Up To Isomorphism - The Error Is A Lie?

Finally we come to the phrase “Up to isomorphism”. We’ve been taking this detour both to understand a key phrase Langan uses in the CTMU and to get a flavor for the background knowledge Langan is drawing from. The phrase “up to isomorphism” simply means that two thins are isomorphic to one another, in this case that there is a bijective interpretation preserving L-Embedding between two models. However, the phrase also acknowledges that the two models may have different underlying sets or mathematical universes.

This phrase is used at the beginning of section 4.5 where Langan writes: “matter-information equivalence: identification (up to isomorphism) of concreate physical reality with information, the abstract currency of perception.” In this one sentence, with background knowledge about L-Embeddings in model theory, we see quite a bit about the CTMU. If matter and information are two different models M and I respectively, and we demand an L-isomorphism between them, then there must be some language L by which they’re related.

It appears, then, that in the “U” diagram (originally from John Wheeler) at the beginning of the section both M and I correspond to the observed side of the “U”, the right side in the illustration.

The phrase is also used in footnote one in section 4.3, this time in reference to the mental side and the material side. Langan writes: “In fact, they [the mental and material sides of reality] are identical up to isomorphism beyond which the mental side, being the more comprehensive of the two, outstrips the material." Let’s call the mental side the cognitive side or C to avoid confusion. This sentence appears to say that C is isomorphic to M, and by transitivity C would also be isomorphic to I. C would then correspond to the eye on the right side of the “U” diagram.

Here’s where the apparent error comes in: To the extent “outstrips” implies greater cardinality, C and M/I cannot be isomorphic if C outstrips or is the more comprehensive of the two. If two models are isomorphic they must have the same cardinality. This seems to imply what Langan was really getting was that M is isomorphic to a submodel of C, or is “isomorphically embedded in” C (term from Chang and Keisler). Langan is, however, very smart, and I am fully prepared to eat my words on this.

Speaking of eating my words, it seems I forgot one little detail. There are three items to be concerned with, not just two. There are three and we’re gonna make it four. Because fuck yeah. Let’s say we have two languages: Ls and Lo, where Lo is a sublanguage of Ls. Then, there could be Lo isomorphisms between C, M, and I, with C being an Ls structure while I and M are merely Lo structures. Meaning, in fact, C and M(or I) are identical “up to an Lo-isomorphism” while C, being an Ls structure, is more comprehensive and outstrips M. There, my words are probably eaten.