On structures in hypergraphs of models of a theory

We define and study structural properties of hypergraphs of models of a theory including lattice ones. Characterizations for the lattice properties of hypergraphs of models of a theory, as well as for structures on sets of isomorphism types of models of a theory, are given

Self-embeddings of models of arithmetic; fixed points, small submodels, and extendability

In this paper we will show that for every cut $I$ of any countable nonstandard model $\mathcal{M}$ of $\mathrm{I}\Sigma_{1}$, each $I$-small $\Sigma_{1}$-elementary submodel of $\mathcal{M}$ is of the form of the set of fixed points of some proper initial self-embedding of $\mathcal{M}$ iff $I$ is a strong cut of $\mathcal{M}$. Especially, this feature will provide us with some equivalent conditions with the strongness of the standard cut in a given countable model $\mathcal{M}$ of $\mathrm{I}\Sigma_{1}$. In addition, we will find some criteria for extendability of initial self-embeddings of countable nonstandard models of $\mathrm{I}\Sigma_{1}$ to larger models

Carnap's early semantics

In jüngerer Zeit hat sich ein verstärktes Interesse an den historischen und technischen Details von Carnaps Philosophie der Logik und Mathematik entwickelt. Meine Dissertation knüpft an diese Entwicklung an und untersucht dessen frühe und formative Beiträge aus den späten 1920er Jahren zu einer Theorie der formalen Semantik. Carnaps zu Lebzeiten unveröffentlichtes Manuskript Untersuchungen zur allgemeinen Axiomatik (Carnap 2000) beinhaltet ein Reihe von erstmals formal entwickelten Definitionen der Begriffe ‚Modell’, ‚Modellerweiterung’, und ‚logischer Folgerung’. Die vorliegende Dissertation entwickelt eine logische und philosophische Analyse dieser semantischen Begriffsbildungen. Darüber hinaus wird Carnaps frühe Semantik in ihrem historisch-intellektuellen Entwicklungskontext diskutiert. Der Fokus der Arbeit liegt in der Thematisierung einiger interpretatorischer Fragen zu dessen implizit gehaltenen Annahmen bezüglich der Variabilität des Diskursuniversums von Modellen sowie zur Interpretation seiner typen-theoretischen logischen Sprache. Mit Bezug auf eine Reihe von historischen Dokumenten aus Carnaps Nachlass, insbesondere zu dem geplanten zweiten Teil der Untersuchungen wird erstens gezeigt, dass dessen Verständnis von Modellen in wesentlichen Punkten heterodox gegenüber dem modernen Begriffsverständnis ist. Zweitens, dass Carnap von einer ‚nonstandard’ Interpretation der logischen Hintergrundtheorie für seine Axiomatik ausgeht. Die Konsequenzen dieser semantischen Annahmen für dessen Konzeptualisierung von metatheoretischen Begriffen werden näher diskutiert. Das erste Kapitel entwickelt eine kritische Analyse von Carnaps Versuch, die axiomatische Definition von Klassen von mathematischen Strukturen mittels des Begriffs von ‚Explizitbegriffen’ formal zu rekonstruieren. Im zweiten Kapitel werden die Implikationen von Carnaps frühem Modellbegriff für seine Theorie von Extremalaxiomen näher beleuchtet. Das letzte Kapitel bildet eine Diskussion der konkreten historischen Einflüsse, insbesondere durch den Mengentheoretiker Abraham Fraenkel, auf Carnaps formale Theorie von Minimalaxiomen.In recent years one was able to witness an intensified interest in the technical and historical details of Carnap’s philosophy of logic and mathematics. In my thesis I will take up this line and focus on his early, formative contributions to a theory of semantics around 1928. Carnap’s unpublished manuscript Untersuchungen zur allgemeinen Axiomatik (Carnap 2000) includes some of the first formal definitions of the genuinely semantic concepts of a model, model extensions, and logical consequence. In the dissertation, I provide a detailed conceptual analysis of their technical details and contextualize Carnap’s results in their historic and intellectual environment. Certain interpretative issues related to his tacit assumptions concerning the domain of a model and the semantics of type theory will be addressed. By referring to unpublished material from Carnap’s Nachlass I will present archival evidence as well as more systematic arguments to the view that Carnap holds a heterodox conception of models and a nonstandard semantics for his type-theoretic logic. Given these semantic background assumptions, their impact on Carnap’s conceptualization of certain aspects of the metatheory of axiomatic theories will be evaluated. The first chapter critically discusses Carnap’s attempt to explicate one of the crucial semantic innovations of formal axiomatics, i.e. the definition of classes of structures, via his notion of ‘Explizitbegriffe’. The second chapter analyses the impact of Carnap’s early theory of model for his theory of extremal axioms. The final chapter reviews the mathematical influences, most importantly by the set theoretician Abraham Fraenkel on Carnap’s specific formalization of minimal axioms

Dependent choice, properness, and generic absoluteness

We show that Dependent Choice is a sufficient choice principle for developing the basic theory of proper forcing, and for deriving generic absoluteness for the Chang model in the presence of large cardinals, even with respect to -preserving symmetric submodels of forcing extensions. Hence, not only provides the right framework for developing classical analysis, but is also the right base theory over which to safeguard truth in analysis from the independence phenomenon in the presence of large cardinals. We also investigate some basic consequences of the Proper Forcing Axiom in, and formulate a natural question about the generic absoluteness of the Proper Forcing Axiom in and. Our results confirm as a natural foundation for a significant portion of classical mathematics and provide support to the idea of this theory being also a natural foundation for a large part of set theory

Some new results on decidability for elementary algebra and geometry

We carry out a systematic study of decidability for theories of (a) real vector spaces, inner product spaces, and Hilbert spaces and (b) normed spaces, Banach spaces and metric spaces, all formalised using a 2-sorted first-order language. The theories for list (a) turn out to be decidable while the theories for list (b) are not even arithmetical: the theory of 2-dimensional Banach spaces, for example, has the same many-one degree as the set of truths of second-order arithmetic. We find that the purely universal and purely existential fragments of the theory of normed spaces are decidable, as is the AE fragment of the theory of metric spaces. These results are sharp of their type: reductions of Hilbert's 10th problem show that the EA fragments for metric and normed spaces and the AE fragment for normed spaces are all undecidable.Comment: 79 pages, 9 figures. v2: Numerous minor improvements; neater proofs of Theorems 8 and 29; v3: fixed subscripts in proof of Lemma 3

The modal logic of arithmetic potentialism and the universal algorithm

I investigate the modal commitments of various conceptions of the philosophy of arithmetic potentialism. Specifically, I consider the natural potentialist systems arising from the models of arithmetic under their natural extension concepts, such as end-extensions, arbitrary extensions, conservative extensions and more. In these potentialist systems, I show, the propositional modal assertions that are valid with respect to all arithmetic assertions with parameters are exactly the assertions of S4. With respect to sentences, however, the validities of a model lie between S4 and S5, and these bounds are sharp in that there are models realizing both endpoints. For a model of arithmetic to validate S5 is precisely to fulfill the arithmetic maximality principle, which asserts that every possibly necessary statement is already true, and these models are equivalently characterized as those satisfying a maximal $\Sigma_1$ theory. The main S4 analysis makes fundamental use of the universal algorithm, of which this article provides a simplified, self-contained account. The paper concludes with a discussion of how the philosophical differences of several fundamentally different potentialist attitudes---linear inevitability, convergent potentialism and radical branching possibility---are expressed by their corresponding potentialist modal validities.Comment: 38 pages. Inquiries and commentary can be made at http://jdh.hamkins.org/arithmetic-potentialism-and-the-universal-algorithm. Version v3 has further minor revisions, including additional reference