ORCID Profile
0000-0003-3734-0933
Current Organisation
University of Leeds
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Publisher: Springer Science and Business Media LLC
Date: 12-02-2013
Publisher: Elsevier BV
Date: 05-2009
Publisher: Cambridge University Press (CUP)
Date: 06-2009
Abstract: We use a reverse Easton forcing iteration to obtain a universe with a definable well-order, while preserving the GCH and proper classes of a variety of very large cardinals. This is achieved by coding using the principle , at a proper class of cardinals κ . By choosing the cardinals at which coding occurs sufficiently sparsely, we are able to lift the embeddings witnessing the large cardinal properties without having to meet any non-trivial master conditions.
Publisher: American Mathematical Society (AMS)
Date: 18-11-2017
DOI: 10.1090/PROC/13190
Abstract: We extend and improve the result of Makkai and Paré (1989) that the powerful image of any accessible functor F F is accessible, assuming there exists a sufficiently large strongly compact cardinal. We reduce the required large cardinal assumption to the existence of L μ , ω L_{\\mu ,\\omega } -compact cardinals for sufficiently large μ \\mu , and also show that under this assumption the λ \\lambda -pure powerful image of F F is accessible. From the first of these statements, we obtain that the tameness of every Abstract Elementary Class follows from a weaker large cardinal assumption than was previously known. We provide two ways of employing the large cardinal assumption to prove each result — one by a direct ultraproduct construction and one using the machinery of elementary embeddings of the set-theoretic universe.
Publisher: Cambridge University Press (CUP)
Date: 30-10-2020
DOI: 10.1017/S1446788719000399
Abstract: We show that the isomorphism problems for left distributive algebras, racks, quandles and kei are as complex as possible in the sense of Borel reducibility. These algebraic structures are important for their connections with the theory of knots, links and braids. In particular, Joyce showed that a quandle can be associated with any knot, and this serves as a complete invariant for tame knots. However, such a classification of tame knots heuristically seemed to be unsatisfactory, due to the apparent difficulty of the quandle isomorphism problem. Our result confirms this view, showing that, from a set-theoretic perspective, classifying tame knots by quandles replaces one problem with (a special case of) a much harder problem.
Publisher: Elsevier BV
Date: 08-2017
Publisher: Cambridge University Press (CUP)
Date: 06-2013
DOI: 10.2178/JSL.7802120
Abstract: We give a sharper version of a theorem of Rosický, Trnková and Adámek [13], and a new proof of a theorem of Rosický [12], both about colimits in categories of structures. Unlike the original proofs, which use category-theoretic methods, we use set-theoretic arguments involving elementary embeddings given by large cardinals such as α-strongly compact and C( n ) -extendible cardinals.
Publisher: Springer Science and Business Media LLC
Date: 28-01-2011
Publisher: Elsevier BV
Date: 2017
Publisher: Wiley
Date: 10-2016
Location: United Kingdom of Great Britain and Northern Ireland
Location: United States of America
Location: United Kingdom of Great Britain and Northern Ireland
Start Date: 2013
End Date: 2016
Funder: Engineering and Physical Sciences Research Council
View Funded ActivityStart Date: 2016
End Date: 2018
Funder: Engineering and Physical Sciences Research Council
View Funded Activity