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We could also completely avoid the axiom set encoding problem by talking only about worlds which could contain intelligent beings and talking about how the intelligent beings would model their world. Let us note that if those intelligent beings are similar enough to us and the optimal set of axioms for their world is infinite then they will never have a complete description of how their world works, but they will be able to build better and better models.  We will Let us  assume that those intelligent beings are continuously trying to find better models for their world and that they are reasonably efficient at this. As a parenthesis, note that until now we restricted the possible world concept several times. The argument below also works with larger possible world concepts as long as those worlds have a few basic properties (e.g. one can make predictions and it can contain intelligent beings) and at the same time it is plausible that our world is such a possible world.