Complete Iff Research Articles

On countably complete topological groups

On countably complete topological groups

LOGICS FROM ULTRAFILTERS

Abstract Ultrafilters play a significant role in model theory to characterize logics having various compactness and interpolation properties. They also provide a general method to construct extensions of first-order logic having these properties. A main result of this paper is that every class $\Omega $ of uniform ultrafilters generates a $\Delta $ -closed logic ${\mathcal {L}}_\Omega $ . ${\mathcal {L}}_\Omega $ is $\omega $ -relatively compact iff some $D\in \Omega $ fails to be $\omega _1$ -complete iff ${\mathcal {L}}_\Omega $ does not contain the quantifier “there are uncountably many.” If $\Omega $ is a set, or if it contains a countably incomplete ultrafilter, then ${\mathcal {L}}_\Omega $ is not generated by Mostowski cardinality quantifiers. Assuming $\neg 0^\sharp $ or $\neg L^{\mu }$ , if $D\in \Omega $ is a uniform ultrafilter over a regular cardinal $\nu $ , then every family $\Psi $ of formulas in ${\mathcal {L}}_\Omega $ with $|\Phi |\leq \nu $ satisfies the compactness theorem. In particular, if $\Omega $ is a proper class of uniform ultrafilters over regular cardinals, ${\mathcal {L}}_\Omega $ is compact.

Силлогистика как логика всех отношений между двумя непустыми множествами

В работе предлагается язык позитивной силлогистики, алфавит которого содержит все силлогистические константы. Сами эти константы понимаются как знаки отношений между двумя непустыми множествами (объемами двух общих терминов). Среди указанных отношений особое место занимают те, которым соответствует в точности одна из пяти диаграмм Эйлера--Жергонна: (1) равенство множеств, (2) строгое включение первого множества во второе, (3) строгое включение второго множества в первое, (4) перекрещивание множеств, (5) несовместимость множеств. Остальные отношения представляют из себя комбинации эйлеровских. Каждая константа $\textit{k}$ в предлагаемом силлогистическом языке кодируется последовательностью чисел от 1 до 5, указывающих на номера тех эйлеровских отношений, при которых простое высказывание форм $\textit{SkP}$ принимает значение <<истина>>. Например, отношение включения одного непустого множества в другое репрезентирует константа 12, а отношение совместимости двух непустых множеств – константа 1234.Задается точная теоретико-множественная семантика данного <<универсального>> языка, соответствующая интуитивному пониманию силлогистических констант. Вводятся понятия выразимости силлогистической константы в <<локальном>> языке, содержащем лишь некоторые из таких констант, и полноты множества исходных констант некоторого <<локального>> языка. Константа $k$ выразима в локальном силлогистическом языке с неким набором исходных силлогистических констант, если и только если в этом локальном языке найдется формула $A$, содержащая в точности два термина $S$ и $P$, такая что $A$ эквивалентна формуле $SkP$ в семантике универсального силлогистического языка. Множество силлогистических констант $\{k_1, k_2,..., k_m\}$ называется $\textit{полным}$, если и только если любая силлогистическая константа выразима в языке с исходными силлогистическими константами $k_1$, $k_2$,..., $k_m$. В качестве примера доказывается полнота множества исходных констант традиционной силлогистики ($\textit{a}$, $\textit{e}$, $\textit{i}$, $\textit{o}$) и множества исходных констант силлогистики Венна (знаков пяти эйлеровских отношений). В универсальном силлогистическом языке строится аксиоматическое исчисление, доказываются метатеоремы о его непротиворечивости и полноте.

Dcpo Models of Choquet Complete and Baire Spaces

A model of a $$T_1$$ topological space X is a poset P such that the set of maximal points of P with the relative Scott topology is homeomorphic to X. In this paper, we prove that (i) a $$T_1$$ space is Baire if and only if it has a dcpo model whose Scott space is Baire; (ii) a $$T_1$$ space is Choquet complete iff it has a dcpo model whose Scott space is Choquet complete; (iii) a $$T_1$$ space has a quasicontinuous dcpo model iff it has a continuous dcpo model.

Logical systems I: Internal calculi

Abstract This is the first of a series of papers on a categorical approach to the logical systems. Its aim is to set forth the necessary foundations for more advanced concepts. The paper shows how the internal models of various lambda calculi arise in any 2-category with a notion of discreteness. We generalise to a 2-categorical setting two famous theorems: one saying that under some mild conditions, an object is internally complete iff it is internally cocomplete; and another saying that an object of a sufficiently cocomplete 2-category cannot be internally (co)complete unless it is degenerated. The first of the theorems specialises to a well-known fact from order theory, and also provides non-trivial results about posets and categories in constructive mathematics. Second of the theorems gives a powerful generalisation of Freyd's Theorem and sheds more light on the difficulty of finding fibrational models for higher polymorphism. As a simple corollary of this theorem, we obtain a variant of Freyd's Theorem for the categories internal to any tensored category. There is also a hidden objective of the paper — reading it backwards should provide a gentle introduction to the 2-categorical concepts through internal categories.

On the metric reflection of a pseudometric space in ZF

We show: (i) The countable axiom of choice $\mathbf{CAC}$ is equivalent to each one of the statements: (a) a pseudometric space is sequentially compact iff its metric reflection is sequentially compact, (b) a pseudometric space is complete iff its metric reflection is complete. (ii) The countable multiple choice axiom $\mathbf{CMC}$ is equivalent to the statement: (a) a pseudometric space is Weierstrass-compact iff its metric reflection is Weierstrass-compact. (iii) The axiom of choice $\mathbf{AC}$ is equivalent to each one of the statements: (a) a pseudometric space is Alexandroff-Urysohn compact iff its metric reflection is Alexandroff-Urysohn compact, (b) a pseudometric space $\mathbf{X}$ is Alexandroff-Urysohn compact iff its metric reflection is ultrafilter compact. (iv) We show that the statement “The preimage of an ultrafilter extends to an ultrafilter” is not a theorem of $\mathbf{ZFA}$.

On some relationships between biorthogonal systems, linear topologies and the weak‐AFPP

In this paper, we obtain new results for the weak‐AFPP in abstract spaces by exploiting biorthogonal systems techniques. Firstly, we investigate the strong‐AFPP on countably infinite dimensional Hausdorff locally convex spaces. Spaces of this class are shown to be sequentially complete iff they have the hereditary FPP for totally bounded, closed convex sets. This might open a research line for the analysis of weak‐AFPP in such frames. In connection, we provide a simple criterion for the containement of ℓ1‐sequences in terms of strongly‐equicontinuous biorthogonal systems. We then establish a few results concerning the existence of Hausdorff finer vector topologies on abstract spaces having as prescribed condition the existence of such systems. The proofs are based on methods of Peck and Porta concerning building of finer vector topologies, and a classical construction of Singer which allows us to prove under rather natural conditions the existence of equicontinuous biorthogonal systems in metrizable locally convex spaces. These results are compatible with the failure of the weak‐AFPP. We also study the inverse problem by proving that every infinite dimensional vector space admits a (non‐locally convex) Hausdorff vector topology which is complete, non‐metrizable and is compatible with a bounded Hamel Schauder basis. It is shown further that such a topology has the ‐AFPP, where is the linear span of coefficient functionals associated to a Hamel basis. Finally, inspired by a result of Shapiro, we observe that if X is a non‐locally convex F‐space with an absolute basis, then the weak‐AFPP is equivalent to the fact that every bounded convex subset of X is compact.

Subcontinuity

Abstract We give interesting characterizations using subcontinuity. Let X, Y be topological spaces. We study subcontinuity of multifunctions from X to Y and its relations to local compactness, local total boundedness and upper semicontinuity. If Y is regular, then F is subcontinuous iff $$\bar F$$ is USCO. A uniform space Y is complete iff for every topological space X and for every net {F a}, F a ⊂ X × Y, of multifunctions subcontinuous at x ∈ X, uniformly convergent to F, F is subcontinuous at x. A Tychonoff space Y is Čech-complete (resp. G m-space) iff for every topological space X and every multifunction F ⊂ X × Y the set of points of subcontinuity of F is a G δ-subset (resp. G m-subset) of X.

A characterization of strongly countably complete topological groups

We prove that a topological group G is strongly countably complete (the notion introduced by Z. Frolík in 1961) iff G contains a closed countably compact subgroup H such that the quotient space G / H is completely metrizable and the canonical mapping π : G → G / H is closed. We also show that every strongly countably complete group is sequentially complete, has countable G δ -tightness, and its completion is a Čech-complete topological group. Further, a pseudocompact strongly countably complete group is countably compact. An example of a pseudocompact topological Abelian group H with the Fréchet–Urysohn property is presented such that H fails to be sequentially complete, thus answering a question posed by Dikranjan, Martín Peinador, and Tarieladze in [Appl. Categor. Struct. 15 (2007) 511–539].

Projective unification in modal logic

A projective unifier for a modal formula A, over a modal logic L, is a unifier σ for A (i.e. a substitution making A a theorem of L) such that the equivalence of σ with the identity map is the consequence of A. Each projective unifier is a most general unifier for A. Let L be a normal modal logic containing S4. We show that every unifiable formula has a projective unifier in L iff L contains S4.3. The syntactic proof is effective. As a corollary, we conclude that all normal modal logics L containing S4.3 are almost structurally complete, i.e. all (structural) admissible rules having unifiable premises are derivable in L. Moreover, L is (hereditarily) structurally complete iff L contains McKinsey axiom M.

Generalized Kripke semantics for Nelson’s logic

A completeness theorem for logics N4 N and N30 is proved. A characterization by classes of N4 N - and N30-models is presented, and it is proved that all logics of four types η(L), η 3(L), η n (L), and η 0(L) are Kripke complete iff so are their respective intuitionistic fragments L. A generalized Kripke semantics is introduced, and it is stated that such is equivalent to an algebraic semantics. The concept of a p-morphism between generalized frames is defined and basic statements on p-morphisms are proved.

Order affine completeness of lattices with Boolean congruence lattices

This paper grew out from attempts to determine which modular lattices of finite height are locally order affine complete. A surprising discovery was that one can go quite far without assuming the modularity itself. The only thing which matters is that the congruence lattice is finite Boolean. The local order affine completeness problem of such lattices ${\mathbf L}$ easily reduces to the case when ${\mathbf L}$ is a subdirect product of two simple lattices ${\mathbf L}_1$ and ${\mathbf L}_2$. Our main result claims that such a lattice is locally order affine complete iff ${\mathbf L}_1$ and ${\mathbf L}_2$ are tolerance trivial and one of the following three cases occurs: 1) ${\mathbf L}={\mathbf L}_1\times {\mathbf L}_2$, 2) ${\mathbf L}$ is a maximal sublattice of the direct product, 3) ${\mathbf L}$ is the intersection of two maximal sublattices, one containing $\langle 0,1\rangle $ and the other $\langle 1,0\rangle $.

Completing circular codes in regular submonoids

Let M be an arbitrary submonoid of the free monoid A∗, and let X⊆M be a variable length code (for short a code). X is weakly M-complete iff any word in M is a factor of some word in X∗ [J. Néraud, C. Selmi, Free monoid theory: Maximality and completeness in arbitrary submonoids, Internat. J. Algebra Comput. 13 (5) (2003) 507–516]. Given a regular submonoid M, and given an arbitrary code X⊆M, we are interested in the existence of a weakly M-complete code Xˆ that contains X. Actually, in [J. Néraud, Completing a code in a regular submonoid, in: Acts of MCU’2004, Lect. Notes Comput. Sci. 3354 (2005) 281–291; J. Néraud, Completing a code in a submonoid of finite rank, Fund. Inform. 74 (2006) 549–562], by presenting a general formula, we have established that, in any case, such a code Xˆ exists. In the present paper, we prove that any regular circular code X⊆M may be embedded into a weakly M-complete one iff the minimal automaton with behavior M has a synchronizing word. As a consequence of our result an extension of the famous theorem of Schützenberger is stated for regular circular codes in the framework of regular submonoids. We study also the behaviour of the subclass of uniformly synchronous codes in connection with these questions.

Compact coverings for Baire locally convex spaces

Very recently Tkachuk has proved that for a completely regular Hausdorff space X the space C p ( X ) of continuous real-valued functions on X with the pointwise topology is metrizable, complete and separable iff C p ( X ) is Baire (i.e. of the second Baire category) and is covered by a family { K α : α ∈ N N } of compact sets such that K α ⊂ K β if α ⩽ β . Our general result, which extends some results of De Wilde, Sunyach and Valdivia, states that a locally convex space E is separable metrizable and complete iff E is Baire and is covered by an ordered family { K α : α ∈ N N } of relatively countably compact sets. Consequently every Baire locally convex space which is quasi-Suslin is separable metrizable and complete.

Mackey Topologies on Vector-Valued Function Spaces

Let \,E\, be an ideal of \,L^{0}\, over a \,\sigma -finite measure space \,(\Omega,\Sigma,\mu) , and let \,(X,\|\cdot\|_X)\, be a real Banach space. Let \,E(X)\, be a subspace of the space \,L^{0}(X)\, of \,\mu -equivalence classes of all strongly \,\Sigma -measurable functions \,f: \Omega\to X\, and consisting of all those \,f\in L^{0}(X)\, for which the scalar function \,\|f(\cdot)\|_X\, belongs to \,E . Let \,E(X)_n^{\sim}\, stand for the order continuous dual of \,E(X) . We examine the Mackey topology \,\tau(E(X),E(X)_n^{\sim}) in case when it is locally solid. It is shown that \,\tau(E(X),E(X)_n^{\sim})\, is the finest Hausdorff locally convex-solid topology on \,E(X)\, with the Lebesgue property. We obtain that the space \,(E(X),\tau(E(X), E(X)_n^{\sim}))\, is complete and sequentially barreled whenever \,E\, is perfect. As an application, we obtain the Hahn-Vitali-Saks type theorem for sequences in \,E(X)_n^{\sim} . In particular, we consider the Mackey topology \,\tau(L^{\Phi}(X), L^{\Phi}(X)_n^{\sim})\, on Orlicz-Bochner spaces \,L^{\Phi}(X) . We show that the space \,(L^{\Phi}(X), \tau(L^{\Phi}(X), L^{\Phi}(X)_n^{\sim}))\, is complete iff \,L^{\Phi}\, is perfect. Moreover, it is shown that the Mackey topology \,\tau(L^{\infty}(X), L^{\infty}(X)_n^{\sim})\, is a mixed topology.

Polynomial interpolation in expanded groups

We call an algebra strictly 1-affine complete iff every unary congruence preserving partial function with finite domain is a restriction of a polynomial. We characterize finite strictly 1-affine complete groups with operations, and, in particular, all finite strictly 1-affine complete groups and commutative rings with unit.

On the intrinsic complexity of learning recursive functions

The intrinsic complexity of learning compares the difficulty of learning classes of objects by using some reducibility notion. For several types of learning recursive functions, both natural complete classes are exhibited and necessary and sufficient conditions for completeness are derived. Informally, a class is complete iff both its topological structure is highly complex while its algorithmic structure is easy. Some self-describing classes turn out to be complete. Furthermore, the structure of the intrinsic complexity is shown to be much richer than the structure of the mind change complexity, though in general, intrinsic complexity and mind change complexity can behave “orthogonally”.

Sequential topological conditions in ℝ in the absence of the axiom of choice

AbstractIt is known that – assuming the axiom of choice – for subsets A of ℝ the following hold: (a) A is compact iff it is sequentially compact, (b) A is complete iff it is closed in ℝ, (c) ℝ is a sequential space. We will show that these assertions are not provable in the absence of the axiom of choice, and that they are equivalent to each

Is there an Analytic Limit of Genuine Modal Realism?

I agree with their first claim. Their answer to Lycan, however, seems to provide arguments that there is no completeness problem for GMR, that is that their claim (b) is false. Divers and Melia draw a helpful analogy between systems of logic and axiomatic characterizations of a set of possible worlds. A characterization, and so the set of possible worlds that fits it, has to be sound and it has to be complete. The characterization is sound iff the set of worlds which fits it contains no impossibilities. The characterization is complete iff for any possibility-to be explained shortly-there is a corresponding possible world. Lycan's objection is discussed in connection with soundness. Lycan (1991) argues that GMR should admit no world which has an impossible individual, since a corresponding world would be an impossibility, the set of possible worlds would not be consistent. This, Lycan argues, implies that the notion of individual employed by GMR has to be taken as possible individual in the first place. Divers and Melia (cf. p. 23) agree with Lycan about excluding impossible individuals, but disagree that this entails that 'individual' means possible individual. They argue for the soundness of local possibility. Local possibilities are possibilities

Sequential completeness and regularity of inductive limits of webbed spaces

Any inductive limit of bornivorously webbed spaces is sequentially complete iff it is regular.