Forking and Dividing in Continuous Logic

We investigate an open question concerning properties of algebraic independence in continuous theories (see Section 4). The rest of the work is essentially a translation to continuous logic of popular notions of independence (in particular, forking and dividing). Much of the time we are simply “copying” classical proofs from well-known sources, while along the way making the necessary adjustments for continuous languages. 1. Forking and Dividing We assume that the reader is familiar with the construction of continuous languages and theories of metric structures. A full introduction to these concepts can be found in [2]. We begin with the usual definition of dividing, which does not need to be altered to work in the continuous setting. Definition 1.1. A partial type π(x, b) divides over C if there is a C-indiscernible sequence (bi)i<ω, with b0 ≡C b, such that ⋃