For an ordinal \, I introduce a variant of the notion of subcompleteness of a forcing poset, which I call \-subcompleteness, and show that this class of forcings enjoys some closure properties that the original class of subcomplete forcings does not seem to have: factors of \-subcomplete forcings are \-subcomplete, and if \ and \ are forcing-equivalent notions, then \ is \-subcomplete iff \ is. I formulate a Two Step Theorem for \-subcompleteness and prove an RCS iteration theorem for \-subcompleteness (...) which is slightly less restrictive than the original one, in that its formulation is more careful about the amount of collapsing necessary. Finally, I show that an adequate degree of \-subcompleteness follows from the \-distributivity of a forcing, for \. (shrink)

I analyze the hierarchies of the bounded and the weak bounded forcing axioms, with a focus on their versions for the class of subcomplete forcings, in terms of implications and consistency strengths. For the weak hierarchy, I provide level-by-level equiconsistencies with an appropriate hierarchy of partially remarkable cardinals. I also show that the subcomplete forcing axiom implies Larson’s ordinal reflection principle atω2, and that its effect on the failure of weak squares is very similar to that of Martin’s Maximum.

This volume includes essays presented at the conference on Emotions and Rationality in Moral Philosophy held at the Universities of Neuchâtel and Bern in October 2005. The authors of this volume share the Humean insight that the ‘sentiments’ have a crucial role to play in elucidating the practice of morality. In a Humean fashion, they warn us against taking an intellectualist view of emotions and reject the rationalist account of morality.

We analyze the notions of indiscernibility and indeterminacy in the light of the Galois theory of field extensions and the generalization to \(K\) -algebras proposed by Grothendieck. Grothendieck’s reformulation of Galois theory permits to recast the Galois correspondence between symmetry groups and invariants as a Galois–Grothendieck duality between \(G\) -spaces and the minimal observable algebras that discern (or separate) their points. According to the natural epistemic interpretation of the original Galois theory, the possible \(K\) -indiscernibilities between the roots of a (...) polynomial \(p(x)\in K[x]\) result from the limitations of the field \(K\) . We discuss the relation between this epistemic interpretation of the Galois–Grothendieck duality and Leibniz’s principle of the identity of indiscernibles. We then use the conceptual framework provided by Klein’s Erlangen program to propose an alternative ontologic interpretation of this duality. The Galoisian symmetries are now interpreted in terms of the automorphisms of the symmetric geometric figures that can be placed in a background Klein geometry. According to this interpretation, the Galois–Grothendieck duality encodes the compatibility condition between geometric figures endowed with groups of automorphisms and the ‘observables’ that can be consistently evaluated at such figures. In this conceptual framework, the Galoisian symmetries do not encode the epistemic indiscernibility between individuals, but rather the intrinsic indeterminacy in the pointwise localization of the figures with respect to the background Klein geometry. (shrink)

Name der Zeitschrift: Jahrbuch für Wissenschaft und Ethik Jahrgang: 18 Heft: 1 Seiten: 125-132.

v. 1. Úvod do filosofie. 1. Filosofická logika.

We extend the results of Hamkins and Thomas concerning the malleability of automorphism tower heights of groups by forcing. We show that any reasonable sequence of ordinals can be realized as the automorphism tower heights of a certain group in consecutive forcing extensions or ground models, as desired. For example, it is possible to increase the height of the automorphism tower by passing to a forcing extension, then increase it further by passing to a ground model, and then decrease it (...) by passing to a further forcing extension, and so on, transfinitely. We make sense of the limit models occurring in such a sequence of models. At limit stages, the automorphism tower height will always be 1. (shrink)

(1997). Information—neither matter nor mind—on the essence and on the evolutionary stage conception of information. World Futures: Vol. 50, No. 1-4, pp. 551-570.

We analyze the notion of indiscernibility in the light of the Galois theory of field extensions and the generalization to K-algebras proposed by Grothendieck. Grothendieck's reformulation of Galois theory permits to recast the Galois correspondence between symmetry groups and invariants as a duality between G-spaces and the minimal observable algebras that separate theirs points. In order to address the Galoisian notion of indiscernibility, we propose what we call an epistemic reading of the Galois-Grothendieck theory. According to this viewpoint, the Galoisian (...) notion of indiscernibility results from the limitations of the `resolving power' of the observable algebras used to discern the corresponding `coarse-grained' states. The resulting Galois-Grothendieck duality is rephrased in the form of what we call a Galois indiscernibility principle. According to this principle, there exists an inverse correlation between the coarsegrainedness of the states and the size of the minimal observable algebra that discern these states. (shrink)