Bibliographie der Schriften von Hilary Putnam [Bibliography of Hilary Putnam's Writings]

Bibliography of the writings by Hilary Putnam: 16 books, 198 articles, 10 translations into German (up to 1994)

Reachability for infinite time Turing machines with long tapes

Infinite time Turing machine models with tape length $\alpha$, denoted $T_\alpha$, strengthen the machines of Hamkins and Kidder [HL00] with tape length $\omega$. A new phenomenon is that for some countable ordinals $\alpha$, some cells cannot be halting positions of $T_\alpha$ given trivial input. The main open question in [Rin14] asks about the size of the least such ordinal $\delta$. We answer this by providing various characterizations. For instance, $\delta$ is the least ordinal with any of the following properties: (a) For some $\xi<\alpha$, there is a $T_\xi$-writable but not $T_\alpha$-writable subset of $\omega$. (b) There is a gap in the $T_\alpha$-writable ordinals. (c) $\alpha$ is uncountable in $L_{\lambda_\alpha}$. Here $\lambda_\alpha$ denotes the supremum of $T_\alpha$-writable ordinals, i.e. those with a $T_\alpha$-writable code of length $\alpha$. We further use the above characterizations, and an analogue to Welch's submodel characterization of the ordinals $\lambda$, $\zeta$ and $\Sigma$, to show that $\delta$ is large in the sense that it is a closure point of the function $\alpha \mapsto \Sigma_\alpha$, where $\Sigma_\alpha$ denotes the supremum of the $T_\alpha$-accidentally writable ordinals

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

Reachability for infinite time Turing machines with long tapes

Infinite time Turing machine models with tape length $\alpha$, denoted $T_\alpha$, strengthen the machines of Hamkins and Kidder [HL00] with tape length $\omega$. A new phenomenon is that for some countable ordinals $\alpha$, some cells cannot be halting positions of $T_\alpha$ given trivial input. The main open question in [Rin14] asks about the size of the least such ordinal $\delta$. We answer this by providing various characterizations. For instance, $\delta$ is the least ordinal with any of the following properties: (a) For some $\xi<\alpha$, there is a $T_\xi$-writable but not $T_\alpha$-writable subset of $\omega$. (b) There is a gap in the $T_\alpha$-writable ordinals. (c) $\alpha$ is uncountable in $L_{\lambda_\alpha}$. Here $\lambda_\alpha$ denotes the supremum of $T_\alpha$-writable ordinals, i.e. those with a $T_\alpha$-writable code of length $\alpha$. We further use the above characterizations, and an analogue to Welch's submodel characterization of the ordinals $\lambda$, $\zeta$ and $\Sigma$, to show that $\delta$ is large in the sense that it is a closure point of the function $\alpha \mapsto \Sigma_\alpha$, where $\Sigma_\alpha$ denotes the supremum of the $T_\alpha$-accidentally writable ordinals

Gap-minimal systems of notations and the constructible hierarchy

If a constructibly countable ordinal alpha is a gap ordinal, then the order type of the set of index ordinals smaller than alpha is exactly alpha. The gap ordinals are the only points of discontinuity of a certain ordinal-valued function. The notion of gap minimality for well ordered systems of notations is defined, and the existence of gap-minimal systems of notations of arbitrarily large constructibly countable length is established

The real numbers in inner models of set theory

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2022, Director: Joan BagariaWe study the structural regularities and irregularities of the reals in inner models of set theory. Starting with $L$, Gödel's constructible universe, our study of the reals is thus two-fold. On the one hand, we study how their generation process is linked to the properties of $L$ and its levels, mainly referring to [18]. We provide detailed proofs for the results of that paper, generalize them in some directions hinted at by the authors, and present a generalization of our own by introducing the concept of an infinite order gap, which is natural and yields some new insights. On the other hand, we present and prove some well-known results that build pathological sets of reals. We generalize this study to $L\left[\#_1\right]$ (the smallest inner model closed under the sharp operation for reals) and $L[\#]$ (the smallest inner model closed under all sharps), for which we provide some introduction and basic facts which are not easily available in the literature. We also discuss some relevant modern results for bigger inner models

The Theory of Countable Analytical Sets

The purpose of this paper is the study of the structure of countable sets in the various levels of the analytical hierarchy of sets of reals. It is first shown that, assuming projective determinacy, there is for each odd n a largest countable ∏_n^1 set of reals, C_n (this is also true for n even, replacing ∏_n^1 by Σ_n^1 and has been established earlier by Solovay for n = 2 and by Moschovakis and the author for all even n > 2). The internal structure of the sets C_n is then investigated in detail, the point of departure being the fact that each C_n is a set of Δ_n^1-degrees, wellordered under their usual partial ordering. Finally, a number of applications of the preceding theory is presented, covering a variety of topics such as specification of bases, ω-models of analysis, higher-level analogs of the constructible universe, inductive definability, etc