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On the other hand, using only finite systems of axioms in this way seems to be some sort of cheating. In order to get a more "honest" system of axioms, we could work with very specific systems of axioms, e.g. we could only talk about worlds which have $\reale^4$ as their space and time, whose objects are things similar to the wave functions used by quantum mechanics and so on.  \section {Intelligent beings / Finite and infinite axiom systems [but [TODO: but  then I would need to move the title above.]} We could also completely avoid the axiom system encoding problem by talking only about worlds which could contain intelligent beings and talking about how the intelligent beings would model their world. This approach may also be a more interesting one, since we may never have a complete and correct model for our world, but we can build better and better models.