Archive for Mathematical Logic最新文献

We study the notion of weak amalgamation in the context of diagonal conjugacy classes. Generalizing results of Kechris and Rosendal, we prove that for every countable structure M, Polish group G of permutations of M, and (n ge 1), G has a comeager n-diagonal conjugacy class iff the family of all n-tuples of G-extendable bijections between finitely generated substructures of M, has the joint embedding property and the weak amalgamation property. We characterize limits of weak Fraïssé classes that are not homogenizable. Finally, we investigate 1- and 2-diagonal conjugacy classes in groups of ball-preserving bijections of certain ordered ultrametric spaces.

In Schwichtenberg (Studies in logic and the foundations of mathematics, vol 90, Elsevier, pp 867–895, 1977), Schwichtenberg fine-tuned Tait’s technique (Tait in The syntax and semantics of infinitary languages, Springer, pp 204–236, 1968) so as to provide a simplified version of Gentzen’s original cut-elimination procedure for first-order classical logic (Gallier in Logic for computer science: foundations of automatic theorem proving, Courier Dover Publications, London, 2015). In this note we show that, limited to the case of classical propositional logic, the Tait–Schwichtenberg algorithm allows for a further simplification. The procedure offered here is implemented on Kleene’s sequent system G4 (Kleene in Mathematical logic, Wiley, New York, 1967; Smullyan in First-order logic, Courier corporation, London, 1995). The specific formulation of the logical rules for G4 allows us to provide bounds on the height of cut-free proofs just in terms of the logical complexity of their end-sequent.

Structural reasoning is simply reasoning that is governed exclusively by structural rules. In this context a proof system can be said to be structural if all of its inference rules are structural. A logic is considered to be structuralizable if it can be equipped with a sound and complete structural proof system. This paper provides a general formulation of the problem of structuralizability of a given logic, giving specific consideration to a family of logics that are based on the Dunn–Belnap four-valued semantics. It is shown how sound and complete structural proof systems can be constructed for a spectrum of logics within different logical frameworks.

Using a countable support product of creature forcing posets, we show that consistently, for uncountably many different functions the associated Yorioka ideals’ uniformity numbers can be pairwise different. In addition we show that, in the same forcing extension, for two other types of simple cardinal characteristics parametrised by reals (localisation and anti-localisation cardinals), for uncountably many parameters the corresponding cardinals are pairwise different.

Let T be a theory. If T eliminates (exists ^infty ), it need not follow that (T^{mathrm {eq}}) eliminates (exists ^infty ), as shown by the example of the p-adics. We give a criterion to determine whether (T^{mathrm {eq}}) eliminates (exists ^infty ). Specifically, we show that (T^{mathrm {eq}}) eliminates (exists ^infty ) if and only if (exists ^infty ) is eliminated on all interpretable sets of “unary imaginaries.” This criterion can be applied in cases where a full description of (T^{mathrm {eq}}) is unknown. As an application, we show that (T^{mathrm {eq}}) eliminates (exists ^infty ) when T is a C-minimal expansion of ACVF.

We present a sequent calculus with the Henkin constants in the place of the free variables. By disposing of the eigenvariable condition, we obtained a proof system with a strong locality property—the validity of each inference step depends only on its active formulas, not its context. Our major outcomes are: the cut elimination via a non-Gentzen-style algorithm without resorting to regularization and the elimination of Skolem functions with linear increase in the proof length for a subclass of derivations with cuts.

In this paper, we introduce the variety of algebras, which we call monadic (ktimes j)-rough Heyting algebras. These algebras constitute an extension of monadic Heyting algebras and in (3times 2) case they coincide with monadic 3-valued Łukasiewicz–Moisil algebras. Our main interest is the characterization of simple and subdirectly irreducible monadic (ktimes j)-rough Heyting algebras. In order to this, an Esakia-style duality for these algebras is developed.

We show that certain type of tree forcings, including Sacks forcing, increases the covering of the strong measure zero ideal ({{mathcal {S}}}{{mathcal {N}}}). As a consequence, in Sacks model, such covering number is equal to the size of the continuum, which indicates that this covering number is consistently larger than any other classical cardinal invariant of the continuum. Even more, Sacks forcing can be used to force that (mathrm {non}({{mathcal {S}}}{{mathcal {N}}})<mathrm {cov}({{mathcal {S}}}{{mathcal {N}}})<mathrm {cof}({{mathcal {S}}}{{mathcal {N}}})), which is the first consistency result where more than two cardinal invariants associated with ({{mathcal {S}}}{{mathcal {N}}}) are pairwise different. Another consequence is that ({{mathcal {S}}}{{mathcal {N}}}subseteq s^0) in ZFC where (s^0) denotes Marczewski’s ideal.

We use the forcing with overlapping extenders (Gitik in Blowing up the power of a singular cardinal of uncountable cofinality, to appear in JSL) to give a direct construction of a model of (lnot )SCH+Reflection.

We study the definability of ultrafilter bases on (omega ) in the sense of descriptive set theory. As a main result we show that there is no coanalytic base for a Ramsey ultrafilter, while in L we can construct (Pi ^1_1) P-point and Q-point bases. We also show that the existence of a ({varvec{Delta }}^1_{n+1}) ultrafilter is equivalent to that of a ({varvec{Pi }}^1_n) ultrafilter base, for (n in omega ). Moreover we introduce a Borel version of the classical ultrafilter number and make some observations.