Results 1  10
of
13
Combined Maximality Principles up to Large Cardinals
, 2008
"... The motivation for this paper is the following: In [Fuc08] I showed that it is inconsistent with ZFC that the maximality principle for closed forcings holds at unboundedly many regular cardinals κ (even only allowing κ itself as a parameter in the maximality principle for <κclosed forcings each tim ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
The motivation for this paper is the following: In [Fuc08] I showed that it is inconsistent with ZFC that the maximality principle for closed forcings holds at unboundedly many regular cardinals κ (even only allowing κ itself as a parameter in the maximality principle for <κclosed forcings each time). So the question is whether it is consistent to have this principle at unboundedly many regular cardinals or at every regular cardinal below some large cardinal κ (instead of ∞), and if so, how strong it is. It turns out that it is consistent in many cases, but the consistency strength is quite high. As a byproduct, assuming the consistency of a supercompact cardinal, I show that it is consistent that the least weakly compact cardinal is indestructible. 1
CATEGORIES OF COMPONENTS AND LOOPFREE CATEGORIES
"... Abstract. Given a groupoid G one has, in addition to the equivalence of categories E from G to its skeleton, a fibration F (Definition 1.11) from G to its set of connected components (seen as a discrete category). From the observation that E and F differ unless G[x, x] = {idx} for every object x of ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Abstract. Given a groupoid G one has, in addition to the equivalence of categories E from G to its skeleton, a fibration F (Definition 1.11) from G to its set of connected components (seen as a discrete category). From the observation that E and F differ unless G[x, x] = {idx} for every object x of G, we prove there is a fibered equivalence (Definition 1.12) from C[Σ1] (Proposition 1.1) to C/Σ (Proposition 1.8) when Σ is a
The Complexity of Quickly ORMDecidable Sets
"... Abstract. The Ordinal Register Machine (ORM) is one of several different machine models for infinitary computability. We classify, by complexity, the sets that can be decided quickly by ORMs. In particular, we show that the arithmetical sets are exactly those sets that can be decided by ORMs in time ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Abstract. The Ordinal Register Machine (ORM) is one of several different machine models for infinitary computability. We classify, by complexity, the sets that can be decided quickly by ORMs. In particular, we show that the arithmetical sets are exactly those sets that can be decided by ORMs in times uniformly less than ω ω. Further, we show that the hyperarithmetical sets are exactly those sets that can be decided by an ORM in time uniformly less than ω CK 1 Key words: Ordinal, ordinal computation, infinite time computation,
On topological models of GLP
, 2009
"... We develop topological semantics of a polymodal provability logic GLP. Our main result states that the bimodal fragment of GLP, although incomplete with respect to relational semantics, is topologically complete. The topological (in)completeness of GLP remains an interesting open problem. 1 ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
We develop topological semantics of a polymodal provability logic GLP. Our main result states that the bimodal fragment of GLP, although incomplete with respect to relational semantics, is topologically complete. The topological (in)completeness of GLP remains an interesting open problem. 1
VISUALIZATION OF ORDINALS ∗
"... We describe the pictorial representations of infinite ordinals used in teaching set theory, and discuss a possible use in naturalistic foundations of mathematics. 1 ..."
Abstract
 Add to MetaCart
We describe the pictorial representations of infinite ordinals used in teaching set theory, and discuss a possible use in naturalistic foundations of mathematics. 1
PreOrders
"... This is a theory of preorders, i.e. reflexive transitive relations, obtained in the first instance by taking the theory Orderings (which means partial orders), removing the antisymmetry axiom, defining a notion of equivalence and substituting equivalence for equality in the conclusions of theorems. ..."
Abstract
 Add to MetaCart
This is a theory of preorders, i.e. reflexive transitive relations, obtained in the first instance by taking the theory Orderings (which means partial orders), removing the antisymmetry axiom, defining a notion of equivalence and substituting equivalence for equality in the conclusions of theorems. Also covers linear preorders.
1.1 Some History........................... 3 1.2 Notation and Terminology.................... 4
"... A model for a set theory with a universal set and its use in the ..."
Membership Structures
"... A exploration of a way of doing set theory in HOL without using axioms. I thought that locales in IsabelleHOL would make this kind of approach workable, and this document is the result of my investigating this hope. However, it quickly became apparent that the limitations on the implementation of l ..."
Abstract
 Add to MetaCart
A exploration of a way of doing set theory in HOL without using axioms. I thought that locales in IsabelleHOL would make this kind of approach workable, and this document is the result of my investigating this hope. However, it quickly became apparent that the limitations on the implementation of locales. particularly in relation to the forms of definition which you can use in a locale, were too severe. Consequently I don’t get very far. I had hoped to do the construction of the PolySets on this kind of set theory, instead of using an axiomatisation of set
Ordinal Completeness of Bimodal Provability Logic GLB Dedicated to Leo Esakia on the occasion of his 75th birthday
"... Abstract. Bimodal provability logic GLB, introduced by G. Japaridze, currently plays an important role in the applications of provability logic to prooftheoretic analysis. Its topological semantics interprets diamond modalities as derived set operators on a scattered bitopological space. We study t ..."
Abstract
 Add to MetaCart
Abstract. Bimodal provability logic GLB, introduced by G. Japaridze, currently plays an important role in the applications of provability logic to prooftheoretic analysis. Its topological semantics interprets diamond modalities as derived set operators on a scattered bitopological space. We study the question of completeness of this logic w.r.t. the most natural space of this kind, that is, w.r.t. an ordinal α equipped with the interval topology and with the socalled club topology. We show that, assuming the axiom of constructibility, GLB is complete for any α ≥ ℵω. On the other hand, from the results of A. Blass it follows that, assuming the consistency of “there is a Mahlo cardinal, ” it is consistent with ZFC that GLB is incomplete w.r.t. any such space. Thus, the question of completeness of GLB w.r.t. natural ordinal spaces turns out to be independent of ZFC. 1