Some topics in set theory

This thesis is divided into two parts. In the first of these we consider Ackermann-type set theories and many of our results concern natural models. We prove a number of results about the existence of natural models of Ackermann's set theory, A, and applications of this work are shown to answer several questions raised by Reinhardt in [56]. A+ (introduced in [56]) is another Ackermann-type set theory and we show that its set theoretic part is precisely ZF. Then we introduce the notion of natural models of A + and show how our results on natural models of A extend to these models. There are a number of results about other Ackermann-type set theories and some of the work which was already known for ZF is extended to A. This includes permutation models, which are shown to answer another of Reinhardt's questions. In the second part we consider the different approaches to set theory; dealing mainly with the more philosophical aspects. We reconsider Cantor's work, suggest that it has frequently been misunderstood and indicate how quasi-constructive set theories seem to use a definite part of Cantor's earlier ideas. Other approaches to set theory are also considered and criticised. The section on NF includes some more technical observations on ordered pairs. There is also an appendix, in which we outline some results on extended ordinal arithmetic.<p

DYNAMIC MODELING AND CLOSED-LOOP CONTROL OF A TWIN ROTOR MIMO SYSTEM

The Twin rotor MIMO system (TRMS) is an aero-dynamical model of helicopter with significant cross-couplings between longitudinal and lateral directional motions. Its behavior in certain aspects resembles the real of a helicopter. Firstly, open loop control is implemented both for tail and main rotor to get the relationship of input and output of the system. Open-loop control is often the preliminary step for development of more complex feedback control laws. Next step was model identification as it is a well established technique for modeling of complex systems whose dynamics are not well understood or difficult to model from the first principles. State feedback controllers were designed by pole placement method for both rotors independently. The model then can be implemented in real-time experiments of the Twin Rotor MIMO System

DYNAMIC MODELING AND CLOSED-LOOP CONTROL OF A TWIN ROTOR MIMO SYSTEM

The Twin rotor MIMO system (TRMS) is an aero-dynamical model of helicopter with significant cross-couplings between longitudinal and lateral directional motions. Its behavior in certain aspects resembles the real of a helicopter. Firstly, open loop control is implemented both for tail and main rotor to get the relationship of input and output of the system. Open-loop control is often the preliminary step for development of more complex feedback control laws. Next step was model identification as it is a well established technique for modeling of complex systems whose dynamics are not well understood or difficult to model from the first principles. State feedback controllers were designed by pole placement method for both rotors independently. The model then can be implemented in real-time experiments of the Twin Rotor MIMO System

A system of axiomatic set theory - Part VII

The reader of Part VI will have noticed that among the set-theoretic models considered there some models were missing which were announced in Part II for certain proofs of independence. These models will be supplied now. Mainly two models have to be constructed: one with the property that there exists a set which is its own only element, and another in which the axioms I-III and VII, but not Va, are satisfied. In either case we need not satisfy the axiom of infinity. Thereby it becomes possible to set up the models on the basis of only I-III, and either VII or Va, a basis from which number theory can be obtained as we saw in Part II. On both these bases the Π0-system of Part VI, which satisfies the axioms I-V and VII, but not VI, can be constructed, as we stated there. An isomorphic model can also be obtained on that basis, by first setting up number theory as in Part II, and then proceeding as Ackermann did. Let us recall the main points of this procedure. For the sake of clarity in the discussion of this and the subsequent models, it will be necessary to distinguish precisely between the concepts which are relative to the basic set-theoretic system, and those which are relative to the model to be define

The search for new axioms

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Linguistics and Philosophy, 2003.Includes bibliographical references (p. 103-106).Abstract The independence results in set theory invite the search for new and justified axioms. In Chapter 1 I set the stage by examining three approaches to justifying the axioms of standard set theory (stage theory, G6del's approach, and reflection principles) and argue that the approach via reflection principles is the most successful. In Chapter 2 I analyse the limitations of ZF and use this analysis to set up a mathematically precise minimal hurdle which any set of new axioms must overcome if it is to effect a significant reduction in incompleteness. In Chapter 3 I examine the standard method of justifying new axioms-reflection principles-and prove a result which shows that no reflection principle (known to be consistent via large cardinals) can overcome the minimal hurdle and yield a significant reduction in incompleteness. In Chapter 4 I introduce a new approach to justifying new axioms-extension principles-and show that such principles can overcome the minimal hurdle and much more, in particular, such principles imply PD and that the theory of second-order arithmetic cannot be altered by set size forcing. I show that in a sense (which I make precise) these principles are inevitable. In Chapter 5 I close with a brief discussion of meta-mathematical justifications stemming from the work of Woodin. These touch on the continuum hypothesis and other questions which are beyond the reach of standard large cardinals.by Peter Koellner.Ph.D

Propositional logic extended with a pedagogically useful relevant implication

First and foremost, this paper concerns the combination of classical propositional logic with a relevant implication. The proposed combination is simple and transparent from a proof theoretic point of view and at the same time extremely useful for relating formal logic to natural language sentences. A specific system will be presented and studied, also from a semantic point of view. The last sections of the paper contain more general considerations on combining classical propositional logic with a relevant logic that has all classical theorems as theorems

A synthetic axiomatization of Map Theory

Includes TOC détaillée, index et appendicesInternational audienceThis paper presents a subtantially simplified axiomatization of Map Theory and proves the consistency of this axiomatization in ZFC under the assumption that there exists an inaccessible ordinal. Map Theory axiomatizes lambda calculus plus Hilbert's epsilon operator. All theorems of ZFC set theory including the axiom of foundation are provable in Map Theory, and if one omits Hilbert's epsilon operator from Map Theory then one is left with a computer programming language. Map Theory fulfills Church's original aim of introducing lambda calculus. Map Theory is suited for reasoning about classical mathematics as well ascomputer programs. Furthermore, Map Theory is suited for eliminating thebarrier between classical mathematics and computer science rather than just supporting the two fields side by side. Map Theory axiomatizes a universe of "maps", some of which are "wellfounded". The class of wellfounded maps in Map Theory corresponds to the universe of sets in ZFC. The first version MT0 of Map Theory had axioms which populated the class of wellfounded maps, much like the power set axiom et.al. populates the universe of ZFC. The new axiomatization MT of Map Theory is "synthetic" in the sense that the class of wellfounded maps is defined inside MapTheory rather than being introduced through axioms. In the paper we define the notion of kappa- and kappasigma-expansions and prove that if sigma is the smallest strongly inaccessible cardinal then canonical kappasigma expansions are models of MT (which proves the consistency). Furthermore, in the appendix, we prove that canonical omega-expansions are fully abstract models of the computational part of Map Theory

The Circle of Concerned African Women Theologians : HIV and the bible.

Thesis (M.A.)-University of KwaZulu-Natal, Pietermaritzburg, 2013.No abstract available