We can safely say that mathematics is our most logically coherent science. But despite its logical nature, mathematics still poses riddles that puzzle the mind. At the beginning of the twentieth century, philosophers and mathematicians Bertrand Russell and Alfred North Whitehead came up with a plan to reorganize mathematics into a fully coherent system. They started with statements that were believed to be true, and then other statements had to follow logically. We use Peano’s axioms to illustrate this method:

• Zero is a number
• Every number has a successor and that successor is also a number
• Zero is not the successor of a number
• Different numbers have different successors
• If zero has a certain property and if it is proven from the assumption that a number has that property that its successor does too, then every number has that property.

Russell’s and Whitehead’s plan worked out smoothly at first. But this changed in 1931, when Kurt Gödel, a mathematical genius, placed a (proverbial) bomb with his incompleteness theorems [1]. Gödel argued that within this system you can formulate propositions that are undecidable. To prove this, Gödel constructed a sentence S in the formal language of an axiomatic system A that claims about itself: S is not provable in A.

If this sentence is true, then it is not provable, because that is what it claims. But this would imply that a true sentence cannot be proven, and system A is therefore incomplete.

However, if the sentence is not true, then it is provable, and a falsehood is provable in A.

Variants are:

– This sentence is not true.
– Pinocchio says: “My nose is growing now”
– The Cretan Epimenides says: “All statements of Cretans are always false”
– This sentence does not exist.
– The following statement is true. The previous one is not true.
– According to Onno, everything always fails, even failure.

These statements are self-references. They refer to themselves, describing themselves with a necessary quality without which they are not the same. They are indivisible. These examples are funny, but Gödel’s incompleteness theorems shocked the mathematical world.

You can complicate this, or keep it simple. Let’s keep it simple. A pair of scissors cut everything except themselves. That is not hard to comprehend. Mathematical theorems can be understood by representing them in a decision tree. Each new statement with the verdict ‘true/false’ cuts the previous statement like a pair of scissors. Until everything is divided and only the judgment, the scissors, remains. This is indivisible. Gödel’s theorem ‘S is not provable in A’ is both ‘true’ and ‘false’. The last step referring to itself is undecidable. Now compare this to the smallest (indivisible) unit of quantum information, the shared information, the entanglement. It is in superposition and therefore indefinite.

Mathematics provides us with models of reality. A calculation or statement in formal mathematics gives exactly the same result when the calculation is run again. There is no emergence in mathematics. In a formal mathematical model, all superposition has been removed. Therefore, formal mathematical models are predictable and calculable. But where is the information that was lost during the transition from reality to mathematical model? It cannot have disappeared. That would be contrary to the presumption of conservation of information. As with Maxwell’s demon, this information is transferred to (and then shared with) the observer who has gained knowledge about the system. In this way reality is converted into a model. Except for the last step. The model ends with shared information in superposition. This one is undecidable: superposition is fundamental. It cannot be reduced any further.

Revisiting the Russell paradox
In Chapter 7 we discussed the Russell paradox, which involves the question whether a collection is able to contain itself. Russell discovered that there are situations where you cannot indicate whether a collection contains itself or not. The Barber in Seville served as an illustration. Let’s discuss another example of Russell’s paradox: the catalog paradox.

Naturally, every library houses many catalogs. We follow a librarian who wants to organize these catalogs. She discovers that there are two kinds of catalogs, those that include themselves, and those that don’t. Therefore, she decides that there should be two collector’s catalogs. Firstly, she creates a catalog that includes all catalogs that contain themselves. Should she also include the title of this collector’s catalog in its list? Of course she does. But when it comes to the collector’s catalog with catalogs that do not contain themselves, she encounters a dilemma. Should the title of this collector’s catalog be included? In doing so she will create something that is no longer a catalog that only contains titles of catalogs that do not contain themselves, and the title suddenly no longer belongs in the overview. But if she doesn’t, she will create a catalog that doesn’t contain itself and then the title does belong in it. Our librarian is basically stuck between a rock and a hard place here. The catalog must contain itself if it does not. And it must not contain itself when it does.

In mathematics, a set is a collection of different objects (elements) which is itself considered a mathematical object. The concept of a set is a basic mathematical concept. It cannot be further reduced to a composite of other more fundamental theoretical mathematical concepts (axioms), but must itself be defined axiomatically. When together, elements form a set. A set in itself is an indivisible whole. The same applies to entangled ‘particles’. Because two entangled ‘particles’ share information, they can no longer be seen separately. They form an indivisible whole.

Set theory is a mathematical model. It encompasses flower collections, catalogs or collections of ideas. It is an attempt to capture everything in a mathematical model. Our project shows that models remove coherence, and thus emergence. And just as we’ve seen with Gödel’s incompleteness theorems, set theory is also inconclusive.

The paradoxes presented by Gödel and Russell are not the same. While Gödel’s incompleteness theorems deal with the elements (numbers or particles), set theory deals with the correspondences, the relationships. But set theory treats those similarities as isolated phenomena. Take another look at the animation/illustrations about particles and relations that you see throughout this book. You may be tempted to view these relations as isolated links, but relations are not connections between particles, they form the overlap, the superposition. So try to resist the temptation to view particles, as well as relations, as isolated phenomena. Because they aren’t. Ultimately, the distinction between particles and relations simply cannot be made. The superposition is fundamental.

Everything begins, and ends, with superposition.

Mathematics is both useful and beautiful, but because it lacks shared information (the coherence) it does not suffice to describe and understand the nature of emergent reality.