*> My answer was that I don't see how Tegmark can make this challenge*
*> effective because the collection of mathematical structures is*
*> not definable in mathematical terms.*

The point could be that there is no collection of mathematical structures that are not definable in mathematical terms. All mathematical structures necessarily have in common an existence. Existence is fundamental. There is no middle ground or alternative. Systems don't partially exist because there is no other state. There isn't even a choice between existing or not existing. Non-existence by definition cannot be. There is no such thing; no meaning to the anomalous idea that something doesn't exist. There is no such alternative. There is only being. That existence becomes what we think of as a mathematical system and all mathematical structures are subsets of the one elementary math. Consequently it is wrong to imagine two mathematical structures that have no relationship to one another or somehow form realities that are ultimately incompatible and irreconcilable. This is to say, there is, always has been, and always will be, a universe. And there is no place the universe is not.