How does trivialism differ from assuming the existence of a Tegmark IV universe?
Tegmark IV is the space of all computable mathematical structures. You can make true and false statements about this space, and there is nothing about it that implies a contradiction. You may think that any coherent empirical claim is true in Tegmark IV, in that anything we say about the world is true of some world. But being true in some world does not make it true in this world. If I say that the sky is green, I am implicitly referring to the sky that I experience, which is blue. That is, I am saying that the sky which is blue is green. So I’m contradicting myself, and the statement is false. You don’t even need to think of alternate universes to reason through this. After all, some planet in our galaxy surely has a green sky.
A spectral argument given in defense of trivialism in the dissertation runs like this...
It all looks shaky, but most obviously, just because every classical proposition may be interpreted in natural language doesn’t mean that every natural language proposition may be interpreted in classical logic. In particular, the aspects of natural language that make it inconsistent probably can’t be translated into classical logic. After all, that’s why we invented classical logic in the first place.
Level IV refers to parallel worlds in distinct mathematical structures, which may have fundamentally different laws of physics.
Trivialism induces a mathematical structure, and so is contained in the level IV multiverse. I think there’s some meta-level confusion in the rest of the first part of your comment.
It all looks shaky, but most obviously, just because every classical proposition may be interpreted in natural language doesn’t mean that every natural language proposition may be interpreted in classical logic.
It’s not clear to me how this claim affects the argument. Asserting the negation of the converse of (c) doesn’t imply anything about (c).
Did these points come up in the dissertation?
The argument is not central to the dissertation. He reports it from a trivialist to establish the existence of at least one trivialist.
Tegmark IV is the space of all computable mathematical structures. You can make true and false statements about this space, and there is nothing about it that implies a contradiction. You may think that any coherent empirical claim is true in Tegmark IV, in that anything we say about the world is true of some world. But being true in some world does not make it true in this world. If I say that the sky is green, I am implicitly referring to the sky that I experience, which is blue. That is, I am saying that the sky which is blue is green. So I’m contradicting myself, and the statement is false. You don’t even need to think of alternate universes to reason through this. After all, some planet in our galaxy surely has a green sky.
It all looks shaky, but most obviously, just because every classical proposition may be interpreted in natural language doesn’t mean that every natural language proposition may be interpreted in classical logic. In particular, the aspects of natural language that make it inconsistent probably can’t be translated into classical logic. After all, that’s why we invented classical logic in the first place.
Did these points come up in the dissertation?
From one of Tegmark’s pop sci papers:
Trivialism induces a mathematical structure, and so is contained in the level IV multiverse. I think there’s some meta-level confusion in the rest of the first part of your comment.
It’s not clear to me how this claim affects the argument. Asserting the negation of the converse of (c) doesn’t imply anything about (c).
The argument is not central to the dissertation. He reports it from a trivialist to establish the existence of at least one trivialist.