On Games on Non-Wellfounded Sets and Stationary Sets

In this thesis we study a few games related to non-wellfounded and stationary sets. Games have turned out to be an important tool in mathematical logic ranging from semantic games defining the truth of a sentence in a given logic to for example games on real numbers whose determinacies have important effects on the consistency of certain large cardinal assumptions. The equality of non-wellfounded sets can be determined by a so called bisimulation game already used to identify processes in theoretical computer science and possible world models for modal logic. Here we present a game to classify non-wellfounded sets according to their branching structure. We also study games on stationary sets moving back to classical wellfounded set theory. We also describe a way to approximate non-wellfounded sets with hereditarily finite wellfounded sets. The framework used to do this is domain theory. In the Banach-Mazur game, also called the ideal game, the players play a descending sequence of stationary sets and the second player tries to keep their intersection stationary. The game is connected to precipitousness of the corresponding ideal. In the pressing down game first player plays regressive functions defined on stationary sets and the second player responds with a stationary set where the function is constant trying to keep the intersection stationary. This game has applications in model theory to the determinacy of the Ehrenfeucht-Fraisse game. We show that it is consistent that these games are not equivalent.

Aspects of Arminian Soteriology in Methodist-Lutheran Ecumenical Dialogues in 20th and 21st Century

The aim of this thesis is to analyse the key ecumenical dialogues between Methodists and Lutherans from the perspective of Arminian soteriology and Methodist theology in general. The primary research question is defined as: "To what extent do the dialogues under analysis relate to Arminian soteriology?" By seeking an answer to this question, new knowledge is sought on the current soteriological position of the Methodist-Lutheran dialogues, the contemporary Methodist theology and the commonalities between the Lutheran and Arminian understanding of soteriology. This way the soteriological picture of the Methodist-Lutheran discussions is clarified. The dialogues under analysis were selected on the basis of versatility. Firstly, the sole world organisation level dialogue was chosen: The Church – Community of Grace. Additionally, the document World Methodist Council and the Joint Declaration on the Doctrine of Justification is analysed as a supporting document. Secondly, a document concerning the discussions between two main-line churches in the United States of America was selected: Confessing Our Faith Together. Thirdly, two dialogues between non-main-line Methodist churches and main-line Lutheran national churches in Europe were chosen: Fellowship of Grace from Norway and Kristuksesta osalliset from Finland. The theoretical approach to the research conducted in this thesis is systematic analysis. The Remonstrant articles of Arminian soteriology are utilised as an analysis tool to examine the soteriological positions of the dialogues. New knowledge is sought by analysing the stances of the dialogues concerning the doctrines of partial depravity, conditional election, universal atonement, resistible grace and conditional perseverance of saints. This way information is also provided for approaching the Calvinist-Arminian controversy from new perspectives. The results of this thesis show that the current soteriological position of the Methodist-Lutheran dialogues is closer to Arminianism than Calvinism. The dialogues relate to Arminian soteriology especially concerning the doctrines of universal atonement, resistible grace and conditional perseverance of saints. The commonalities between the Lutheran and Arminian understanding of soteriology exist mainly in these three doctrines as they are uniformly favoured in the dialogues. The most discussed area of soteriology is human depravity, in which the largest diversity of stances occurs as well. On the other hand, divine election is the least discussed topic. The overall perspective, which the results of the analysis provide, indicates that the Lutherans could approach the Calvinist churches together with the Methodists with a wider theological perspective and understanding when the soteriological issues are considered as principal. Human depravity is discovered as the area of soteriology which requires most work in future ecumenical dialogues. However, the detected Lutheran hybrid notion on depravity (a Calvinist-Arminian mixture) appears to provide a useful new perspective for Calvinist-Arminian ecumenism and offers potentially fruitful considerations to future ecumenical dialogues.