Definability of second order generalized quantifiers

Acknowledgements I wish to express my sincere gratitude to my supervisor Professor Jouko Väänänen for introducing me to this subject and for his inspiring guidance and support during my doctoral studies. I am also grateful to Professor Lauri Hella and Docent Kerkko Luosto for carefully reading the manuscript and making valuable comments on it. I am indebted to the Graduate School of Mathematical Analysis and Logic and to the Finnish Academy of Science and Letters (Vilho, Yrjö and Kalle Väisälä Foundation) for financial support. I want to thank my friends and colleagues at the Department of Mathematics and Statistics and the participants of the Finite Model Theory Seminar for their support and intrest. Especially, I want to thank Ph.Lic. Pekka Pankka and Ph.Lic. Jussi Laitila. I express my best thanks to my parents Raili and Esko for their continuous encouragement and support. My thanks are also due to my brother Jarmo. Finally, I am grateful to Ilona for her love and her important support during my doctoral studies.