Measurable cardinals and good Σ1(κ)\Sigma_1(\kappa)-wellorderings

We study the influence of the existence of large cardinals on the existence of wellorderings of power sets of infinite cardinals $\kappa$ with the property that the collection of all initial segments of the wellordering is definable by a $\Sigma_1$-formula with parameter $\kappa$. A short argument shows that the existence of a measurable cardinal $\delta$ implies that such wellorderings do not exist at $\delta$-inaccessible cardinals of cofinality not equal to $\delta$ and their successors. In contrast, our main result shows that these wellorderings exist at all other uncountable cardinals in the minimal model containing a measurable cardinal. In addition, we show that measurability is the smallest large cardinal property that interferes with the existence of such wellorderings at uncountable cardinals and we generalize the above result to the minimal model containing two measurable cardinals.Comment: 14 page

An integer construction of infinitesimals: Toward a theory of Eudoxus hyperreals

A construction of the real number system based on almost homomorphisms of the integers Z was proposed by Schanuel, Arthan, and others. We combine such a construction with the ultrapower or limit ultrapower construction, to construct the hyperreals out of integers. In fact, any hyperreal field, whose universe is a set, can be obtained by such a one-step construction directly out of integers. Even the maximal (i.e., On-saturated) hyperreal number system described by Kanovei and Reeken (2004) and independently by Ehrlich (2012) can be obtained in this fashion, albeit not in NBG. In NBG, it can be obtained via a one-step construction by means of a definable ultrapower (modulo a suitable definable class ultrafilter).Comment: 17 pages, 1 figur

Model Theory for a Compact Cardinal

We would like to develop model theory for T, a complete theory in L_{theta,theta}(tau) when theta is a compact cardinal. We already have bare bones stability theory and it seemed we can go no further. Dealing with ultrapowers (and ultraproducts) naturally we restrict ourselves to "D a theta-complete ultrafilter on I, probably (I,theta)-regular". The basic theorems of model theory work and can be generalized (like Los theorem), but can we generalize deeper parts of model theory? The first section tries to sort out what occurs to the notion of stable T for complete L_{theta,theta}-theories T. We generalize several properties of complete first order T, equivalent to being stable (see [Sh:c]) and find out which implications hold and which fail. In particular, can we generalize stability enough to generalize [Sh:c, Ch. VI]? Let us concentrate on saturation in the local sense (types consisting of instances of one formula). We prove that at least we can characterize the T's (of cardinality < theta for simplicity) which are minimal for appropriate cardinal lambda > 2^kappa +|T| in each of the following two senses. One is generalizing Keisler order which measures how saturated are ultrapowers. Another asks: Is there an L_{theta,theta}-theory T_1 supseteq T of cardinality |T| + 2^theta such that for every model M_1 of T_1 of cardinality > lambda, the tau(T)-reduct M of M_1 is lambda^+-saturated. Moreover, the two versions of stable used in the characterization are different

Asymptotically Hilbertian Modular Banach Spaces: Examples of Uncountable Categoricity

We give a criterion ensuring that the elementary class of a modular Banach space E (that is, the class of Banach spaces, some ultrapower of which is linearly isometric to an ultrapower of E) consists of all direct sums E\oplus_m H, where H is an arbitrary Hilbert space and \oplus_m denotes the modular direct sum. Also, we give several families of examples in the class of Nakano direct sums of finite dimensional normed spaces that satisfy this criterion. This yields many new examples of uncountably categorical Banach spaces, in the model theory of Banach space structures.Comment: 20 page

Inner Model from the Cofinality Quantifier

This thesis discusses the inner model obtained from the cofinality quantifier introduced in the paper Inner Models From Extended Logics: Part 1 by Juliette Kennedy, Menachem Magidor and Jouko Väänänen, to appear in the Journal of Mathematical Logic. The paper is a contribution to inner model theory, presenting many different inner models obtained by replacing first order logic by extended logics in the definition of the constructible hierarchy. We will focus on the model C* obtained from the logic that extends first order logic by the cofinality quantifier for \omega. The goal of this thesis is to present two major theorems of the paper and the theory that is needed to understand their proofs. The first theorem states that the Dodd-Jensen core model of V is contained in C*. The second theorem, the Main Theorem of the thesis, is a characterization of C* assuming V = L[U]. Chapters 2-5 present the theory needed to understand the proofs. Our presentation in these chapters mostly follows standard sources but we present the proofs of many lemmas in much greater detail than our source material. Chapter 2 presents the basics of iterated ultrapowers. If a model M of ZFC^- satisfies “U is a normal ultrafilter on \kappa” for some ordinal \kappa, then we can construct its ultrapower by U. We can take the ultrapower of the resulting model M1 and then continue taking ultrapowers at successor ordinals and direct limits at limit ordinals. If the constructed iterated ultrapowers M_\alpha are well-founded for all ordinals \alpha, the model M is called iterable. Chapter 3 presents L[A], the class of sets constructible relative to a set or class A. The hierarchy L_\alpha[A] is a generalization of the constructible hierachy L_\alpha. The difference is that the formulas defining the successor level L(\alpha+1)[A] can use A \ L_\alpha[A] as a unary predicate. The Main Theorem uses the model L[U], where U is a normal measure on some cardinal \kappa. Chapter 4 presents the basics of Prikry forcing, a notion of forcing defined from a measurable cardinal. The sequence of critical points of the iterable ultrapowers of L[U] generates a generic set for the Prikry forcing defined from the critical point of the \omega-th iterated ultrapower. Chapter 5 presents the theory of the Dodd-Jensen core model which is an important inner model. The core model is based on the Jensen hierarchy J_\alpha^A which produces L[A] as the union of all levels. The theory is concerned with so called premice which are levels of the J-hierarchy J_\alpha^U satisfying “U is a normal ultrafilter on \kappa” for some ordinal \kappa. A mouse is a premouse satisfying some specific properties and the core model K is the union of all mice. The last chapter presents the approach of the paper in detail. We present the definition of C(L*), the class of sets constructible using an extended logic L*, and the exact definition of C*. Then we present the proofs of the two major theorems mentioned above. The chapter naturally follows the paper but presents the proofs in greater detail and adds references to lemmas in the previous chapters that are needed for the arguments in the proofs