On colimits and elementary embeddings

We give a sharper version of a theorem of Rosicky, Trnkova and Adamek, and a new proof of a theorem of Rosicky, both about colimit preservation between 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 alpha-strongly compact and C^(n)-extendible cardinals.Comment: 17 page

The infinite random simplicial complex

We study the Fraisse limit of the class of all finite simplicial complexes. Whilst the natural model-theoretic setting for this class uses an infinite language, a range of results associated with Fraisse limits of structures for finite languages carry across to this important example. We introduce the notion of a local class, with the class of finite simplicial complexes as an archetypal example, and in this general context prove the existence of a 0-1 law and other basic model-theoretic results. Constraining to the case where all relations are symmetric, we show that every direct limit of finite groups, and every metrizable profinite group, appears as a subgroup of the automorphism group of the Fraisse limit. Finally, for the specific case of simplicial complexes, we show that the geometric realisation is topologically surprisingly simple: despite the combinatorial complexity of the Fraisse limit, its geometric realisation is homeomorphic to the infinite simplex.Comment: 33 page

Subcompact cardinals, squares, and stationary reflection

We generalise Jensen's result on the incompatibility of subcompactness with square. We show that alpha^+-subcompactness of some cardinal less than or equal to alpha precludes square_alpha, but also that square may be forced to hold everywhere where this obstruction is not present. The forcing also preserves other strong large cardinals. Similar results are also given for stationary reflection, with a corresponding strengthening of the large cardinal assumption involved. Finally, we refine the analysis by considering Schimmerling's hierarchy of weak squares, showing which cases are precluded by alpha^+-subcompactness, and again we demonstrate the optimality of our results by forcing the strongest possible squares under these restrictions to hold.Comment: 18 pages. Corrections and improvements from referee's report mad

The formal failure and social success of logic

Is formal logic a failure? It may be, if we accept the context-independent limits imposed by Russell, Frege, and others. In response to difficulties arising from such limitations I present a Toulmin-esque social recontextualization of formal logic. The results of my project provide a positive view of formal logic as a success while simultaneously reaffirming the social and contextual concerns of argumentation theorists, critical thinking scholars, and rhetoricians

Small u_kappa and large 2^kappa for supercompact kappa

In a recent preprint, Garti and Shelah state that the techniques of a paper of Dzamonja and Shelah can be used to force u_kappa to be kappa^+ for supercompact kappa with 2^kappa arbitrarily large. In this expository article we spell out the details of how this can be done.Comment: Expository article going through a specific application of the technique introduced in arXiv:math/0102043. 11 page

Parallel functional differentiation of an invasive annual plant on two continents.

Rapid local adaptation frequently occurs during the spread of invading species. It remains unclear, however, how consistent, and therefore potentially predictable, such patterns of local adaptation are. One approach to this question is to measure patterns of local differentiation in functional traits and plasticity levels in invasive species in multiple regions. Finding consistent patterns of local differentiation in replicate regions suggests that these patterns are adaptive. Further, this outcome indicates that the invading species likely responds predictably to selection along environmental gradients, even though standing genetic variation is likely to have been reduced during introduction. We studied local differentiation in the invasive annual plant Erodium cicutarium in two invaded regions, California and Chile. We collected seeds from across strong gradients in precipitation and temperature in Mediterranean-climate parts of the two regions (10 populations per region). We grew seeds from maternal families from these populations through two generations and exposed the second generation to contrasting levels of water and nutrient availability. We measured growth, flowering time and leaf functional traits across these treatments to obtain trait means and plasticity measures. We found strong differentiation among populations in all traits. Plants from drier environments flowered earlier, were less plastic in flowering time and reached greater size in all treatments. Correlations among traits within regions suggested a coordinated evolutionary response along environmental gradients associated with growing season length. There was little divergence in traits and trait intercorrelations between regions, but strongly parallel divergence in traits within regions. Similar, statistically consistent patterns of local trait differentiation across two regions suggest that local adaptation to environmental gradients has aided the spread of this invasive species, and that the formation of ecotypes in newly invaded environments has been relatively consistent and predictable