Annals of Pure and Applied Logic最新文献

It is an open question whether compositional truth with the principle of propositional soundness: “All arithmetical sentences which are propositional tautologies are true” is conservative over Peano Arithmetic. In this article, we show that the principle of propositional soundness imposes some saturation-like properties on the truth predicate, thus showing significant limitations to the possible conservativity proof.

We develop a framework for nonstandard analysis that gives foundations to the interplay between external and internal iterations of the star map, and we present a few examples to show the strength and flexibility of such a nonstandard technique for applications in combinatorial number theory.

Feferman [9] defines an impredicative system $T_0$ of explicit mathematics, which is proof-theoretically equivalent to the subsystem

We provide a general and syntactically defined family of sequent calculi, called semi-analytic, to formalize the informal notion of a “nice” sequent calculus. We show that any sufficiently strong (multimodal) substructural logic with a semi-analytic sequent calculus enjoys the Craig Interpolation Property, CIP. As a positive application, our theorem provides a uniform and modular method to prove the CIP for several multimodal substructural logics, including many fragments and variants of linear logic. More interestingly, on the negative side, it employs the lack of the CIP in almost all substructural, superintuitionistic and modal logics to provide a formal proof for the well-known intuition that almost all logics do not have a “nice” sequent calculus. More precisely, we show that many substructural logics including $U{L}^{-}$, $\mathrm{MTL}$, $R$, Ł_n (for $n⩾3$), G_n (for $n⩾4$), and almost all extensions of $\mathrm{IMTL}$, $Ł$, $\mathrm{BL}$, $R{M}^e$, $\mathrm{IPC}$, $\mathrm{S4}$, and $\mathrm{Grz}$ (except for at most 1, 1, 3, 8, 7, 37, and 6 of them, respectively) do not have a semi-analytic calculus.

We classify intermediate logics according to their unification types. There are exactly two minimal intermediate logics with hereditary finitary unification: the least logic with hereditary unitary unification and the least logic with hereditary projective proximity (a notion close to projective approximation of Ghilardi [17], [18]), see Figure 4. They are locally tabular and are union splittings in the lattice Ext INT. There are exactly four maximal intermediate logics with nullary unification (see Figure 21) and they are tabular. Any intermediate logic with neither hereditary unitary unification nor with hereditary projective proximity is included in one of the four logics. There are logics with finitary/unitary (but not hereditary finitary) unification scattered among the majority of those with nullary unification, see Figure 23. Our main tools are the characterization of locally tabular logics with finitary (or unitary) unification, by their Kripke models [12], [13] and splittings.

We study a version of the Vitali covering theorem, which we call $\text{WHBU}$ and which is a direct weakening of the Heine-Borel theorem for uncountable coverings, called $\text{HBU}$. We show that $\text{WHBU}$ is central to measure theory by deriving it from various central approximation results related to Littlewood's three principles. A natural question is then how hard it is to prove $\text{WHBU}$ (in the sense of Kohlenbach's higher-order Reverse Mathematics), and how hard it is to compute the objects claimed to exist by $\text{WHBU}$ (in the sense of Kleene's computation schemes S1-S9). The answer to both questions is ‘extremely hard’, as follows: on one hand, in terms of the usual scale of (conventional) comprehension axioms, $\text{WHBU}$ is only provable using Kleene's ${\exists }^3$, which implies full second-order arithmetic. On the other hand, realisers (aka witnessing functionals) for $\text{WHBU}$, so-called Λ-functionals, are computable from Kleene's ${\exists }^3$, but not from weaker comprehension functionals. Despite this hardness, we show that $\text{WHBU}$, and certain Λ-functionals, behave much better than $\text{HBU}$ and the associated class of realisers, called Θ-functionals. In particular, we identify a specific Λ-functional called ${\Lambda }_{\text{S}}$ which adds no computational power to the Suslin functional, in contrast to Θ-functionals. Finally, we introduce a hierarchy involving Θ-functionals and $\text{HBU}$.

Here we give a complete list of the groups definable in Presburger arithmetic up to a finite index subgroup.

Given a weakly compact cardinal κ, we give an axiomatization of intuitionistic first-order logic over $L_{{\kappa }^{+},\kappa }$ and prove it is sound and complete with respect to Kripke models. As a consequence we get the disjunction and existence properties for that logic. This generalizes the work of Nadel in [8] for intuitionistic logic over $L_{{\omega }_1,\omega }$. When κ is a regular cardinal such that ${\kappa }^{<\kappa }=\kappa$, we deduce, by an easy modification of the proof, a complete axiomatization of intuitionistic first-order logic over $L_{{\kappa }^{+},\kappa ,\kappa }$, the language with disjunctions of at most κ formulas, conjunctions of less than κ formulas and quantification on less than κ many variables. In particular, this applies to any regular cardinal under the Generalized Continuum Hypothesis.

Let $\mu <\kappa <\lambda$ be three infinite cardinals, the first two being regular. We show that if there is no inner model with large cardinals, $u(\kappa ,\lambda )$ is regular, where $u(\kappa ,\lambda )$ denotes the least size of a cofinal subset in $\left(P_{\kappa }\right(\lambda ),\subseteq )$, and $\mathrm{cf}\left(\lambda \right)\ne \mu$, then (a) the μ-club filters on $P_{\kappa }\left(\lambda \right)$ and $P_{\kappa }\left(u\right(\kappa ,\lambda \left)\right)$ are isomorphic, and (b) the ideal dual to the μ-club filter on $P_{\kappa }\left(\lambda \right)$ (and hence the restriction of the nonstationary ideal on $P_{\kappa }\left(\lambda \right)$ to sets of uniform cofinality μ) is not $I_{\kappa ,\lambda }$-$b_{u(\kappa ,\lambda )}$-saturated.