Propositional extensions of $L_ω_1_ω$

Richard Gostanian; Karel Hrbacek

Displaying similar documents to “Propositional extensions of $L_{ω} 1_{ω}$ ”

The $ℒ_{n}^{m}$ -propositional calculus

Carlos Gallardo, Alicia Ziliani (2015)

Mathematica Bohemica

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T. Almada and J. Vaz de Carvalho (2001) stated the problem to investigate if these Łukasiewicz algebras are algebras of some logic system. In this article an affirmative answer is given and the $ℒ_{n}^{m}$ -propositional calculus, denoted by $ℓ_{n}^{m}$ , is introduced in terms of the binary connectives $\to$ (implication), $↠$ (standard implication), $\land$ (conjunction), $\lor$ (disjunction) and the unary ones $f$ (negation) and $D_{i}$ , $1 \leq i \leq n - 1$ (generalized Moisil operators). It is proved that $ℓ_{n}^{m}$ belongs to the class of standard systems...

On sentences provable in impredicative extensions of theories

Zygmunt Ratajczyk

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CONTENTS0. Introduction.......................................................................... 51. Preliminaries............................................................................... 72. Basic facts to be used in the sequel....................................... 113. Predicates OD(.,.) and CL(.,.).................................................... 174. Predicate Sels............................................................................. 185. Strong $\sum_{n}^{1}$ -collection...........................................................

On the solvability of systems of linear equations over the ring $ℤ$ of integers

Horst Herrlich, Eleftherios Tachtsis (2017)

Commentationes Mathematicae Universitatis Carolinae

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We investigate the question whether a system ${(E_{i})}_{i \in I}$ of homogeneous linear equations over $ℤ$ is non-trivially solvable in $ℤ$ provided that each subsystem ${(E_{j})}_{j \in J}$ with $| J | \leq c$ is non-trivially solvable in $ℤ$ where $c$ is a fixed cardinal number such that $c < | I |$ . Among other results, we establish the following. (a) The answer is ‘No’ in the finite case (i.e., $I$ being finite). (b) The answer is ‘No’ in the denumerable case (i.e., $| I | = ℵ_{0}$ and $c$ a natural number). (c) The answer in case that $I$ is uncountable and $c \leq ℵ_{0}$ is ‘No...

A generalization of a formalized theory of fields of sets on non-classical logics

Helena Rasiowa

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Contents Introduction.................................................................................................................................................. 3 § 1. System $𝒮$ of a propositional calculus...................................................................... 4 § 2. System $𝒮 *$ ..................................................................................................................... 5 § 3. $𝒮 *$ -algebras.....................................................................................................................

On preimages of ultrafilters in ZF

Horst Herrlich, Paul Howard, Kyriakos Keremedis (2016)

Commentationes Mathematicae Universitatis Carolinae

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We show that given infinite sets $X, Y$ and a function $f : X \to Y$ which is onto and $n$ -to-one for some $n \in ℕ$ , the preimage of any ultrafilter $ℱ$ of $Y$ under $f$ extends to an ultrafilter. We prove that the latter result is, in some sense, the best possible by constructing a permutation model $ℳ$ with a set of atoms $A$ and a finite-to-one onto function $f : A \to ω$ such that for each free ultrafilter of $ω$ its preimage under $f$ does not extend to an ultrafilter. In addition, we show that in $ℳ$ there exists an ultrafilter compact...

On certain non-constructive properties of infinite-dimensional vector spaces

Eleftherios Tachtsis (2018)

Commentationes Mathematicae Universitatis Carolinae

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In set theory without the axiom of choice ( $AC$ ), we study certain non-constructive properties of infinite-dimensional vector spaces. Among several results, we establish the following: (i) None of the principles AC $^{LO}$ (AC for linearly ordered families of nonempty sets)—and hence AC $^{WO}$ (AC for well-ordered families of nonempty sets)— $DC (< κ)$ (where $κ$ is an uncountable regular cardinal), and “for every infinite set $X$ , there is a bijection $f : X \to {0, 1} \times X$ ”, implies the statement “there exists a field $F$ such that...

On the $T$ -conditionality of $T$ -power based implications

Zuming Peng (2022)

Kybernetika

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It is well known that, in forward inference in fuzzy logic, the generalized modus ponens is guaranteed by a functional inequality called the law of $T$ -conditionality. In this paper, the $T$ -conditionality for $T$ -power based implications is deeply studied and the concise necessary and sufficient conditions for a power based implication $I^{T}$ being $T$ -conditional are obtained. Moreover, the sufficient conditions under which a power based implication $I^{T}$ is $T^{*}$ -conditional are discussed, this discussions...

Balcar's theorem on supports

Lev Bukovský (2018)

Commentationes Mathematicae Universitatis Carolinae

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In A theorem on supports in the theory of semisets [Comment. Math. Univ. Carolinae 14 (1973), no. 1, 1–6] B. Balcar showed that if $σ \subseteq D \in M$ is a support, $M$ being an inner model of ZFC, and $𝒫 (D ∖ σ) \cap M = r ` ` σ$ with $r \in M$ , then $r$ determines a preorder " $⪯$ " of $D$ such that $σ$ becomes a filter on $(D, ⪯)$ generic over $M$ . We show that if the relation $r$ is replaced by a function $𝒫 (D ∖ σ) \cap M = f_{- 1} (σ)$ , then there exists an equivalence relation " $\sim$ " on $D$ and a partial order on $D / \sim$ such that $D / \sim$ is a complete Boolean algebra, $σ / \sim$ is a generic filter and ${[f (u)]}_{\sim} = - \sum (u / \sim)$ for...

$C^{*}$ -points vs $P$ -points and $P^{♭}$ -points

Jorge Martinez, Warren Wm. McGovern (2022)

Commentationes Mathematicae Universitatis Carolinae

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In a Tychonoff space $X$ , the point $p \in X$ is called a $C^{*}$ -point if every real-valued continuous function on $C ∖ {p}$ can be extended continuously to $p$ . Every point in an extremally disconnected space is a $C^{*}$ -point. A classic example is the space $𝐖^{*} = ω_{1} + 1$ consisting of the countable ordinals together with $ω_{1}$ . The point $ω_{1}$ is known to be a $C^{*}$ -point as well as a $P$ -point. We supply a characterization of $C^{*}$ -points in totally ordered spaces. The remainder of our time is aimed at studying when a point in a product space...

Spaces with property $(D C (ω_{1}))$

Wei-Feng Xuan, Wei-Xue Shi (2017)

Commentationes Mathematicae Universitatis Carolinae

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We prove that if $X$ is a first countable space with property $(D C (ω_{1}))$ and with a $G_{δ}$ -diagonal then the cardinality of $X$ is at most $𝔠$ . We also show that if $X$ is a first countable, DCCC, normal space then the extent of $X$ is at most $𝔠$ .

$H^{\infty}$ calculus and dilatations

Andreas M. Fröhlich, Lutz Weis (2006)

Bulletin de la Société Mathématique de France

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We characterise the boundedness of the $H^{\infty}$ calculus of a sectorial operator in terms of dilation theorems. We show e. g. that if $- A$ generates a bounded analytic $C_{0}$ semigroup $(T_{t})$ on a UMD space, then the $H^{\infty}$ calculus of $A$ is bounded if and only if $(T_{t})$ has a dilation to a bounded group on $L^{2} ([0, 1], X)$ . This generalises a Hilbert space result of C.LeMerdy. If $X$ is an $L^{p}$ space we can choose another $L^{p}$ space in place of $L^{2} ([0, 1], X)$ .

Displaying similar documents to “Propositional extensions of L ω 1 ω ”

Displaying similar documents to “Propositional extensions of $L_{ω} 1_{ω}$ ”