lattice into its Boolean fragments. The decomposition is not unique but its maximal components are uniquely determined. We define an ordering relation between these maximal Boolean fragments of a given lattice and use this link ordering to describe the structure of the lattice.
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