RealityModels
Menu

Glossary: mathematical universe, information and the regress of containers

Terms that recur in discussions of Tegmark's mathematical universe hypothesis, the background of every simulation in the project (see README.md), and in the "regress of containers" argument: nothing produces a bit, the bit needs a space that contains it, the space needs a theory that describes it, and so on. Section 1 summarises the argument, the following sections define the terms grouped by area. References in square brackets point to the bibliography at the end. Italian version: mathematical-universe-glossary.it.md.

1. The argument in brief

  1. Nothing produces a bit. To be nothing, nothing must exclude something. An exclusion is a distinction, and a distinction is a bit. Pure nothing is therefore logically unstable [Spencer-Brown 1969]. Set theory carries this out literally: from the empty set one builds 0, 1, 2 and all of arithmetic (von Neumann ordinals).
  2. A bit needs a container. A bit is a choice between two alternatives, so the state space {0, 1} precedes the value. The content of a string is defined only relative to a machine that reads it (algorithmic information). The truth of a language is defined only in a metalanguage [Tarski 1933] and no system certifies itself [Gödel 1931]. Every floor needs the floor above.
  3. The tower. Value → state space → set theory → logic → ... The regress is vicious only if every floor must be produced before the next, that is, if it is a process in time. If the floors coexist outside time, the tower is the cumulative hierarchy V of set theorists, where mathematicians have worked for a century without anything breaking.
  4. The three exits. No container is needed (Tegmark: structures do not sit in boxes, they simply are); close the loop (Wheeler, Müller: bits generate the observer who gives bits their meaning); the regress is real and benign (reality is open upward, "the whole" is not an object).
  5. The objection to Tegmark. If every structure needs a metastructure, "the set of all structures" is contradictory (Cantor, Russell) and the MUH does not close. The retreat to the CUH repairs nothing: computability does not remove the interpreter, it only names it (the universal machine), which is itself a structure.

2. Hypotheses about the nature of reality

MUH (Mathematical Universe Hypothesis). Max Tegmark's thesis: physical reality is a mathematical structure, not merely described by one [Tegmark 2008, 2014]. It follows from the ERH (External Reality Hypothesis, there is an external reality independent of observers) through the argument that a complete description free of human "baggage" (words, intuitions) can be nothing but a mathematical structure. Consequence: every mathematical structure exists physically on the same footing as ours.

CUH (Computable Universe Hypothesis). A restriction of the MUH proposed by Tegmark himself [2008]: only mathematical structures defined by computable relations exist physically, that is, relations a Turing machine can evaluate in a finite number of steps. Its purpose is to avoid undecidable structures and the problems tied to Gödel. The price: it excludes the continuum and much of physics as it is written, and moves the problem onto the machine that runs the computation (see universal machine).

Level IV multiverse. The highest of Tegmark's four multiverse levels [2003]: I, regions beyond the cosmic horizon; II, other inflationary bubbles with different constants; III, Everett's branches (see many-worlds-convergence-theory.en.md); IV, all mathematical structures. Level IV is the MUH taken as a multiverse: our universe is one structure among the others.

Many interacting worlds (MIW). Hall, Deckert and Wiseman's proposal [2014]: quantum mechanics as the statistics of N classical worlds, each with definite positions, that repel each other; the wave function is emergent and collapse is the selection of the sub-ensemble of worlds compatible with the record. Unlike Everett's branches the worlds never split or merge. Theory and simulation: many-interacting-worlds-theory.en.md.

Mathematical structure. A set of abstract elements with relations among them (for example: the integers with addition, a graph, a differentiable manifold). Two structures with the same relations are the same structure, whatever the elements are called. In the MUH "being contained in" is a relation between structures, not the position of an object in a space.

Structural realism. A position in philosophy of science: what theories capture of reality is the structure of relations, not the intrinsic nature of objects [Ladyman 1998]. In the "ontic" version relations are all there is. The MUH is its limiting case.

Block universe. Spacetime exists all at once, past, present and future, and the flow of time is a property of the point of view, not of the world. Theory and simulations: block-universe-relativity-theory.en.md. In the regress argument the block matters because, if the floors of the tower coexist outside time like slices of spacetime, the regress is not a process that has to "finish".

It from bit. John A. Wheeler's slogan [1990]: every "it", every physical thing, derives in the end from yes/no answers to questions posed by observers, that is, from bits. The physical world is information before it is matter.

Self-excited circuit. Wheeler's diagram, a U with an eye on one arm looking at the other: the universe produces observers, and observers through their measurements give the universe its reality [Wheeler 1983]. Also called the "participatory universe". It is the second exit from the regress: not a tower but a loop, in which the container of the bits is the observer that emerges from the bits.

Law without law. Wheeler's phrase for a world in which physical laws are not given a priori but emerge. Taken up by Markus Müller [2020] in a formal theory: what is fundamental are observer states, and the probability of the next state is given by Solomonoff's algorithmic probability; a regular external world emerges as a compact description. It closes the loop without presupposing an external physical space.

3. Information and computation

Bit. The smallest unit of information: the answer to a question with two equally possible outcomes [Shannon 1948]. A bit is not an isolated symbol but a choice between two alternatives: without the set of alternatives there is no bit. This is the foothold of step 2 of the argument.

State space. The set of all possible configurations of a system. For one bit it is {0, 1}; for n bits it has 2ⁿ elements; for a quantum system it is a Hilbert space. The value of a state makes sense only relative to the space that contains it, which is therefore logically prior.

Algorithmic information (Kolmogorov complexity). The information content of a string is the length of the shortest program that produces it on a fixed universal machine [Solomonoff 1964; Kolmogorov 1965; Chaitin 1966]. Changing the machine changes the measure by at most an additive constant (invariance theorem): the dependence on the interpreter is softened but not removed, and for short strings the constant dominates.

Universal (Turing) machine. A machine that can simulate any other Turing machine by reading its description [Turing 1936]. It is the "interpreter" that algorithmic information presupposes and that the CUH names without removing: the universal machine is itself a mathematical structure, and the question "what contains it?" starts again.

Computable. A function is computable if some Turing machine calculates it for every input in a finite number of steps. A relation is decidable if its characteristic function is computable. The CUH asks for decidable relations.

Interpreter. Any system that assigns a meaning to a string of symbols: a machine, a formal language, an observer. In the regress every floor acts as interpreter for the one below.

4. Logic and set theory

Distinction (Laws of Form). For George Spencer-Brown [1969] the primordial act is "draw a distinction": separating an inside from an outside. His notation has a single symbol, the mark, which serves at once as distinction and as operator. The first possible distinction is nothing/not-nothing, and it is the content of step 1.

Empty set and von Neumann ordinals. The set with no elements, ∅. From it the natural numbers are defined by nesting alone: 0 = ∅, 1 = {∅}, 2 = {∅, {∅}}, n + 1 = n ∪ {n} [von Neumann 1923]. With the Zermelo-Fraenkel axioms the integers, rationals, reals and all ordinary mathematics follow. This is the technical sense in which "nothing is the only free brick".

Leibniz's medal. In 1697 Leibniz proposed to Duke Rudolph Augustus of Brunswick a medal celebrating binary arithmetic with the motto Omnibus ex nihilo ducendis sufficit unum, "to derive everything from nothing, one is enough" [Leibniz 1697]. He saw in 0 and 1 an image of creation.

Principles of identity and non-contradiction. Everything is identical to itself; a proposition and its negation are not both true. They are the hidden assumption of step 1: for nothing to "exclude" something these rules are already needed, so the regress starts below the bit, not above it.

Carroll's tortoise. In Lewis Carroll's dialogue [1895] the Tortoise accepts the premises "A" and "if A then B" but refuses to conclude B until the rule "if A and (if A then B) then B" is added, and then asks for the rule that justifies that one, without end. It shows that a rule of inference cannot be reduced to a premise: every rule needs a metarule.

Undefinability of truth (Tarski). In a sufficiently rich formal language there is no predicate "is true" that holds for all its sentences; the truth of the sentences of a language is defined only in a more expressive metalanguage [Tarski 1933]. Every floor needs the floor above.

Incompleteness theorems (Gödel). First: every consistent formal system containing arithmetic has true sentences it does not prove. Second: such a system cannot prove its own consistency [Gödel 1931]. A system does not certify itself; certification, if any, comes from a stronger system.

Cumulative hierarchy V. The construction of all sets in stages: V₀ = ∅, V₍α+1₎ = power set of Vα, and at limits the union of what was built before [Zermelo 1930]. Each stage is a set contained in the next; the totality V is not a set but a proper class. It is the tower of containers realised in current mathematics.

Proper class. A collection too large to be a set: the class of all sets, of all ordinals, of all structures. One can talk about it but not treat it as an object, on pain of paradox.

Cantor's theorem. No set is in one-to-one correspondence with its power set, which always has greater cardinality [Cantor 1891]. Consequence: there is no set of all sets, since its subsets would outnumber it.

Russell's paradox. The set of all sets that do not belong to themselves belongs to itself if and only if it does not [Russell 1902]. It brought down naive set theory and motivated the Zermelo-Fraenkel axioms. Together with Cantor, it forbids "the totality of all structures" from being itself a structure.

Grothendieck universe. A set U closed under the usual operations (power set, unions, pairs) inside which all ordinary mathematics can be done as if U were "everything" [Grothendieck et al. 1972]. Its existence is equivalent to that of a strongly inaccessible cardinal. Every universe is contained in a larger one: no floor is the last.

Large cardinals. Infinite cardinals whose existence does not follow from the Zermelo-Fraenkel axioms and must be postulated (inaccessible, measurable and others) [Kanamori 2003]. Each large cardinal axiom raises the tower by one floor and proves the consistency of the floor below. This is how mathematicians live in a hierarchy with no roof.

5. Philosophical terms

Ex nihilo. "From nothing". Leibniz's question "why is there something rather than nothing?" [Leibniz 1714] is the starting point of the argument; step 1 answers that nothing is not a stable option.

Infinite regress. A chain of justifications, causes or containers with no first term. It is vicious if the existence or justification of each link depends on the completion of the chain, which never completes; it is benign if the links coexist and none waits for the others. The distinction decides whether the tower of containers is a problem or a description.

Self-reference. A system that describes, contains or justifies itself. It appears negatively in the paradoxes (Russell) and in the limits of formal systems (Gödel, Tarski), and positively in the "loop" exit of Wheeler and Müller, where it must explain why the loop holds.

Metalanguage, metarule, metastructure. The language in which one speaks about a language, the rule that governs the use of a rule, the structure in which a structure is defined. The prefix "meta" marks the floor above in the tower.

Outside time. To exist without existence being a dated event. Mathematics is the typical case: 2 did not "come after" 1. It is the condition under which the regress stops being a process and becomes a hierarchy.

Bibliography

Mathematical universe and multiverse

Information and the observer

Logic and sets