I was asked by a reader to take a look at yet another crackpot theory of everything. This time, it's the Cognitive Theoretic Model of the Universe. This one is as cranky as any, but it's actually got some interestingly silly math to it.

Stripped down to its basics, the CTMU is just yet another postmodern "perception defines the universe" idea. Nothing unusual about it on that level. What makes it interesting is that it tries to take a set-theoretic approach to doing it.

The real universe has always been theoretically treated as an object, and specifically as the composite type of object known as a set. But an object or set exists in space and time, and reality does not. Because the real universe by definition contains all that is real, there is no "external reality" (or space, or time) in which it can exist or have been "created". We can talk about lesser regions of the real universe in such a light, but not about the real universe as a whole. Nor, for identical reasons, can we think of the universe as the sum of its parts, for these parts exist solely within a spacetime manifold identified with the whole and cannot explain the manifold itself. This rules out pluralistic explanations of reality, forcing us to seek an explanation at once monic (because nonpluralistic) and holistic (because the basic conditions for existence are embodied in the manifold, which equals the whole). Obviously, the first step towards such an explanation is to bring monism and holism into coincidence.

Right from the start, we can see the beginnings of how he's going to use a supposedly set-theoretic notion. And also, right from the beginning, we can see exactly the kind of semantic games he's going to play. He manages to say pretty much nothing about the universe - all he's doing is playing with the semantics of the words "Universe", "real", "holistic", etc.

I particularly love this next bit.

When theorizing about an all-inclusive reality, the first and most important principle is containment, which simply tells us what we should and should not be considering. Containment principles, already well known in cosmology, generally take the form of tautologies; e.g., "The physical universe contains all and only that which is physical." The predicate "physical", like all predicates, here corresponds to a structured set, "the physical universe" (because the universe has structure and contains objects, it is a structured set). But this usage of tautology is somewhat loose, for it technically amounts to a predicate-logical equivalent of propositional tautology called autology, meaning self-description. Specifically, the predicate physical is being defined on topological containment in the physical universe, which is tacitly defined on and descriptively contained in the predicate physical, so that the self-definition of "physical" is a two-step operation involving both topological and descriptive containment. While this principle, which we might regard as a statement of "physicalism", is often confused with materialism on the grounds that "physical" equals "material", the material may in fact be only a part of what makes up the physical. Similarly, the physical may only be a part of what makes up the real. Because the content of reality is a matter of science as opposed to mere semantics, this issue can be resolved only by rational or empirical evidence, not by assumption alone.

After a particularly egregious exercise in english semantics, in which he does nothing but play with word meanings, coming nowhere near actually saying anything, but using lots of impressive-looking words, he concludes that it "is a matter of science as opposed to mere semantics". Rich!

He spends some more time rambling about semantics of words like "physicalism", "materialism", and "containment", before finally getting to the part that's got any math content at all.

Now for a brief word on sets. Mathematicians view set theory as fundamental. Anything can be considered an object, even a space or a process, and wherever there are objects, there is a set to contain them. This "something" may be a relation, a space or an algebraic system, but it is also a set; its relational, spatial or algebraic structure simply makes it a structured set. So mathematicians view sets, broadly including null, singleton, finite and infinite sets, as fundamental objects basic to meaningful descriptions of reality. It follows that reality itself should be a set...in fact, the largest set of all. But every set, even the largest one, has a powerset which contains it, and that which contains it must be larger (a contradiction). The obvious solution: define an extension of set theory incorporating two senses of "containment" which work together in such a way that the largest set can be defined as "containing" its powerset in one sense while being contained by its powerset in the other. Thus, it topologically includes itself in the act of descriptively including itself in the act of topologically including itself..., and so on, in the course of which it obviously becomes more than just a set.

First - he gets the definition of set wrong. He's talking about naive set theory, which we know is unsound. And in fact, he's talking about exactly the kinds of inclusion issues that lead to the unsoundness of naive set theory!

Then he uses semantic word-games to argue that the universe can't be a set according to set theory, because the universe is the largest thing there is, but set theory says that you can always create something larger by taking a powerset. What does he conclude from this pointless exercise? That playing word-games doesn't tell you anything about the universe? No, that makes too much sense. That naive set theory perhaps isn't a great model for the physical universe? No, still too much sense. No, he concludes that this problem of word-games means that set theory is wrong, and must be expanded to include the contradiction of the largest thing being both smaller than its powerset and larger than its powerset.

Yes, the solution is to take an unsound mathematical theory, and make it doubly unsound.

In the Cognitive-Theoretic Model of the Universe or CTMU, the set of all sets, and the real universe to which it corresponds, take the name (SCSPL) of the required extension of set theory. SCSPL, which stands for Self-Configuring Self-Processing Language, is just a totally intrinsic, i.e. completely self-contained, language that is comprehensively and coherently (self-distributively) self-descriptive, and can thus be model-theoretically identified as its own universe or referent domain. Theory and object go by the same name because unlike conventional ZF or NBG set theory, SCSPL hologically infuses sets and their elements with the distributed (syntactic, metalogical) component of the theoretical framework containing and governing them, namely SCSPL syntax itself, replacing ordinary set-theoretic objects with SCSPL syntactic operators. The CTMU is so-named because the SCSPL universe, like the set of all sets, distributively embodies the logical syntax of its own descriptive mathematical language. It is thus not only self-descriptive in nature; where logic denotes the rules of cognition (reasoning, inference), it is self-cognitive as well. (The terms "SCSPL" and "hology" are explained further below; to skip immediately to the explanations, just click on the above links.)

(His text refers to "the above links", but in fact, the document doesn't contain any links.)

This is pure muddle. It's hard to figure out what he even thinks he's doing. It's clear that he believes he's inventing a new kind of set theory, which he calls a "self-processing language", and he goes on to get very muddled about the differences between syntax and semantics, and between a model and what it models. I have no idea what he means by "replacing set-theoretic objects with syntactic operators" - but I do know that what he wrote makes no sense - it's sort of like saying "I'm going to fix the sink in my bathroom by replacing the leaky washer with the color blue", or "I'm going to fly to the moon by correctly spelling my left leg."

From there who moves to adding a notion of time, which he seems to believe can be done using nothing but set theory. Unfortunately, that makes no sense at all: he wants to somehow say that sets have time properties, without modifying the sets, modeling the time property, or in fact anything at all - once again, he just throws around lots of terminology in meaningless ways:

An act is a temporal process, and self-inclusion is a spatial relation. The act of self-inclusion is thus "where time becomes space"; for the set of all sets, there can be no more fundamental process. No matter what else happens in the evolving universe, it must be temporally embedded in this dualistic self-inclusion operation. In the CTMU, the self-inclusion process is known as conspansion and occurs at the distributed, Lorentz-invariant conspansion rate c, a time-space conversion factor already familiar as the speed of light in vacuo (conspansion consists of two alternative phases accounting for the wave and particle properties of matter and affording a logical explanation for accelerating cosmic expansion). When we imagine a dynamic self-including set, we think of a set growing larger and larger in order to engulf itself from without. But since there is no "without" relative to the real universe, external growth or reference is not an option; there can be no external set or external descriptor. Instead, self-inclusion and self-description must occur inwardly as the universe stratifies into a temporal sequence of states, each state topologically and computationally contained in the one preceding it (where the conventionally limited term computation is understood to refer to a more powerful SCSPL-based concept, protocomputation, involving spatiotemporal parallelism). On the present level of discourse, this inward self-inclusion is the conspansive basis of what we call spacetime.

I can't make head or tails out of this. It's just word-games, trying to throw in as many fancy-sounding terms as possible. What on earth does Lorentz invariance have to do with this muddle? LI means something quite specific, and he's done nothing to connect any of this rubbish to it. He's just throwing around words: "conspansion", "lorentz invariance", "protocomputation".

But it gets worse. We get yet more of his confusion about just what "syntax" means:

Every object in spacetime includes the entirety of spacetime as a state-transition syntax according to which its next state is created. This guarantees the mutual consistency of states and the overall unity of the dynamic entity the real universe. And because the sole real interpretation of the set-theoretic entity "the set of all sets" is the entire real universe, the associated foundational paradoxes are resolved in kind (by attributing mathematical structure like that of the universe to the pure, uninterpreted set-theoretic version of the set of all sets). Concisely, resolving the set-of-all-sets paradox requires that (1) an endomorphism or self-similarity mapping D:S-->rÎS be defined for the set of all sets S and its internal points r; (2) there exist two complementary senses of inclusion, one topological [S Ét D(S)] and one predicative [D(S) Éd S], that allow the set to descriptively "include itself" from within, i.e. from a state of topological self-inclusion (where Ét denotes topological or set-theoretic inclusion and Éd denotes descriptive inclusion, e.g. the inclusion in a language of its referents); and (3) the input S of D be global and structural, while the output D(S) = (r Éd S) be internal to S and play a syntactic role. In short, the set-theoretic and cosmological embodiments of the self-inclusion paradox are resolved by properly relating the self-inclusive object to the descriptive syntax in terms of which it is necessarily expressed, thus effecting true self-containment: "the universe (set of all sets) is that which topologically contains that which descriptively contains the universe (set of all sets)."

Yes, lucky us, more wordplay!

The thing to notice here is right in the first sentence: "Every object in spacetime includes the entirety of spacetime as a state-transition syntax". Spacetime isn't a syntax. Like I said before, it's like talking about spelling your leg. An object can't be a syntax. A syntax is a method of writing down a sequence of symbols that expresses some logical statement. An object in spacetime can't "include the universe as a state transition syntax".

What I think he's trying to say here is that we can describe objects in the universe as state transition systems, in which the state of an object plus the state of the universe can be used to compute the next state of the object. But he doesn't understand that a syntax and a system are different things. And he seems to think that the idea of describing the universe as a state transition system is somehow profound and original. It's not. I've read papers proposing state-transition semantics for the universe dating back to the 1950s, and I'd be surprised if people like von Neumann hadn't though of it even earlier than that.

The rest of that paragraph is yet more of his silly word-games, trying to cope with the self-created paradox of inclusion and size in his mangled set theory.

At this point, I'm going to stop bothering to quote any more of his stuff. The basic point of his argument, and the basic problems that pervade it are all abundantly clear after this much, and you've already experienced as much fun as your going to by laughing at his foolishness.

To recap: this "theory" of his has three problems, each of which is individually enough to discard it; with the three of them together, it's a virtual masterpiece of crap.

1. The "theory" consists mostly of word-games - arguing about the meanings of words like "universe" and "inclusion", withou actually explaining anything about how the universe works. It's a theory with no predictive or descriptive value.
2. The "theory" is defined by creating a new version of set theory, whose axioms are never stated, and whose specific goal guarantees that it will be an unsound theory. Unsound mathematical theories are useless: every possible statement is provable in an unsound theory.
3. The author doesn't understand the difference between syntax and semantics, between objects and models, or between statements and facts - and because of that, the basic statements in his theory are utterly meaningless.