Mathematics

A function is strongly non-recursive (SNR) if it is eventually different from each recursive function. We obtain hierarchy results for the mass problems associated with computing such functions with varying growth bounds. In particular, there is no least and no greatest Muchnik degree among those of the form SNR f consisting of SNR functions bounded by varying recursive bounds f . We show that the connection between SNR functions and canonically immune sets is, in a sense, as strong as that between DNR (diagonally non-recursive) functions and effectively immune sets. Finally, we introduce pandemic numberings, a set-theoretic dual to immunity.

We describe the extension of normal iteration strategies with appropriate condensation properties to strategies for stacks of normal trees, with full normalization. Given a regular uncountable cardinal Ω and an (m,Ω+1) -iteration strategy Σ for a premouse M such that Σ and M both have appropriate condensation properties, we extend Σ to a strategy Σ ??for the (m,Ω,Ω+1 ) ??-iteration game such that for all λ<Ω and all stacks T ??= ??T α ??α<λ via Σ ??, consisting of normal trees T α , each of length <Ω , there is a corresponding normal tree X via Σ with M T ????= M X ??, along with agreement of iteration maps when there are no drops in model or degree on main branches. We also use the methods to analyze the comparison of multiple iterates via such a common strategy.

In the former article "Formal mathematical systems including a structural induction principle" we have presented a unified theory for formal mathematical systems including recursive systems closely related to formal grammars, including the predicate calculus as well as a formal induction principle. In this paper we present some further results and examples in order to illustrate how this theory works.

Assume C is the class of all linear orders L such that L is not a countable union of well ordered sets, and every uncountable subset of L contains a copy of ω 1 . We show it is consistent that C has minimal elements. This answers an old question due to Galvin.

We prove consistency of intensional Martin-Löf type theory (MLTT) with formal Church's thesis (CT), which was open for at least fifteen years. The difficulty in proving the consistency is that a standard method of realizability à la Kleene does not work for the consistency, though it validates CT, as it does not model MLTT; specifically, the realizability does not validate MLTT's congruence rule on pi-types (known as the ξ -rule). We overcome this point and prove the consistency by novel realizability à la game semantics, which is based on the author's previous work.

We develop various Ehrenfeucht-Fra\"ıssé games for distances between metric structures. We study two forms of distances: pseudometrics stemming from mapping spaces onto each other with some form of approximate isomorphism, and metrics stemming from measuring the distances between two spaces isometrically embedded into a third space. Using an infinitary version of Henson's positive bounded logic with approximations, we form Scott sentences capturing fixed distances to a given space. The Scott sentences of separable spaces are in L ? 1 ? for 0-distances and in L ? 2 ? for positive distances.

This paper has two parts. The first is concerned with a variant of a family of games introduced by Holy and Schlicht, that we call \emph{Welch games}. Player II having a winning strategy in the Welch game of length ω on κ is equivalent to weak compactness. Winning the game of length 2 κ is equivalent to κ being measurable. We show that for games of intermediate length γ , II winning implies the existence of precipitous ideals with γ -closed, γ -dense trees. The second part shows the first is not vacuous. For each γ between ω and κ + , it gives a model where II wins the games of length γ , but not γ + . The technique also gives models where for all ω 1 <γ≤κ there are κ -complete, normal, κ + -distributive ideals having dense sets that are γ -closed, but not γ + -closed.

We present some results about the burgeoning research area concerning set theory of the kappa-reals. We focus on some notions of measurability coming from generalizations of Silver and Miller trees. We present analogies and mostly differences from the classical setting.

The Curry-Howard correspondence is often described as relating proofs (in intutionistic natural deduction) to programs (terms in simply-typed lambda calculus). However this narrative is hardly a perfect fit, due to the computational content of cut-elimination and the logical origins of lambda calculus. We revisit Howard's work and interpret it as an isomorphism between a category of proofs in intuitionistic sequent calculus and a category of terms in simply-typed lambda calculus. In our telling of the story the fundamental duality is not between proofs and programs but between local (sequent calculus) and global (lambda calculus or natural deduction) points of view on a common logico-computational mathematical structure.

We analyze several natural Goodstein principles which themselves are defined with respect to the Ackermann function and the extended Ackermann function. These Ackermann functions are well established canonical fast growing functions labeled by ordinals not exceeding ε 0 . Among the Goodsteinprinciples under consideration, the giant ones, will be proof-theoretically strong (being unprovable in PA in the Ackermannian case and being unprovable in ID 1 in the extended Ackermannian case) whereas others, the illusionary giant ones, will turn out to be comparatively much much weaker although they look strong at first sight.