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The coefficients of the regular continued fraction for random numbers are distributed by the Gauss-Kuzmin distribution according to Khinchin's law. Their geometric mean converges to Khinchin's constant and their rational approximation speed is Khinchin's speed. It is an open question whether these theorems also apply to algebraic numbers of degree $>2$. Since they apply to almost all numbers it is, however, commonly inferred that it is most likely that non quadratic algebraic numbers also do so. We argue that this inference is not well grounded. There is strong numerical evidence that Khinchin's speed is too fast. For Khinchin's law and Khinchin's constant the numerical evidence is unclear. We apply the Kullback Leibler Divergence (KLD) to show that the Gauss-Kuzmin distribution does not fit well for algebraic numbers of degree $>2$. Our suggestion to truncate the Gauss-Kuzmin distribution for finite parts fits slightly better but its KLD is still much larger than the KLD of a random number. So, if it converges the convergence is non uniform and each algebraic number has its own bound. We conclude that there is no evidence to apply the theorems that hold for random numbers to algebraic numbers.

Modern logic engines widely fail to decide axiom sets that are satisfiable only in an infinite domain. This paper specifies an algorithm that automatically generates a database of independent infinity axiom sets with fewer than 1000 characters. It starts with complete theories of pure first-order logic with only one binary relation (FOL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{R}$$\end{document}) and generates further infinity axiom sets S of FOL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{R}$$\end{document} with fewer than 1000 characters such that no other infinity axiom set with fewer than 1000 characters exists in the database that implies S. We call the generated infinity axiom sets S “superpostulates”. Any formula that is derivable from (satisfiable) superpostulates is also satisfiable. Thus far, we have generated a database with 2346 infinity superpostulates by running our algorithm. This paper ends by identifying three practical uses of the algorithmic generation of such a database: (i) for systematic investigations of infinity axiom sets, (ii) for deciding infinity axiom sets and (iii) for the development of saturation algorithms.

Enrico Bombieri proved that the ABC Conjecture implies Roth's theorem in 1994. This paper concerns the other direction. In making use of Bombieri's and Van der Poorten's explicit formula for the coefficients of the regular continued fractions of algebraic numbers, we prove that Roth's theorem implies a weakened non-effective version of the ABC Conjecture in certain cases relating to roots.

Enrico Bombieri proved that the ABC Conjecture implies Roth's theorem in 1994. This paper concerns the other direction. In making use of Bombieri's and Van der Poorten's explicit formula for the coefficients of the regular continued fractions of algebraic numbers, we prove that Roth's theorem implies a weakened non-effective version of the ABC Conjecture in certain cases relating to roots.

This paper argues that the distribution of the coefficients of the regular continued fraction should be considered for each algebraic number of degree >2 separately. For random numbers the coefficients are distributed by the Gauss-Kuzmin distribution (also called Khinchin's law). We apply the Kullback Leibler Divergence (KLD) to show that the Gauss-Kuzmin distribution does not fit well for algebraic numbers of degree > 2. Our suggestion to truncate the Gauss-Kuzmin distribution for finite parts fits slightly better, but its KLD is still much larger than the KLD of a random number. We consider differences regarding Khinchin's constant and Khinchin's approximation speed between random and algebraic numbers and conclude that laws concerning the random numbers do not automatically carry over to the algebraic numbers.

This paper describes a decision procedure for disjunctions of conjunctions of anti-prenex normal forms of pure first-order logic (FOLDNFs) that do not contain $\vee$ within the scope of quantifiers. The disjuncts of these FOLDNFs are equivalent to prenex normal forms whose quantifier-free parts are conjunctions of atomic and negated atomic formulae (= Herbrand formulae). In contrast to the usual algorithms for Herbrand formulae, neither skolemization nor unification algorithms with function symbols are applied. Instead, a procedure is described that rests on nothing but equivalence transformations within pure first-order logic (FOL). This procedure involves the application of a calculus for negative normal forms (the NNF-calculus) with $A \dashv\vdash A \wedge A$ (= $\wedge$I) as the sole rule that increases the complexity of given FOLDNFs. The described algorithm illustrates how, in the case of Herbrand formulae, decision problems can be solved through a systematic search for proofs that reduce the number of applications of the rule $\wedge$I to a minimum in the NNF-calculus. In the case of Herbrand formulae, it is even possible to entirely abstain from applying $\wedge$I. Finally, it is shown how the described procedure can be used within an optimized general search for proofs of contradiction and what kind of questions arise for a $\wedge$I-minimal proof strategy in the case of a general search for proofs of contradiction., Comment: 30 pages, 2 figures

Auf der 2. Tagung für Erkenntnislehre der exakten Wissenschaft in Königsberg 1930 sollten die unterschiedlichen Ansätze in den Grundlagen der Mathematik dargestellt werden: Carnap referierte über den Logizismus, Heyting über den Intuitionismus, Neumann über den Formalismus (vgl. Hahn et al. 1931). Interessanterweise wurde neben diesen Ansätzen noch ein vierter Ansatz dargestellt: Der Standpunkt Wittgensteins, über den Friedrich Waismann vortrug. Waismann tat dies mit der Unterstützung Wittgensteins, was zeigt, dass Wittgenstein keineswegs den Anspruch ablehnte, eine alternative Position in den Grundlagenfragen der Mathematik einzunehmen. Das Schicksal, das Waismanns Versuch erfuhr, Wittgensteins Standpunkt zu kommunizieren, ist symptomatisch: Waismann gelang es im Unterschied zu den anderen Referenten nicht, seinen Vortrag in eine publizierfähige Form zu bringen. Ein erhaltenes Skript (vgl. Waismann 1982) ist unvollständig und beschäftigt sich vornehmlich mit Wittgensteins Kritik am Logizismus; eine eigenständige Motivation für einen alternativen Standpunkt entfaltet Waismann nicht. In der anschliessenden Diskussion der Referate wurde Wittgensteins Standpunkt übergangen, da er noch nicht in einer "spruchreifen Form" (Hahn et al. 1931, S. 141) vorlag. Die Rezeption von Wittgensteins Standpunkt scheiterte auch bei gutwilligen und mit der Materie vertrauten Forschern, da es nicht gelang, diesen als eine konstruktive Alternative zu identifizieren und verständlich zu machen.

Von Gerechtigkeitstheorien wird gemeinhin erwartet, dass diese handlungsleitend sein können. Gleichzeitig dominierten in den vergangenen Jahrzehnten sogenannte ideale Theorien den Diskurs um Gerechtigkeit, die auf kontrafaktischen Annahmen basieren, welche nicht die Realität abbilden. Gemäß der Mainstream-Auffassung konfligiert dieser ideale Charakter nicht mit dem Desideratum der Handlungsleitung: Ideale Theorien zeichnen die (vollkommen) gerechte Gesellschaft als ein Ziel, dem es entgegenzustreben gilt. Um dieses Ziel zu erreichen, brauche es darüber hinaus eine nicht-ideale Theorie, die mehr an der empirischen Wirklichkeit ausgerichtet ist und es als Ziel hat, bestehende Ungerechtigkeiten durch entsprechende Handlungsanweisungen zu beseitigen. Dieser Auffassung nach ist ideale Theorie somit zwar nicht selbst handlungsleitend, wohl aber eine notwendige Bedingung für Handlungsleitung. Dass ideale Theorie dieser Funktion gerecht werden kann, stößt hierbei zunehmend auf Widerspruch. Dabei lassen sich im Wesentlichen drei Gruppen von Fundamentalkritiken unterscheiden, die alle drei in der Konklusion münden, dass ideale Theorie nicht zur Handlungsleitung unter Realbedingungen beitragen kann: Erstens, Kritiken, denen entsprechend ideale Theorie nicht zur Genese von Handlungsanweisungen fähig und nötig ist; zweitens, Kritiken, denen entsprechend ideale Theorie nicht anwendbar auf das Hier-und-Jetzt ist; drittens, Kritiken, entsprechend denen idealer Theorie der Bezug zur Gerechtigkeit fehlt. In der vorliegenden Arbeit verteidige ich ideale Theorie gegen jede dieser drei Kritiken. Dafür (re-)konstruiere ich als erstes die Unterscheidung in ideale und nicht-ideale Theorie, stelle dann die Kritiken dar und formuliere schließlich entsprechende Entgegnungen. Dabei akzeptiere ich das Desideratum der Handlungsleitung und argumentiere dafür, dass alle drei Kritiken scheitern., Theories of justice are commonly expected to be able to guide actions. At the same time, the debate led on justice during the last decades was dominated by so called ideal theories. These theories are based on contrafactual assumptions, which do not portray the reality. According to the mainstream-view, this ideal character is not detrimental for the desideratum of action-guidingness: ideal theories describe the (fully) just society as an aim one ought to strive for. To accomplish this goal, however, a non-ideal theory is needed, which is rooted in the empirical reality and seeks to eliminate injustices through recommendations of actions. On this account, ideal theory is not action-guiding itself, albeit a necessary condition for action-guidingness. However, the idea that ideal theory can fulfil this function is increasingly disagreed with. Three main groups of fundamental critiques can be distinguished, which all three lead to the conclusion that ideal theory cannot contribute to action-guidingness under real-life conditions: Firstly, critiques, according to which ideal theory is neither able nor necessary to create recommendations for actions; secondly, critiques, according to which ideal theory is not applicable to the here-and-now; thirdly, critiques, according to which ideal theory does not connect to justice. In this thesis I defend ideal theory against each one of these three critiques. To do so, I, firstly, (re-)construct the distinction between ideal and non-ideal theory. Thereafter, I present the three critiques and formulate replies to them. In doing so, I accept the desideratum of action-guidingness and argue that all three critiques fail.