[eli_gottlieb]

>You cannot simply dream up whatever structure you want and have it mean what you want and behave how you want. See Godel's incompleteness theorems.

That is not at all what the Incompleteness Theorems actually say. They say literally nothing whatsoever about what sorts of structures you can implement inside a given foundational theory, except that there will always be more, because given any foundational theory, you can construct two more foundational theories as extensions (one in which the Goedel statement is unprovable, and one in which the theory believes it's inconsistent).