Lindstrom's theorem states that any extension of first order logic (FOL) more expressible than FOL fails to have either compactness or Lowenheim-Skolem. When I first read Lindstrom's theorem my first reaction was: "Does it mean incompleteness of any more expressible extension of FOL? And why this obviously important question isn't discussed in standard logic texts?" Standard extensions (such as second order logic) in fact are incomplete. After some attempts of proving incompleteness I found reference to Vaught's paper in which he proves completeness of extension of FOL by adding quantifier Qx = "there are uncountably many x such that…" It would be very interesting (at least for me) to understand complete extensions in general. Such an extensions may be of great importance, for instance, for computer science, because FOL isn't enough expressive for its purposes.
So, my questions are:
1) Do you know some other examples of complete FOL extensions?
2) Are there some results concerning characterization of complete FOL extensions?
3) Do you know any people, papers, books… studying general properties of FOL extensions?
Thanks in advance.
Best Answer
Shelah has some nice results on complete extensions of first order logic. For example, he introduced cofinality quantifiers in which the truth value of a given formula (Qxy)Phi(x,y,a) is determined by the cofinality of the linear order of pairs (x,y) satisfying Phi. It turned out that such logics are complete (in addition to other nice model-theoretic properties). You may find these results (and much more) in the following paper: http://www.ams.org/journals/tran/1975-204-00/S0002-9947-1975-0376334-6/S0002-9947-1975-0376334-6.pdf
You should check the book "Model theoretic logics" by Feferman and Baldwin (although it's a pretty old book). As far as name-dropping goes, you should also check the works of Barwise. You may also check this nice survey paper: http://www.math.ucla.edu/~asl/bsl/1001/1001-004.ps