Acknowledgements I want to express my sincere gratitude to my supervisor Professor Jouko Väänänen for supporting me during all these years of getting acquainted with the intriguing field of set theory. I am also grateful to all other members of the Helsinki Logic Group for interesting discussions and guidance. Especially I wish to thank Do-cent Tapani Hyttinen who patiently has worked with all graduate students regardless of whether they study under his supervision or not. Professor Saharon Shelah deserve special thanks for all collaboration and sharing of his insight in the subject. The officially appointed readers Professor Boban Veličkovi´c and Docent Kerkko Luosto have done a careful and minute job in reading the thesis. In effect they have served as referees for the second paper and provided many valuable comments. Finally I direct my warmest gratitude and love to my family to whom I also wish to dedicate this work. My wife Carmela and my daughters Jolanda and Vendela have patiently endured the process and have always stood me by.