A five element basis for the uncountable linear orders

In this paper I will show that it is relatively consistent with the usual axioms of mathematics (ZFC) together with a strong form of the axiom of infinity (the existence of a supercompact cardinal) that the class of uncountable linear orders has a five element basis. In fact such a basis follows from the Proper Forcing Axiom, a strong form of the Baire Category Theorem. The elements are X, omega_1, omega_1^*, C, C^* where X is any suborder of the reals of cardinality aleph_1 and C is any Countryman line. This confirms a longstanding conjecture of Shelah.Comment: 21 page

More minimal non-σ\sigma-scattered linear orders

Assuming an instance of the Brodsky-Rinot proxy principle holding at a regular uncountable cardinal $\kappa$, we construct $2^\kappa$-many pairwise non-embeddable minimal non-$\sigma$-scattered linear orders of size $\kappa$. In particular, in G\"odel's constructible universe $L$, these linear orders exist for any regular uncountable cardinal $\kappa$ that is not weakly compact. This extends a recent result of Cummings, Eisworth and Moore that takes care of all the successor cardinals of $L$. At the level of $\aleph_1$, their work answered an old question of Baumgartner by constructing from $\diamondsuit$ a minimal Aronszajn line that is not Souslin. Our use of the proxy principle yields the same conclusion from a weaker assumption which holds for instance in the generic extension after adding a single Cohen real to a model of $CH$.Comment: 22 pages, comments are welcom

Characterization and Transformation of Countryman Lines and R-Embeddable Coherent Trees in ZFC

Ph.DDOCTOR OF PHILOSOPH

On Nudity. An Introduction to Nonsense

none1noUna indagine filosofica della categoria del nonsense, come modo di disvelamento di un nuovo accesso ai testi. Ne esce il nudo, ciò che è già davanti agli occhi, nella prospettiva di decifrare "le cose come sono".openArena, Leonardo VittorioArena, LEONARDO VITTORI

Set Theory

This workshop included selected talks on pure set theory and its applications, simultaneously showing diversity and coherence of the subject

Gardner-Webb Review, Volume 2, 2000

Annual publication associated with the Life of the Scholar Multidisciplinary Conference. This publication presents selected essays written by undergraduate students at Gardner-Webb University

The oldest new woodland on earth: Recognising, mapping, naming and narrating the Great Western Woodlands

The Great Western Woodlands (GWW) cover an area of 160,000 km2 of largely intact semi-arid woodland in inland south-western Australia. The highly biodiverse GWW is a large-scale ecosystem and a refuge for native species endangered elsewhere, but faces many challenges, including poor fire management, mining and mining exploration impacts, proposed clearing for agriculture, introduced species and climate change. This paper traces the way in which stories about the region have powerfully shaped different groups' dealings with it. In Western Australia, settler society's long-standing focus on the agricultural zone of the Wheatbelt and the mineral wealth of the goldfields as 'productive' landscapes produced a dominant narrative about conquering nature, physical labour and economic wealth that marginalised the ecologies and First Peoples of the GWW. More recently, a network of local settler and Indigenous people, NGOs, scientists and conservationists have begun to produce a new narrative with the cultural and natural values of the woodland at its heart, as a foundation for better understanding, managing and protecting the GWW. Reflecting on the historical framing of a particular region reveals the important cultural-ecological work performed by regional narratives