Tristram de Piro
Using methods developed by Robinson, we find a complete theory suitable for a first order description of infintesimal neighborhoods. We use this to construct a specialisation having universal properties and to find a recursively enumerable model in which the algebraic version of Bezout's theorem is provable by non-standard methods.
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It is proved that all recursively enumerable sets of natural numbers can be represented by arithmetic formulas (of two kinds) with only 3 quantifiers.
Kolmogorov complexity and instance complexity of recursively enumerable sets
我们研究了Kolmogorov复杂性和实例复杂性对r.e.性质的影响。集.我们证明了r.e.的初始段的Kolmogorov复杂度的2logn上界是已知的。集合是最优的，并表征了r. e.达到这个境界的集合。本文的主要部分是研究r.e.的实例复杂度。集.我们构造了一个非递归的r.e.集合的实例复杂度是Kolmogorov复杂度的对数。这反驳了Ko、Orponen、Schoning和Watanabe的一个猜想。在另一个极端，我们证明了所有的wtt-完备集和所有的Q-完备集都有无穷多个硬实例。
A real number αis called recursively enumerable if there exists a computable, increasing sequence of rational numbers which converges to α. The randomness of a recursively enumerable real αcan be characterized in various ways using each of the notions; program-size complexity, Martin-Löf test, Chaitin's Ωnumber, the domination and Ω-likeness of α, the universality of a computable, increasing sequence of rational numbers which converges to α, and universal probability. In this paper, we generalize these characterizations of randomness over the notion of partial randomness by parameterizing each…
Godel's First Incompleteness Theorem is generalized to definable theories, which are not necessarily recursively enumerable, by using a couple of syntactic-semantic notions, one is the consistency of a theory with the set of all true $Π_n$-sentences or equivalently the $Σ_n$-soundness of the theory, and the other is $n$-consistency the restriction of $ω$-consistency to the $Σ_n$-formulas. It is also shown that Rosser's Incompleteness Theorem does not generally hold for definable non-recursively enumerable theories, whence Godel-Rosser's Incompleteness Theorem is optimal in a sense. Though the …
Given a finite lattice $L$ that can be embedded in the recursively enumerable (r.e.) Turing degrees $\mathcal{R}_{\mathrm{T}}$, it is not known how one can characterize the degrees $\mathbf{d}\in\mathcal{R}_{\mathrm{T}}$ below which $L$ can be bounded. Two important characterizations are of the $L_7$ and $M_3$ lattices, where the lattices are bounded below $\mathbf{d}$ if and only if $\mathbf{d}$ contains sets of ``\emph{fickleness}'' $>ω$ and $\geqω^ω$ respectively. We work towards finding a lattice that characterizes the levels above $ω^2$, the first non-trivial level after $ω$. We considere…
A longstanding open problem is whether there exists a non syntactical model of the untyped lambda-calculus whose theory is exactly the least lambda-theory (l-beta). In this paper we investigate the more general question of whether the equational/order theory of a model of the (untyped) lambda-calculus can be recursively enumerable (r.e. for brevity). We introduce a notion of effective model of lambda-calculus calculus, which covers in particular all the models individually introduced in the literature. We prove that the order theory of an effective model is never r.e.; from this it follows tha…
We argue that Godel's completeness theorem is equivalent to completability of consistent theories, and Godel's incompleteness theorem is equivalent to the fact that this completion is not constructive, in the sense that there are some consistent and recursively enumerable theories which cannot be extended to any complete and consistent and recursively enumerable theory. Though any consistent and decidable theory can be extended to a complete and consistent and decidable theory. Thus deduction and consistency are not decidable in logic, and an analogue of Rice's Theorem holds for recursively en…
The central result of this paper is the small-is-very-small principle for restricted sequential theories. The principle says roughly that whenever the given theory shows that a property has a small witness, i.e. a witness in every definable cut, then it shows that the property has a very small witness: i.e. a witness below a given standard number. We draw various consequences from the central result. For example (in rough formulations): (i) Every restricted, recursively enumerable sequential theory has a finitely axiomatized extension that is conservative w.r.t. formulas of complexity $\leq n$…
We show that there is no theory that is minimal with respect to interpretability among recursively enumerable essentially undecidable theories.
A real αis called recursively enumerable ("r.e." for short) if there exists a computable, increasing sequence of rationals which converges to α. It is known that the randomness of an r.e. real αcan be characterized in various ways using each of the notions; program-size complexity, Martin-Löf test, Chaitin Ωnumber, the domination and Ω-likeness of α, the universality of a computable, increasing sequence of rationals which converges to α, and universal probability. In this paper, we generalize these characterizations of randomness over the notion of partial randomness by parameterizing each of …
We develop the abstract framework for a proof-theoretic analysis of theories with scope beyond ordinal numbers, resulting in an analog of Ordinal Analysis aimed at the study of theorems of complexity $Π^1_2$. This is done by replacing the use of ordinal numbers by particularly uniform, wellfoundedness preserving functors in the category of linear orders. Generalizing the notion of a proof-theoretic ordinal, we define the functorial $Π^1_2$ norm of a theory and prove its existence and uniqueness for $Π^1_2$-sound theories. From this, we further abstract a definition of the $Σ^1_2$- and $Π^1_2$-…
For a set $X\subseteq \mathbb{N}$, we define the $X$-torsion of a group $G$ to be all elements $g\in G$ with $g^{n}=e$ for some $n\in X$. With $X$ recursively enumerable, we give two independent proofs (group-theoretic, and model-theoretic) that there exists a universal finitely presented $X$-torsion-free group; one which contains all finitely presented $X$-torsion-free groups. We also show that, if $X$ is recursively enumerable, then the set of finite presentations of $X$-torsion-free groups is $Π_{2}^{0}$-complete in Kleene's arithmetic hierarchy.
Inquisitive first-order logic is neither compact nor recursively axiomatizable
本文研究了探究性一阶逻辑（InqBQ）的元理论性质，证明其蕴含关系不满足紧凑性，且有效式集不可递归枚举。
Let R be a recursive subring of a number field. We show that recursively enumerable sets are diophantine for the polynomial ring R[Z].
Since the axioms in (Consi-CoS) are not recursively enumerable, NACT* is no axiom system in the classical sense . Therefore we construct a series of partial systems which form a recursive axiom system too. Starting with the "dichotomic" systems NACT# and its variant NACT#4, we are going on to the "disjunctive" systems NACT+ and NACT+4, and eventually to NACT+Strat. After that we discuss the medium classes of these systems. Finally we present the inconsistent NSA-systems based on Not-SelfApplicability and explain their help for computational set theory.
We present a version of Gödel's Second Incompleteness Theorem for recursively enumerable consistent extensions of a fixed axiomatizable theory, by incorporating some bi-theoretic version of the derivability conditions. We also argue that Tarski's theorem on the Undefinability of Truth is Gödel's First Incompleteness Theorem relativized to definable oracles; a unification of these two theorems is given.
We define a fragment of monadic infinitary second-order logic corresponding to an abstract separation property. We use this to define the concept of a separation subclass. We use model theoretic techniques and games to show that separation subclasses whose axiomatisations are recursively enumerable in our second-order fragment can also be recursively axiomatised in their original first-order language. We pin down the expressive power of this formalism with respect to first-order logic, and investigate some questions relating to decidability and computational complexity. As applications of thes…
In 1950, B.A. Trakhtenbrot showed that the set of first-order tautologies associated to finite models is not recursively enumerable. In 1999, P. Hájek generalized this result to the first-order versions of Łukasiewicz, Gödel and Product logics. In this paper we extend the analysis to the first-order axiomatic extensions of MTL. Our main result is the following. Let L be an axiomatic extension L of MTL s.t. TAUT$_\text{L}$ is decidable, and whose corresponding variety is generated by a chain: for every generic L-chain $\mathcal{A}$ the set fTAUT$^\mathcal{A}_{\forall}$ (the set of first-order t…
First order complexity of finite random structures
本文提出有限随机结构的一阶复杂度概念，建立复杂度类层级和随机一阶归约工具，推导新的逻辑极限律，并证明使0-1律成立的语句集合不可递归枚举。
First-Order Intuitionistic Linear Logic and Hypergraph Languages
本文研究一阶直觉主义线性逻辑ILL1及其乘法片段MILL1与超图文法理论的关系，引入超图一阶逻辑范畴语法，证明相关文法的表达能力，并为MILL1开发超图语言模型且证明其全称-蕴涵片段的完全性。
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