The arithmetical hierarchy in the setting of ω1\omega_1

We continue work from (Greenberg and Knight) on computable structure theory in the setting of $\omega_1$, where the countable ordinals play the role of natural numbers, and countable sets play the role of finite sets. In the present paper, we define the arithmetical hierarchy through all countable levels (not just the finite levels). We consider two different ways of doing this—one based on the standard definition of the hyperarithmetical hierarchy, and the other based on the standard definition of the effective Borel hierarchy. For each definition, we define computable infinitary formulas through all countable levels, and we obtain analogues of the well-known results from (Ash and Knight, 1989) and (Chisholm, 1990) saying that a relation is relatively intrinsically $\Sigma^0_\alpha$ just in case it is definable by a computable $\Sigma_\alpha$ formula. Although we obtain the same results for the two definitions of the arithmetical hierarchy, we conclude that the definition resembling the standard definition of the hyperarithmetical hierarchy seems preferable

The Power of Naive Truth

While non-classical theories of truth that take truth to be transparent have some obvious advantages over any classical theory that evidently must take it as non-transparent, several authors have recently argued that there's also a big disadvantage of non-classical theories as compared to their “external” classical counterparts: proof-theoretic strength. While conceding the relevance of this, the paper argues that there is a natural way to beef up extant internal theories so as to remove their proof-theoretic disadvantage. It is suggested that the resulting internal theories should seem preferable to their external counterparts

The Lost Melody Phenomenon

A typical phenomenon for machine models of transfinite computations is the existence of so-called lost melodies, i.e. real numbers $x$ such that the characteristic function of the set $\{x\}$ is computable while $x$ itself is not (a real having the first property is called recognizable). This was first observed by J. D. Hamkins and A. Lewis for infinite time Turing machine, then demonstrated by P. Koepke and the author for $ITRM$s. We prove that, for unresetting infinite time register machines introduced by P. Koepke, recognizability equals computability, i.e. the lost melody phenomenon does not occur. Then, we give an overview on our results on the behaviour of recognizable reals for $ITRM$s. We show that there are no lost melodies for ordinal Turing machines or ordinal register machines without parameters and that this is, under the assumption that $0^{\sharp}$ exists, independent of $ZFC$. Then, we introduce the notions of resetting and unresetting $\alpha$-register machines and give some information on the question for which of these machines there are lost melodies