Inner model operators in L(R) Original Research Article

An inner model operator is a function M such that given a Turing degree d, M(d) is a countable set of reals, d⊆M(d), and M(d) has certain closure properties. The notion was introduced by Steel. In the context of AD, we study inner model operators M such that for a.e. d, there is a wellorder of M(d) in L(M(d)). This is related to the study of mice which are below the minimal inner model with ω Woodin cardinals. As a technical tool, we show that the alternative fine structure theory developed by Mitchell and Steel for mice with Woodin cardinals is equivalent (in some sense) to the traditional fine structure theory developed by Jensen for L.