In the context that a particular evolving universe is a recursively enumerable theorem cascade of a particular finite consistent FAS what happens in a branching cascade?
Each theorem in a cascade has an AIT complexity greater than its preimage. If the FAS has a finite AIT complexity and is fixed then the cascade must eventually stop when the complexity of its individual theorems becomes equal to the max complexity supported by the fixed FAS i.e. the complexity of the FAS plus a constant. The endpoint theorem must be one that has no successor under the rules of the cascade. Such a theorem has a low complexity since I just described it. Thus we seem to have a contradiction - a cascade of ever more complex theorems must stop and can only stop on a very simple theorem. This seems to have two consequents: 1) Any branching of the cascade results in one of the two new universes trying to reach the low complexity stop i.e. a new running contradiction. 2) But the other universe also has to stop given the fixed FAS complexity i.e. another new running contradiction. The only way out seems to be that there is no branching and the FAS must become more complex as the cascade continues. An increase in complexity of the FAS has a possible interpretation: The cascade is reinitialized with the new theorem acting as the initiating axiom and the added content of the FAS comes from a random oracle. Hal
