We refine the definition of II-computability of [12] so that oracles have a “consistent”, but natural, behaviour. We prove a Kleene Normal Form Theorem and closure of semi-recursive relations under ∃1. We also show that in this more inclusive computation theory Post's theorem in the arithmetical hierarchy still holds.

I propose that empirical procedures, like computational procedures, are justified in terms of truth-finding efficiency. I contrast the idea with more standard philosophies of science and illustrate it by deriving Ockham's razor from the aim of minimizing dramatic changes of opinion en route to the truth.

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 possible-worlds semantics for predicates of (...) sentences has not been developed very far. By introducing possible-worlds semantics for the language of arithmetic plus the unary predicate □, we tackle both problems. Given a frame (W, R) consisting of a set W of worlds and a binary relation R on W, we investigate whether we can interpret □ at every world in such a way that □ $\ulcorner A \ulcorner$ holds at a world ᵆ ∊ W if and only if A holds at every world $\upsilon$ ∊ W such that ᵆR $\upsilon$ . The arithmetical vocabulary is interpreted by the standard model at every world. Several 'paradoxes' (like Montague's Theorem, Gödel's Second Incompleteness Theorem, McGee's Theorem on the ω-inconsistency of certain truth theories, etc.) show that many frames, e.g., reflexive frames, do not allow for such an interpretation. We present sufficient and necessary conditions for the existence of a suitable interpretation of □ at any world. Sound and complete semi-formal systems, corresponding to the modal systems K and K4, for the class of all possible-worlds models for predicates and all transitive possible-worlds models are presented. We apply our account also to nonstandard models of arithmetic and other languages than the language of arithmetic. (shrink)

This paper presents an approach to truth and the Liar paradox which combines elements of context dependence and hierarchy. This approach is developed formally, using the techniques of model theory in admissible sets. Special attention is paid to showing how starting with some ideas about context drawn from linguistics and philosophy of language, we can see the Liar sentence to be context dependent. Once this context dependence is properly understood, it is argued, a hierarchical structure emerges which is neither ad (...) hoc nor unnatural. (shrink)

We prove two of the inequalities needed to obtain the following result on the ordinal values of ptykes of type 2, which are definable in Gödel'sT. LetG be a dilator satisfyingG(0)=ω, ∀x:G(x)≧x, and ∀η<Ω:G(η)<Ω, and letg be the ordinal function induced byG. Then sup{A(G)∣A ptyx of type 2 definable in Gödel'sT} = sup{x∣x is∑ 1 g -definable without parameters provably in KP(G)} =J (2 +Id) g (ω) (0) = the “Bachmann-Howard ordinal relative tog”. KP(G) is obtained from Kripke-Platek set theory (...) without urelements KP by adjoining a two-place relation symbolG and axioms expressing thatG is the graph of a total function from ordinals to ordinals.J D g is an iteration hierarchy defined relative tog by primitive recursion on dilators. (2 +Id)(ω) is the dilator $$\mathop {\sup }\limits_{n< \omega } (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{2} + Id)_{(n)} $$ with (2 +Id)(0)≔1 and $$(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{2} + Id)_{(n + 1)} : = (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{2} + Id)^{(2 + Id)_{(n)} } $$ . The “Bachmann-Howard ordinal relative tog” is the closure ordinal of a Bachmann hierarchy of lengthε Ω + 1, which is built on an iteration ofg as initial function.For the caseG=(1+Id)·ω, KP(G) is equivalent to Jäger's theory KPu, and the “Bachmann-Howard ordinal relative tog” is the usual “Bachmann-Howard ordinal”. For the caseG=Ξ1 KP(G) can be replaced by Jäger's theory KPi, andg can be replaced by the functionλx. x +.x + is the successor admissible ofx, andΞ 1 is the sum of all recursive dilators in an arbitrary enumeration. (shrink)

An effectively closed set, or ${\Pi^{0}_{1}}$ class, may viewed as the set of infinite paths through a computable tree. A numbering, or enumeration, is a map from ω onto a countable collection of objects. One numbering is reducible to another if equality holds after the second is composed with a computable function. Many commonly used numberings of ${\Pi^{0}_{1}}$ classes are shown to be mutually reducible via a computable permutation. Computable injective numberings are given for the family of ${\Pi^{0}_{1}}$ classes and (...) for the subclasses of decidable and of homogeneous ${\Pi^{0}_{1}}$ classes. However no computable numberings exist for small or thin classes. No computable numbering of trees exists that includes all computable trees without dead ends. (shrink)

How do ordinals measure the strength and computational power of formal theories? This paper is concerned with the connection between ordinal representation systems and theories established in ordinal analyses. It focusses on results which explain the nature of this connection in terms of semantical and computational notions from model theory, set theory, and generalized recursion theory.

We give a new elementary proof of the comparison theorem relating $\sum^1_{n + 1}-\mathrm{AC}\uparrow$ and $\Pi^1_n -\mathrm{CA}\uparrow$ ; the proof does not use Skolem theories. By the same method we prove: a) $\sum^1_{n + 1}-\mathrm{DC} \uparrow \equiv (\Pi^1_n -CA)_{ , for suitable classes of sentences; b) $\sum^1_{n+1}-DC \uparrow$ proves the consistency of (Π 1 n -CA) ω k, for finite k, and hence is stronger than $\sum^1_{n+1}-AC \uparrow$ . a) and b) answer a question of Feferman and Sieg.

Michael Dummett's argument for intuitionism can be criticized for the implicit reliance on the existence of what might be called absolutely undecidable statements. Neil Tennant attacks epistemic optimism, the view that there are no such statements. I expose what seem serious flaws in his attack, and I suggest a way of defending the use of classical logic in arithmetic that circumvents the issue of optimism. I would like to thank an anonymous referee for helpful comments. CiteULike Connotea Del.icio.us What's this?

Let U be an admissible structure. A cPCd(U) class is the class of all models of a sentence of the form $\neg\exists\bar{K} \bigwedge \Phi$ , where K̄ is an U-r.e. set of relation symbols and φ is an U-r.e. set of formulas of L∞ω that are in U. The main theorem is a generalization of the following: Let U be a pure countable resolvable admissible structure such that U is not Σ-elementarily embedded in HYP(U). Then a class K of countable (...) structures whose universes are sets of urelements is a cPCd(U) class if and only if for some Σ formula σ (with parameters from U), M is in K if and only if M is a countable structure with universe a set of urelements and $(\mathrm{HYP}_\mathfrak{U}(\mathfrak{M}), \mathfrak{U}, \mathfrak{M}) \models \sigma$ , where HYPU(M), the smallest admissible set above M relative to U, is a generalization of HYP to structures with similarity type Σ over U that is defined in this article. Here we just note that when Lα is admissible, HYPLα(M) is Lβ(M) for the least β ≥ α such that Lβ(M) is admissible, and so, in particular, that HYPHF(M) is just HYP(M) in the usual sense when M has a finite similarity type. The definition of HYPU(M) is most naturally formulated using Adamson's notion of a +-admissible structure (1978). We prove a generalization from admissible to +-admissible structures of the well-known truncation lemma. That generalization is a key theorem applied in the proof of the generalized Spector-Gandy theorem. (shrink)

In this work we show that every structure [MATHEMATICAL SCRIPT CAPITAL A] can be expanded to a partial structure [MATHEMATICAL SCRIPT CAPITAL A]* with universal functions for the class of polynomials on [MATHEMATICAL SCRIPT CAPITAL A]*. We can embed [MATHEMATICAL SCRIPT CAPITAL A]* monomorphically in a total structure [MATHEMATICAL SCRIPT CAPITAL A]º that preserves universal functions of [MATHEMATICAL SCRIPT CAPITAL A]* and that is universal among such structures, i.e. [MATHEMATICAL SCRIPT CAPITAL A]º can be homomorphically embedded in every total structure (...) that preserves universal functions of [MATHEMATICAL SCRIPT CAPITAL A]*. Universal functions are the starting point for developing recursion theoretic tools in an [MATHEMATICAL SCRIPT CAPITAL A]* that satisfies some simple additional conditions. (shrink)

This article reports that some robustness of the notions of predicativity and of autonomous progression is broken down if as the given infinite total entity we choose some mathematical entities other than the traditionalω. Namely, the equivalence between normal transfinite recursion scheme and newdependent transfinite recursionscheme, which does hold in the context of subsystems of second order number theory, does not hold in the context of subsystems of second order set theory where the universeVof sets is treated as the given (...) totality. (shrink)