-
11A wellorder of the reals with saturatedJournal of Symbolic Logic 84 (4): 1466-1483. 2019.We show that, assuming the existence of the canonical inner model with one Woodin cardinal $M_1 $, there is a model of $ZFC$ in which the nonstationary ideal on $\omega _1 $ is $\aleph _2 $-saturated and whose reals admit a ${\rm{\Sigma }}_4^1 $-wellorder.
-
11An inner model for global dominationJournal of Symbolic Logic 74 (1): 251-264. 2009.In this paper it is shown that the global statement that the dominating number for k is less than $2^k $ for all regular k, is internally consistent, given the existence of $0^\# $ . The possible range of values for the dominating number for k and $2^k $ which may be simultaneously true in an inner model is also explored
-
10X1. Introduction. In 1938, K. Gödel defined the model L of set theory to show the relative consistency of Cantor's Continuum Hypothesis. L is defined as a union L= (review)Bulletin of Symbolic Logic 3 (4): 453-468. 1997.We present here an approach to the fine structure of L based solely on elementary model theoretic ideas, and illustrate its use in a proof of Global Square in L. We thereby avoid the Lévy hierarchy of formulas and the subtleties of master codes and projecta, introduced by Jensen [3] in the original form of the theory. Our theory could appropriately be called ”Hyperfine Structure Theory”, as we make use of a hierarchy of structures and hull operations which refines the traditional Lα -or Jα-seque…Read more
-
10Definability of Satisfaction in Outer ModelsIn Carolin Antos, Sy-David Friedman, Radek Honzik & Claudio Ternullo (eds.), The Hyperuniverse Project and Maximality, Birkhäuser. pp. 135-160. 2018.Let M be a transitive model of ZFC. We say that a transitive model of ZFC, N, is an outer model of M if M ⊆ N and ORD ∩ M = ORD ∩ N. The outer model theory of M is the collection of all formulas with parameters from M which hold in all outer models of M. Satisfaction defined with respect to outer models can be seen as a useful strengthening of first-order logic. Starting from an inaccessible cardinal κ, we show that it is consistent to have a transitive model M of ZFC of size κ in which the oute…Read more
-
10Generic absolutenessAnnals of Pure and Applied Logic 108 (1-3): 3-13. 2001.We explore the consistency strength of Σ31 and Σ41 absoluteness, for a variety of forcing notions.
-
10Structural Properties of the Stable CoreJournal of Symbolic Logic 88 (3): 889-918. 2023.The stable core, an inner model of the form $\langle L[S],\in, S\rangle $ for a simply definable predicate S, was introduced by the first author in [8], where he showed that V is a class forcing extension of its stable core. We study the structural properties of the stable core and its interactions with large cardinals. We show that the $\operatorname {GCH} $ can fail at all regular cardinals in the stable core, that the stable core can have a discrete proper class of measurable cardinals, but t…Read more
-
9The completeness of isomorphismIn Dieter Spreen, Hannes Diener & Vasco Brattka (eds.), Logic, Computation, Hierarchies, De Gruyter. pp. 157-164. 2014.
-
7Annual Meeting of the Association for Symbolic Logic, Durham, 1992Journal of Symbolic Logic 58 (1): 370-382. 1993.
-
7Explaining Maximality Through the Hyperuniverse ProgrammeIn Carolin Antos, Sy-David Friedman, Radek Honzik & Claudio Ternullo (eds.), The Hyperuniverse Project and Maximality, Birkhäuser. pp. 185-204. 2018.The iterative concept of set is standardly taken to justify ZFC and some of its extensions. In this paper, we show that the maximal iterative concept also lies behind a class of further maximality principles expressing the maximality of the universe of sets V in height and width. These principles have been heavily investigated by the first author and his collaborators within the Hyperuniverse Programme. The programme is based on two essential tools: the hyperuniverse, consisting of all countable…Read more
-
7The Search for New Axioms in the Hyperuniverse ProgrammeIn Carolin Antos, Sy-David Friedman, Radek Honzik & Claudio Ternullo (eds.), The Hyperuniverse Project and Maximality, Birkhäuser. pp. 161-183. 2018.The Hyperuniverse Programme, introduced in Arrigoni and Friedman :77–96, 2013), fosters the search for new set-theoretic axioms. In this paper, we present the procedure envisaged by the programme to find new axioms and the conceptual framework behind it. The procedure comes in several steps. Intrinsically motivated axioms are those statements which are suggested by the standard concept of set, i.e. the ‘maximal iterative concept’, and the programme identifies higher-order statements motivated by…Read more
-
6Internal consistency for embedding complexityJournal of Symbolic Logic 73 (3): 831-844. 2008.In a previous paper with M. Džamonja, class forcings were given which fixed the complexity (a universality covering number) for certain types of structures of size λ together with the value of 2λ for every regular λ. As part of a programme for examining when such global results can be true in an inner model, we build generics for these class forcings
-
6Annual Meeting of the Association for Symbolic LogicJournal of Symbolic Logic 49 (4): 1441-1449. 1984.
-
6The Search for New Axioms in the Hyperuniverse ProgrammeIn Francesca Boccuni & Andrea Sereni (eds.), Objectivity, Realism, and Proof. FilMat Studies in the Philosophy of Mathematics, Springer International Publishing. pp. 165-188. 2016.The Hyperuniverse Programme, introduced in Arrigoni and Friedman, fosters the search for new set-theoretic axioms. In this paper, we present the procedure envisaged by the programme to find new axioms and the conceptual framework behind it. The procedure comes in several steps. Intrinsically motivated axioms are those statements which are suggested by the standard concept of set, i.e. the ‘maximal iterative concept’, and the programme identifies higher-order statements motivated by the maximal i…Read more
-
4James E. Baumgartner. On the size of closed unbounded sets. Annals of pure and applied logic, vol. 54 , pp. 195–227 (review)Bulletin of Symbolic Logic 7 (4): 538-539. 2001.
-
1Theorem 1 (Easton's Theorem). There is a forcing extension L [G] of L in which GCH fails at every regular cardinal. Assume that the universe V of all sets is rich in the sense that it contains inner models with large cardinals. Then what is the relationship between Easton's model L [G] and V? In particular, are these models compatible (review)Bulletin of Symbolic Logic 12 (4). 2006.
-
Δ1-definabilityAnnals of Pure and Applied Logic 89 (1): 93-99. 1997.We isolate a condition on a class A of ordinals sufficient to Δ1-code it by a real in a class-generic extension of L. We then apply this condition to show that the class of ordinals of L-cofinality ω is Δ1 in a real of L-degree strictly below O#.
-
Generalizations of Gödel's universe of constructible setsIn Kurt Gödel, Solomon Feferman, Charles Parsons & Stephen G. Simpson (eds.), Kurt Gödel: Essays for His Centennial, Association For Symbolic Logic. 2010.