Alessandro Andretta

Generalised Iteration Trees

A theorem of Gaifman states that any internal linear iteration whose length belongs to the model it is applied to has a well-founded direct limit. We have isolated a notion of "generalized iteration trees" for which a similar result is possible, at least if the length of the tree is $\omega$. These iterations are more general than the objects introduced by Martin and Steel over three decades ago in that the extender $E_n$ used to construct $M_{n+1}$ need not to belong to the last model $M_n$. In other words $E_n \in M_{d(n+1)}$, with $d(n+1) \leq n$. We isolate a simple property of the function $d$ characterizing continuous ill-foundedness of generalized iteration trees. Any generalized iteration trees satisfying this property is not continuously ill-founded. Conversely, any tree order with a $d$ function failing such property can be realized as a continuously ill-founded iteration tree on V. This is joint work with John Steel.