Annals of Pure and Applied Logic最新文献

A bi-Heyting algebra validates the Gödel-Dummett axiom $(p\to q)\vee (q\to p)$ iff the poset of its prime filters is a disjoint union of co-trees (i.e., order duals of trees). Bi-Heyting algebras of this kind are called bi-Gödel algebras and form a variety that algebraizes the extension $\mathrm{bi-GD}$ of bi-intuitionistic logic axiomatized by the Gödel-Dummett axiom. In this paper we initiate the study of the lattice $\Lambda \left(\mathrm{bi-GD}\right)$ of extensions of $\mathrm{bi-GD}$.

We develop the methods of Jankov-style formulas for bi-Gödel algebras and use them to prove that there are exactly continuum many extensions of $\mathrm{bi-GD}$. We also show that all these extensions can be uniformly axiomatized by canonical formulas. Our main result is a characterization of the locally tabular extensions of $\mathrm{bi-GD}$. We introduce a sequence of co-trees, called the finite combs, and show that a logic in $\Lambda \left(\mathrm{bi-GD}\right)$ is locally tabular iff it contains at least one of the Jankov formulas associated with the finite combs. It follows that there exists the greatest nonlocally tabular extension of $\mathrm{bi-GD}$ and consequently, a unique pre-locally tabular extension of $\mathrm{bi-GD}$. These results contrast with the case of the intermediate logic axiomatized by the Gödel-Dummett axiom, which is known to have only countably many extensions, all of which are locally tabular.

We investigate some Weihrauch problems between ${\mathrm{ATR}}_2$ and $C_{{\omega }^{\omega }}$. We show that the fixed point theorem for monotone operators on the Cantor space (a weaker version of the Knaster-Tarski theorem) is not Weihrauch reducible to ${\mathrm{ATR}}_2$. Furthermore, we introduce the ω-model reflection ${\mathrm{ATR}}_2^{\mathrm{rfn}}$ of ${\mathrm{ATR}}_2$ and show that it is an upper bound for problems provable from the axiomatic system ${\mathrm{ATR}}_0$ which are of the form $\forall X\left(\theta \right(X)\to \exists Y\eta (X,Y\left)\right)$ with arithmetical formulas $\theta ,\eta$. We also show that Weihrauch degrees of relativized least fixed point theorems for monotone operators on the Cantor space form a linear hierarchy between ${\mathrm{ATR}}_2^{\mathrm{rfn}}$ and $C_{{\omega }^{\omega }}$.

We consider Presburger arithmetic extended by the sine function, call this extension sine-Presburger arithmetic (sin-PA), and systematically study decision problems for sets of sentences in sin-PA. In particular, we detail a decision algorithm for existential sin-PA sentences under assumption of Schanuel's conjecture. This procedure reduces decisions to the theory of the ordered additive group of real numbers extended by sine, which is decidable under Schanuel's conjecture. On the other hand, we prove that four alternating quantifier blocks suffice for undecidability of sin-PA sentences. To do so, we explicitly interpret the weak monadic second-order theory of the grid, which is undecidable, in sin-PA.

We consider cyclic proof systems in which derivations are graphs rather than trees. Such systems typically come with a condition that isolates which derivations are admitted as proofs, known as the soundness condition. This soundness condition frequently takes the form of either a global trace condition, a property dependent on all infinite paths in the proof-graph, or a reset condition, a ‘local’ condition depending on the simple cycles only which, as a result, is typically stable under more proof transformations.

In this article we present a general method for constructing cyclic proof systems with reset conditions from systems with global trace conditions. In contrast to previous approaches, this method of generation is entirely independent of logic's semantics, only relying on combinatorial aspects of the notion of ‘trace’ and ‘progress’. We apply this method to present reset proof systems for three cyclic proof systems from the literature: cyclic arithmetic, cyclic Gödel's T and cyclic tableaux for the modal μ-calculus.

We prove some technical results on definable types in p-adically closed fields, with consequences for definable groups and definable topological spaces. First, the code of a definable n-type (in the field sort) can be taken to be a real tuple (in the field sort) rather than an imaginary tuple (in the geometric sorts). Second, any definable type in the real or imaginary sorts is generated by a countable union of chains parameterized by the value group. Third, if X is an interpretable set, then the space of global definable types on X is strictly pro-interpretable, building off work of Cubides Kovacsics, Hils, and Ye [7], [8]. Fourth, global definable types can be lifted (in a non-canonical way) along interpretable surjections. Fifth, if G is a definable group with definable f-generics (dfg), and G acts on a definable set X, then the quotient space $X/G$ is definable, not just interpretable. This explains some phenomena observed by Pillay and Yao [24]. Lastly, we show that interpretable topological spaces satisfy analogues of first-countability and curve selection. Using this, we show that all reasonable notions of definable compactness agree on interpretable topological spaces, and that definable compactness is definable in families.

We show that there are models of ${\mathrm{MA}}_{{\omega }_1}$ where the ${\Sigma }_3^1$-uniformization property holds. Further we show that “$\mathrm{BPFA}$+ ${\aleph }_1$ is not inaccessible to reals” outright implies that the ${\Sigma }_3^1$-uniformization property is true.

We prove that there are strong minimal pairs in the enumeration degrees and that the degrees of the left and right sides of strong minimal pairs include ${\Sigma }_2^0$ degrees, although it is unknown if there is a strong minimal pair in the ${\Sigma }_2^0$ enumeration degrees. We define a stronger type of minimal pair we call a strong super minimal pair, and show that there are none of these in the enumeration degrees, answering a question of Lempp et al. [6]. We leave open the question of the existence of a super minimal pair in the enumeration degrees.

A field K in a ring language $L$ is finitely undecidable if $\text{Cons}\left(\Sigma \right)$ is undecidable for every nonempty finite $\Sigma \subseteq \text{Th}(K;L)$. We adapt arguments originating with Cherlin-van den Dries-Macintyre/Ershov (for PAC fields) and Haran (for PRC fields) to prove all PAC and PRC fields are finitely undecidable. We describe the difficulties that arise in adapting the proof to PpC fields, and show no bounded PpC field is finitely axiomatisable. This work is drawn from the author's PhD thesis [44, Chapter 4].

A well-known version of Rowbottom's theorem for supercompactness ultrafilters leads naturally to notions of two-cardinal Ramseyness and corresponding normal ideals introduced herein. Generalizing results of Baumgartner, Feng and the first author, from the cardinal setting to the two-cardinal setting, we study hierarchies associated with a particular version of two-cardinal Ramseyness and a strong version of two-cardinal ineffability, as well as the relationships between these hierarchies and a natural notion of transfinite two-cardinal indescribability.