Archive for Mathematical Logic最新文献

This paper investigates properties of (sigma )-closed forcings which generate ultrafilters satisfying weak partition relations. The Ramsey degree of an ultrafilter ({mathcal {U}}) for n-tuples, denoted (t({mathcal {U}},n)), is the smallest number t such that given any (lge 2) and coloring (c:[omega ]^nrightarrow l), there is a member (Xin {mathcal {U}}) such that the restriction of c to ([X]^n) has no more than t colors. Many well-known (sigma )-closed forcings are known to generate ultrafilters with finite Ramsey degrees, but finding the precise degrees can sometimes prove elusive or quite involved, at best. In this paper, we utilize methods of topological Ramsey spaces to calculate Ramsey degrees of several classes of ultrafilters generated by (sigma )-closed forcings. These include a hierarchy of forcings due to Laflamme which generate weakly Ramsey and weaker rapid p-points, forcings of Baumgartner and Taylor and of Blass and generalizations, and the collection of non-p-points generated by the forcings ({mathcal {P}}(omega ^k)/mathrm {Fin}^{otimes k}). We provide a general approach to calculating the Ramsey degrees of these ultrafilters, obtaining new results as well as streamlined proofs of previously known results. In the second half of the paper, we calculate pseudointersection and tower numbers for these (sigma )-closed forcings and their relationships with the classical pseudointersection number ({mathfrak {p}}).

(mathbf {Z}) is a new type of non-finitist inference, i.e., an inference that involves treating some infinite collection as completed, designed for contraction free logic with unrestricted abstraction. It has been introduced in Petersen (Studia Logica 64:365–403, 2000) and shown to be consistent within a system (mathbf {{}L^iD{}}{}) (_{uplambda }) of contraction free logic with unrestricted abstraction. In Petersen (Arch Math Log 42(7):665–694, 2003) it was established that adding ( mathbf {Z}) to (mathbf {{}L^iD{}}{}) (_{uplambda }) is sufficient to prove the totality of primitive recursive functions but it was also indicated that this would not extend to 2-recursive functions such as the Ackermann–Péter function, for instance. The purpose of the present paper is to expand the underlying idea in the construction of (mathbf {Z}) to gain a stronger notion, conveniently labeled (mathbf {Z}_2), which is sufficient to prove a form of nested double induction and thereby the totality of 2-recursive functions.

We introduce a class of proper posets which is preserved under countable support iterations, includes (omega ^omega )-bounding, Cohen, Miller, and Mathias posets associated to filters with the Hurewicz covering properties, and has the property that the ground model reals remain splitting and unbounded in corresponding extensions. Our results may be considered as a possible path towards solving variations of the famous Roitman problem.

We give a new characterization of SOP (the strict order property) in terms of the behaviour of formulas in any model of the theory as opposed to having to look at the behaviour of indiscernible sequences inside saturated ones. We refine a theorem of Shelah, namely a theory has OP (the order property) if and only if it has IP (the independence property) or SOP, in several ways by characterizing various notions in functional analytic style. We point out some connections between dividing lines in first order theories and subclasses of Baire 1 functions, and give new characterizations of some classes and new classes of first order theories.

In the language (lbrace 0, 1, circ , preceq rbrace ), where 0 and 1 are constant symbols, (circ ) is a binary function symbol and (preceq ) is a binary relation symbol, we formulate two theories, ( textsf {WD} ) and ( {textsf {D}}), that are mutually interpretable with the theory of arithmetic ( {textsf {R}} ) and Robinson arithmetic ({textsf {Q}} ), respectively. The intended model of ( textsf {WD} ) and ( {textsf {D}}) is the free semigroup generated by (lbrace {varvec{0}}, {varvec{1}} rbrace ) under string concatenation extended with the prefix relation. The theories ( textsf {WD} ) and ( {textsf {D}}) are purely universally axiomatised, in contrast to ( {textsf {Q}} ) which has the (varPi _2)-axiom (forall x ; [ x = 0 vee exists y ; [ x = Sy ] ] ).

In Minker and Rajasekar (J Log Program 9(1):45–74, 1990), Minker proposed a semantics for negation-free disjunctive logic programs that offers a natural generalisation of the fixed point semantics for definite logic programs. We show that this semantics can be further generalised for disjunctive logic programs with classical negation, in a constructive modal-theoretic framework where rules are built from claims and hypotheses, namely, formulas of the form (Box varphi ) and (Diamond Box varphi ) where (varphi ) is a literal, respectively, yielding a “base semantics” for general disjunctive logic programs. Model-theoretically, this base semantics is expressed in terms of a classical notion of logical consequence. It has a complete proof procedure based on a general form of the cut rule. Usually, alternative semantics of logic programs amount to a particular interpretation of nonclassical negation as “failure to derive.” The counterpart in our framework is to complement the original program with a set of hypotheses required to satisfy specific conditions, and apply the base semantics to the resulting set. We demonstrate the approach for the answer set semantics. The proposed framework is purely classical in mainly three ways. First, it uses classical negation as unique form of negation. Second, it advocates the computation of logical consequences rather than of particular models. Third, it makes no reference to a notion of preferred or minimal interpretation.

The main result of the paper states that there is a finitely presented group G with decidable word problem where detection of finite subsets of G which generate amenable subgroups is not decidable.

Answering in negative a question of M. Hrušák, we construct a Borel ideal not extendable to any (F_sigma ) ideal and such that it is not Katětov above the ideal (mathrm {conv}).

We investigate possible cardinalities of maximal antichains in the poset of copies (langle {mathbb {P}}(mathbb X),subseteq rangle ) of a countable ultrahomogeneous relational structure ({{mathbb {X}}}). It turns out that if the age of ({{mathbb {X}}}) has the strong amalgamation property, then, defining a copy of ({{mathbb {X}}}) to be large iff it has infinite intersection with each orbit of ({{mathbb {X}}}), the structure ({{mathbb {X}}}) can be partitioned into countably many large copies, there are almost disjoint families of large copies of size continuum and, hence, there are (maximal) antichains of size continuum in the poset ({{mathbb {P}}}({{mathbb {X}}})). Finally, we show that the posets of copies of all countable ultrahomogeneous partial orders contain maximal antichains of cardinality continuum and determine which of them contain countable maximal antichains. That holds, in particular, for the generic (universal ultrahomogeneous) poset.

Dyrda and Prucnal gave a Hilbert-style axiomatization for the ({wedge ,vee })-fragment of classical propositional logic. Their proof of completeness follows a different approach to the standard one proving the completeness of classical propositional logic. In this note, we present an alternative proof of Dyrda and Prucnal’s result following the standard arguments which prove the completeness of classical propositional logic.