Archive for Mathematical Logic最新文献

We investigate the degree spectra of computable relations on canonically ordered natural numbers ((omega ,<)) and integers ((zeta ,<)). As for ((omega ,<)), we provide several criteria that fix the degree spectrum of a computable relation to all c.e. or to all (Delta _2) degrees; this includes the complete characterization of the degree spectra of so-called computable block functions that have only finitely many types of blocks. Compared to Bazhenov et al. (in: LIPIcs, vol 219, pp 8:1–8:20, 2022), we obtain a more general solution to the problem regarding possible degree spectra on ((omega ,<)), answering the question whether there are infinitely many such spectra. As for ((zeta ,<)), we prove the following dichotomy result: given an arbitrary computable relation R on ((zeta ,<)), its degree spectrum is either trivial or it contains all c.e. degrees. This result, and the proof techniques required to solve it, extend the analogous theorem for ((omega ,<)) obtained by Wright (Computability 7:349–365, 2018), and provide initial insight to Wright’s question whether such a dichotomy holds on computable ill-founded linear orders. This article is an extended version of Bazhenov et al. (in: LIPIcs, vol 219, pp 8:1–8:20, 2022).

In recent research, the prime and irreducible elements of strong finite factorization domains were studied. It was shown that strongly computable strong finite factorization domains (SCSFFD) have necessarily computable irreducible elements and a computable division algorithm. However, the question of how to best classify this class of structures is left unanswered. This work provides a classification for SCSFFDs by showing the existence of a computable norm where norm-form equations can be solved computably. This classification provides the intuition to extend further the notion of strong computability to finite factorization domains in general.

We investigate the covering numbers of some ideals on ({^{omega }}{2}{}) associated with tree forcings. We prove that the covering of the Sacks ideal remains small in the Silver and uniform Sacks model, respectively, and that the coverings of the uniform Sacks ideal and the Mycielski ideal, ({mathfrak {C}_{2}}), remain small in the Sacks model.

In this article we investigate the complexity of isomorphisms between scattered linear orders of constructive ranks. We give the general upper bound and prove that this bound is sharp. Also, we construct examples showing that the categoricity level of a given scattered linear order can be an arbitrary ordinal from 3 to the upper bound, except for the case when the ordinal is the successor of a limit ordinal. The existence question of the scattered linear orders whose categoricity level equals the successor of a limit ordinal is still open.

We study Basic Arithmetic, (textsf{BA}) introduced by Ruitenburg (Notre Dame J Formal Logic 39:18–46, 1998). (textsf{BA}) is an arithmetical theory based on basic logic which is weaker than intuitionistic logic. We show that the class of the provably total recursive functions of (textsf{BA}) is a proper sub-class of the primitive recursive functions. Three extensions of (textsf{BA}), called (textsf{BA}+mathsf U), (mathsf {BA_{mathrm c}}) and (textsf{EBA}) are investigated with relation to their provably total recursive functions. It is shown that the provably total recursive functions of these three extensions of (textsf{BA}) are exactly the primitive recursive functions. Moreover, among other things, it is shown that the well-known MRDP theorem does not hold in (textsf{BA}), (textsf{BA}+mathsf U), (mathsf {BA_{mathrm c}}), but holds in (textsf{EBA}).

By using algebraic properties of (commutative unital) indecomposable polynomial rings we achieve results concerning their first-order theory, namely: interpretability of arithmetic and a uniform proof of undecidability of their full theory, both in the language of rings without parameters. This vastly extends the scope of a method due to Raphael Robinson, which deals with a restricted class of polynomial integral domains.

We investigate the problem of punctual (fully primitive recursive) presentability of algebraic structures up to primitive recursive and computable isomorphism. We show that for mono-unary structures and undirected graphs, if a structure is not punctually categorical then it has infinitely many punctually non-isomorphic punctual presentations. We also show that the punctual degrees of any computably almost rigid structure as well as the order ((mathbb {Z},<)) are dense. Finally we characterise the Boolean algebras which have a punctually 1-decidable presentation that is computably isomorphic to a 1-decidable presentation.

We formalize various counting principles and compare their strengths over (V^{0}). In particular, we conjecture the following mutual independence between:

• a uniform version of modular counting principles and the pigeonhole principle for injections,

• a version of the oddtown theorem and modular counting principles of modulus p, where p is any natural number which is not a power of 2,

• and a version of Fisher’s inequality and modular counting principles.

Then, we give sufficient conditions to prove them. We give a variation of the notion of PHP-tree and k-evaluation to show that any Frege proof of the pigeonhole principle for injections admitting the uniform counting principle as an axiom scheme cannot have o(n)-evaluations. As for the remaining two, we utilize well-known notions of p-tree and k-evaluation and reduce the problems to the existence of certain families of polynomials witnessing violations of the corresponding combinatorial principles with low-degree Nullstellensatz proofs from the violation of the modular counting principle in concern.

On the set of all first-order complete theories (T(sigma )) of a language (sigma ) we define a binary operation ({cdot }) by the rule: (Tcdot S= {{,textrm{Th},}}({Atimes Bmid Amodels T ,,text {and},, Bmodels S})) for any complete theories (T, Sin T(sigma )). The structure (langle T(sigma );cdot rangle ) forms a commutative semigroup. A subsemigroup S of (langle T(sigma );cdot rangle ) is called an absorption’s formula definable semigroup if there is a complete theory (Tin T(sigma )) such that (S=langle {Xin T(sigma )mid Xcdot T=T};cdot rangle ). In this event we say that a theory T absorbs S. In the article we show that for any absorption’s formula definable semigroup S the class ({{,textrm{Mod},}}(S)={Ain {{,textrm{Mod},}}(sigma )mid Amodels T_0,,text {for some},, T_0in S}) is axiomatizable, and there is an idempotent element (Tin S) that absorbs S. Moreover, ({{,textrm{Mod},}}(S)) is finitely axiomatizable provided T is finitely axiomatizable. We also prove that ({{,textrm{Mod},}}(S)) is a quasivariety (variety) provided T is an universal (a positive universal) theory. Some examples are provided.

It has long been known that (classical) Peano arithmetic is, in some strong sense, “equivalent” to the variant of (classical) Zermelo–Fraenkel set theory (including choice) in which the axiom of infinity is replaced by its negation. The intended model of the latter is the set of hereditarily finite sets. The connection between the theories is so tight that they may be taken as notational variants of each other. Our purpose here is to develop and establish a constructive version of this. We present an intuitionistic theory of the hereditarily finite sets, and show that it is definitionally equivalent to Heyting Arithmetic HA, in a sense to be made precise. Our main target theory, the intuitionistic small set theory SST is remarkably simple, and intuitive. It has just one non-logical primitive, for membership, and three straightforward axioms plus one axiom scheme. We locate our theory within intuitionistic mathematics generally.