Model Theoretic Complexity of Automatic Structures

We study the complexity of automatic structures via well-established concepts from both logic and model theory, including ordinal heights (of well-founded relations), Scott ranks of structures, and Cantor-Bendixson ranks (of trees). We prove the following results: 1) The ordinal height of any automatic well- founded partial order is bounded by \omega^\omega ; 2) The ordinal heights of automatic well-founded relations are unbounded below the first non-computable ordinal; 3) For any computable ordinal there is an automatic structure of Scott rank at least that ordinal. Moreover, there are automatic structures of Scott rank the first non-computable ordinal and its successor; 4) For any computable ordinal, there is an automatic successor tree of Cantor-Bendixson rank that ordinal.Comment: 23 pages. Extended abstract appeared in Proceedings of TAMC '08, LNCS 4978 pp 514-52

On the structure of classical realizability models of ZF

The technique of "classical realizability" is an extension of the method of "forcing"; it permits to extend the Curry-Howard correspondence between proofs and programs, to Zermelo-Fraenkel set theory and to build new models of ZF, called "realizability models". The structure of these models is, in general, much more complicated than that of the particular case of "forcing models". We show here that the class of constructible sets of any realizability model is an elementary extension of the constructibles of the ground model (a trivial fact in the case of forcing, since these classes are identical). It follows that Shoenfield absoluteness theorem applies to realizability models.Comment: 17 page

DEVELOPMENT OF EUROPEAN UNION AND JOINING PERSPECTIVE OF CROATIA

In this paper we researched European Union starting with the Agreement from Maastrich from year 1992, even though the European Union has a long traditional history and its origin is founded on regulations of economical integrations in Europe beginning from the 1950’s through the Roman treaty from year 1957 and the forming of the European Union Committee in year 1965. Further we follow her expansion and introduction of the European economic and monetary policy, to last, the joining perspective of Croatia. According to the Agreement from Maastrich, European Union lies on three posts: 1) Legal-political and regulative post, 2) Economical post, where the forming of European economical and monetary policy is in the first plan, especially the introducing of Euro as the unique European currency, 3) Post of Mutual foreign security policy within European Union. In that context we need to highlight the research conducted here and in European Union, including the world, regarding development of European Union and its economical, legal, political and cultural, as well as foreign diplomatic results, which are all perspectives of European Union. All the scientists and researches which were involved in exploring the development of EU with its modern tendencies and development perspective, agree that extraordinary results are achieved regards to economical, legal, political, foreign-security and diplomatic views, even tough many repercussions exist in progress of some particular members and within the EU as a whole. The biggest controversy arises in the perspective and expanding of European Union regarding ratification of the Constitution of EU from particular country members, but especially after the referendum was refused from two European countries, France and Netherlands. According to some estimates, the Constitution of EU would have difficulty to be adopted in Switzerland and some other Scandinavian countries, but also in Great Britain and other very developed countries. However the European Community and European Union were developing and expanding towards third European countries, regardless of Constitutional non-existence, where we can assume that if and when the Constitution of EU will be ratified, the EU will further develop as one of the most modern communities. This will enable economical development, especially development of European business, unique European market and free trade of goods and services, market of financial capital and labour market in free movement of labour. Being that EU has become one of the most largest dominating markets in the world, it offers a possibility to all new members to divide labour by using modern knowledge and high technology which insure economical, social and political prosperity. This results to forming a society of European countries which will guarantee all rights and freedom of development for all nations and ethnic groups. As well as, all European countries with somewhat less sovereignty, but in international relations will be stronger and significant, not only in sense of economics, but also in politics and military diplomatic relations. Therefore, Croatia has no choice and perspective if she does not join the European Union till year 2010, but until than it needs to create its strategy of economical and scientific-technological development, including demographic development, which will insure equal progress of Croatia as an equal member of European Union

Every countable model of set theory embeds into its own constructible universe

The main theorem of this article is that every countable model of set theory M, including every well-founded model, is isomorphic to a submodel of its own constructible universe. In other words, there is an embedding $j:M\to L^M$ that is elementary for quantifier-free assertions. The proof uses universal digraph combinatorics, including an acyclic version of the countable random digraph, which I call the countable random Q-graded digraph, and higher analogues arising as uncountable Fraisse limits, leading to the hypnagogic digraph, a set-homogeneous, class-universal, surreal-numbers-graded acyclic class digraph, closely connected with the surreal numbers. The proof shows that $L^M$ contains a submodel that is a universal acyclic digraph of rank $Ord^M$. The method of proof also establishes that the countable models of set theory are linearly pre-ordered by embeddability: for any two countable models of set theory, one of them is isomorphic to a submodel of the other. Indeed, they are pre-well-ordered by embedability in order-type exactly $\omega_1+1$. Specifically, the countable well-founded models are ordered by embeddability in accordance with the heights of their ordinals; every shorter model embeds into every taller model; every model of set theory $M$ is universal for all countable well-founded binary relations of rank at most $Ord^M$; and every ill-founded model of set theory is universal for all countable acyclic binary relations. Finally, strengthening a classical theorem of Ressayre, the same proof method shows that if $M$ is any nonstandard model of PA, then every countable model of set theory---in particular, every model of ZFC---is isomorphic to a submodel of the hereditarily finite sets $HF^M$ of $M$. Indeed, $HF^M$ is universal for all countable acyclic binary relations.Comment: 25 pages, 2 figures. Questions and commentary can be made at http://jdh.hamkins.org/every-model-embeds-into-own-constructible-universe. (v2 adds a reference and makes minor corrections) (v3 includes further changes, and removes the previous theorem 15, which was incorrect.

The EU's Normative Role: The Use of Political Conditionality in Relations with Cuba, China and Zimbabwe

The European Union (EU) is a unique foreign policy actor, because of its history and the diversity of its member states. The EU has developed a distinct approach to promoting human rights and democracy in third states. Its focus lies on positive engagement through constructive dialogue. This preference for using positive rather than negative engagement is the subject of this thesis. It is argued that it is necessary for the EU to judge and conducts its relations with each individual state on a case by case basis. Case specific considerations need to be taken into account as it provides the foundation on which the relationship is based on and continues to shape the negotiations and any use of political conditionality throughout the relationship. This does not diminish the EU¨s normative basis that continues to be well founded in the international human rights treaties. This thesis proceeds to analyse three distinct cases by focusing on the evolution of the Unions relationship with the Peoples Republic of China, Cuba and Zimbabwe. They demonstrate the fragile nature of political dialogue regarding human rights and democratisation and underline the need to constructively engage the recipient state