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Eliminating deficiencies: the complete picture.

We now prove by induction on the number of possible deficiencies that every deficiency in x w.r.t. y can be eliminated in an adequate way. (For the time being we only consider the harder case when intermediate worlds do exist.) Adequate means that all the Sxsuccessors of y that are defined for this purpose contain all the $\Box$-formulas of x' ( xReBx'Ry). The new worlds should not provoke new deficiencies in x. And all of them should lie B-critically above x.

If there is a deficiency in x w.r.t. y and there is no intermediate world, the situation becomes even easier. Again the process of eliminating this deficiency in x w.r.t. y shall be referred to as the process of making an Sxy-block. If in a general setting the process of making an Sxy-block is mentioned, either one of these two is meant, depending on the presence of intermediate worlds. The process of making an Sxy-block in the case of no intermediate worlds can inductively be described as follows.

As there is only a finite number of possible deficiencies, this process must come to an end.


next up previous contents
Next: The actual construction. Up: Deficiencies Previous: Deficiencies
Joost Joosten
2000-02-07