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The general notions of object and metalanguage are discussed and as a special case of this relation an arbitrary first order language with an infinite model is expanded by a predicate symbol T0 which is interpreted as truth predicate for . Then the expanded language is again augmented by a new truth predicate T1 for the whole language plus T0. This process is iterated into the transfinite to obtain the Tarskian hierarchy of languages. It is shown that there are natural (...) 

We study the structure of Σ11 equivalence relations on hyperarithmetical subsets of ω under reducibilities given by hyperarithmetical or computable functions, called hreducibility and FFreducibility, respectively. We show that the structure is rich even when one fixes the number of properly equation imagei.e., Σ11 but not equation image equivalence classes. We also show the existence of incomparable Σ11 equivalence relations that are complete as subsets of ω × ω with respect to the corresponding reducibility on sets. We study complete Σ11 (...) 

Pour le philosophe intéressé aux structures et aux fondements du savoir théorétique, à la constitution d'une « métathéorétique «, θεωρíα., qui, mieux que les « Wissenschaftslehre » fichtéenne ou husserlienne et pardelà les débris de la métaphysique, veut dans une intention nouvelle faire la synthèse du « théorétique », la logique mathématique se révèle un objet privilégié. 

Complexity is pleased to announce the installment of Prof Hiroki Sayama as its new Chief Editor. In this Editorial, Prof Sayama describes his feelings about his recent appointment, discusses some of the journal’s journey and relevance to current issues, and shares his vision and aspirations for its future. 

The appropriate methodology for psychological research depends on whether one is studying mental algorithms or their implementation. Mental algorithms are abstract specifications of the steps taken by procedures that run in the mind. Implementational issues concern the speed and reliability of these procedures. The algorithmic level can be explored only by studying acrosstask variation. This contrasts with psychology's dominant methodology of looking for withintask generalities, which is appropriate only for studying implementational issues.The implementationalgorithm distinction is related to a number of (...) 

Higman essentially showed that if A is any language then SUBSEQ(A) is regular, where SUBSEQ(A) is the language of all subsequences of strings in A. Let s1, s2, s3, . . . be the standard lexicographic enumeration of all strings over some finite alphabet. We consider the following inductive inference problem: given A(s1), A(s2), A(s3), . . . . learn, in the limit, a DFA for SUBSEQU). We consider this model of learning and the variants of it that are usually (...) 

This dissertation examines aspects of the interplay between computing and scientific practice. The appropriate foundational framework for such an endeavour is rather real computability than the classical computability theory. This is so because physical sciences, engineering, and applied mathematics mostly employ functions defined in continuous domains. But, contrary to the case of computation over natural numbers, there is no universally accepted framework for real computation; rather, there are two incompatible approaches computable analysis and BSS model, both claiming to formalise algorithmic (...) 

A quasiorder Q induces two natural quasiorders on P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{69pt} \begin{document}$${\mathcal{P}}$$\end{document}, but if Q is a wellquasiorder, then these quasiorders need not necessarily be wellquasiorders. Nevertheless, GoubaultLarrecq, pp. 453–462, 2007) showed that moving from a wellquasiorder Q to the quasiorders on P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{69pt} \begin{document}$${\mathcal{P}}$$\end{document} preserves wellquasiorderedness in a topological sense. Specifically, GoubaultLarrecq proved that the upper topologies of the induced quasiorders on P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} (...) 

Locally finite omega languages were introduced by Ressayre [Formal languages defined by the underlying structure of their words. J Symb Log 53(4):1009–1026, 1988]. These languages are defined by local sentences and extend ωlanguages accepted by Büchi automata or defined by monadic second order sentences. We investigate their topological complexity. All locally finite ωlanguages are analytic sets, the class LOC ω of locally finite ωlanguages meets all finite levels of the Borel hierarchy and there exist some locally finite ωlanguages which are (...) 



Computational complexity theory is a subfield of computer science originating in computability theory and the study of algorithms for solving practical mathematical problems. Amongst its aims is classifying problems by their degree of difficulty — i.e., how hard they are to solve computationally. This paper highlights the significance of complexity theory relative to questions traditionally asked by philosophers of mathematics while also attempting to isolate some new ones — e.g., about the notion of feasibility in mathematics, the $\mathbf{P} \neq \mathbf{NP}$ (...) 









Computationalism holds that our grasp of notions like ‘computable function’ can be used to account for our putative ability to refer to the standard model of arithmetic. Tennenbaum's Theorem has been repeatedly invoked in service of this claim. I will argue that not only do the relevant class of arguments fail, but that the result itself is most naturally understood as having the opposite of a referencefixing effect — i.e., rather than securing the determinacy of numbertheoretic reference, Tennenbaum's Theorem points (...) 





. We describe the important role that the conjectures and questions posed at the end of the two editions of Gerald Sacks's Degrees of Unsolvability have had in the development of recursion theory over the past thirty years. 

The aim of this paper is to comprehensively question the validity of the standard way of interpreting Chaitin's famous incompleteness theorem, which says that for every formalized theory of arithmetic there is a finite constant c such that the theory in question cannot prove any particular number to have Kolmogorov complexity larger than c. The received interpretation of theorem claims that the limiting constant is determined by the complexity of the theory itself, which is assumed to be good measure of (...) 



Earman (1993) distinguishes three notions of empirical indistinguishability and offers a rigorous framework to investigate how each of these notions relates to the problem of underdetermination of theory choice. He uses some of the results obtained in this framework to argue for a version of scientific anti realism. In the present paper we first criticize Earman's arguments for that position. Secondly, we propose and motivate a modification of Earman's framework and establish several results concerning some of the notions of indistinguishability (...) 



We will study several weak axiom systems that use the Subtraction and Division primitives (rather than Addition and Multiplication) to formally encode the theorems of Arithmetic. Provided such axiom systems do not recognize Multiplication as a total function, we will show that it is feasible for them to verify their Semantic Tableaux, Herbrand, and CutFree consistencies. If our axiom systems additionally do not recognize Addition as a total function, they will be capable of recognizing the consistency of their Hilbertstyle deductive (...) 

Chaitin’s incompleteness result related to random reals and the halting probability has been advertised as the ultimate and the strongest possible version of the incompleteness and undecidability theorems. It is argued that such claims are exaggerations. 

It is shown that the Fodor's interpretation of the frame problem is the central indication that his version of the Modularity Thesis is incompatible with computationalism. Since computationalism is far more plausible than this thesis, the latter should be rejected. 

Presented here is a new result concerning the computational power of socalled SADn computers, a class of Turingmachinebased computers that can perform some nonTuring computable feats by utilising the geometry of a particular kind of general relativistic spacetime. It is shown that SADn can decide nquantifier arithmetic but not (n+1)quantifier arithmetic, a result that reveals how neatly the SADn family maps into the Kleene arithmetical hierarchy. Introduction Axiomatising computers The power of SAD computers Remarks regarding the concept of computability. 



It is commonly held that the natural numbers sequence 0, 1, 2,... possesses a unique structure. Yet by a well known model theoretic argument, there exist nonstandard models of the formal theory which is generally taken to axiomatize all of our practices and intentions pertaining to use of the term “natural number.” Despite the structural similarity of this argument to the influential set theoretic indeterminacy argument based on the downward L ̈owenheimSkolem theorem, most theorists agree that the number theoretic version (...) 

A natural problem from elementary arithmetic which is so strongly undecidable that it is not even Trial and Error decidable (in other words, not decidable in the limit) is presented. As a corollary, a natural, elementary arithmetical property which makes a diﬀerence between intuitionistic and classical theories is isolated. 



We define the bounded jump of $A$ by $A^{b}=\{x\in \omega \mid \exists i\leq x[\varphi_{i}\downarrow \wedge\Phi_{x}^{A\upharpoonright \!\!\!\upharpoonright \varphi_{i}}\downarrow ]\}$ and let $A^{nb}$ denote the $n$th bounded jump. We demonstrate several properties of the bounded jump, including the fact that it is strictly increasing and orderpreserving on the bounded Turing degrees. We show that the bounded jump is related to the Ershov hierarchy. Indeed, for $n\geq2$ we have $X\leq_{bT}\emptyset ^{nb}\iff X$ is $\omega^{n}$c.e. $\iff X\leq_{1}\emptyset ^{nb}$, extending the classical result that $X\leq_{bT}\emptyset '\iff (...) 

We prove the following three theorems on the enumeration degrees of ∑20 sets. Theorem A: There exists a nonzero noncuppable ∑20 enumeration degree. Theorem B: Every nonzero Δ20enumeration degree is cuppable to 0′e by an incomplete total enumeration degree. Theorem C: There exists a nonzero low Δ20 enumeration degree with the anticupping property. 

We study the connection between factors of the Medvedev lattice and constructive logic. The algebraic properties of these factors determine logics lying in between intuitionistic propositional logic and the logic of the weak law of the excluded middle (also known as De Morgan, or Jankov, logic). We discuss the relation between the weak law of the excluded middle and the algebraic notion of joinreducibility. Finally we discuss autoreducible degrees. 

If □ is conceived as an operator, i.e., an expression that gives applied to a formula another formula, the expressive power of the language is severely restricted when compared to a language where □ is conceived as a predicate, i.e., an expression that yields a formula if it is applied to a term. This consideration favours the predicate approach. The predicate view, however, is threatened mainly by two problems: Some obvious predicate systems are inconsistent, and possibleworlds semantics for predicates of (...) 

The paper offers a solution to the semantic paradoxes, one in which we keep the unrestricted truth schema “True↔A”, and the object language can include its own metalanguage. Because of the first feature, classical logic must be restricted, but full classical reasoning applies in “ordinary” contexts, including standard set theory. The more general logic that replaces classical logic includes a principle of substitutivity of equivalents, which with the truth schema leads to the general intersubstitutivity of True with A within the (...) 



We study the Lindenbaum algebra ${\fancyscript{A}}$ (WKL o, RCA o) of sentences in the language of secondorder arithmetic that imply RCA o and are provable from WKL o. We explore the relationship between ${\Sigma^1_1}$ sentences in ${\fancyscript{A}}$ (WKL o, RCA o) and ${\Pi^0_1}$ classes of subsets of ω. By applying a result of Binns and Simpson (Arch. Math. Logic 43(3), 399–414, 2004) about ${\Pi^0_1}$ classes, we give a specific embedding of the free distributive lattice with countably many generators into ${\fancyscript{A}}$ (...) 

We discuss the interplay between the axiomatic and the semantic approach to truth. Often, semantic constructions have guided the development of axiomatic theories and certain axiomatic theories have been claimed to capture a semantic construction. We ask under which conditions an axiomatic theory captures a semantic construction. After discussing some potential criteria, we focus on the criterion of ℕcategoricity and discuss its usefulness and limits. 

We begin by distinguishing computationalism from a number of other theses that are sometimes conflated with it. We also distinguish between several important kinds of computation: computation in a generic sense, digital computation, and analog computation. Then, we defend a weak version of computationalism—neural processes are computations in the generic sense. After that, we reject on empirical grounds the common assimilation of neural computation to either analog or digital computation, concluding that neural computation is sui generis. Analog computation requires continuous (...) 