The starting point for the book is Gödel’s First Incompleteness Theorem. Kurt Gödel was an Austrian mathematician who destroyed the very bedrock of the most fundamental work of mathematics at the time, Principia Mathematica, a vast edifice built by Alfred North Whitehead and Bertrand Russell. Gödel’s genius was to show how to write a statement in a sufficiently complex mathematical system, call it system X, which could refer to itself. Most mathematical systems with which we are familiar are syntactically rich enough to express such a statement. Written in English, Gödel’s statement is equivalent to labeling a statement G where the text of the statement is “G cannot be proved in system X”. To prove a statement in a mathematical system, you start with one of the system’s axioms, which are basic truths of the system, so basic that they are not proved themselves, and then you use the axioms, step by step, to arrive at the required statement. Statements that can be proved in this way are called “theorems”.

Gödel’s initially innocuous-looking statement is, in fact, cataclysmic. If you can prove that G can, indeed, be proved in system X, then this would make the statement G, (that is, that G cannot be proved), false. So you would have been able to use the mathematics of system X to prove something (namely, G) that is false, like proving 2 + 2 = 5. Therefore, if we assume that system X is not an irrational system (that is, in mathematical terms, we assume for the following discussion that X is a consistent system), we must conclude that Gödel’s statement, “G cannot be proved in system X”, is, indeed, true. This was so destructive to the then-current ideas of mathematics because two fundamental principles were that (i) you could not prove something that is false (like 2 + 2 = 5) in a mathematical system and (ii) you could, in principle, prove any true statement that the mathematical system was able to express. Gödel’s theorem violates the second principle. The statement G, which can be expressed in system X and which we have seen to be true, says that it cannot be proved. This means that we cannot prove a true statement (namely G) which can be expressed in system X, and this transgresses the second of the two fundamental principles. Because of the existence of this true, unprovable statement (and also an indefinitely large number of related statements), system X is said to be “incomplete”, which is the origin of the name of Gödel’s Incompleteness Theorem.

The reason why the theorem appears so paradoxical is that, while G is true, only we can see it – the truth of G cannot be demonstrated in system X. Early in the book, firstly in the Maidenhead Café and then in the Coffee House, Lucy and David use coffee cups to help them visualize Gödel’s Theorem. The image is reproduced here as a picture, which helps to resolve the paradox for us.

Among other objects, the picture shows two cups on a table. The table is labeled W, standing for system W – which may be thought of as representing ourselves – We. The leftmost cup contains only statements that can be proved in system W, like 2 + 2 = 4. The rightmost cup contains only statements that cannot be proved in W, such as 2 + 2 ≠ 4 (which, of course, is not provable in any mathematical system that we are familiar with). The table also contains a tray on which are another two cups. The tray represents system X, which is meant to be a mathematical subsystem of system W. System X is not so complex as system W, but is sufficiently so for our purposes. The cup on the left on the tray again contains only statements that can be proved in system X, like 2 + 2 = 4, and the cup on the right again contains only statements that cannot be proved in system X, like 2 + 2 ≠ 4. None of this is controversial, but notice that, as a general rule, if a statement can be proved, and is therefore to be found in the left cup, then its negation, its opposite, is then to be found in the right cup.

But look what happens in the case of Gödel’s statement, G. We have seen, above, that G cannot be proved, and so we find the statement G (which is spelt out fully on the yellow label: “G cannot be proved in X”) correctly placed in the cup on the right of the tray, in system X. You might think that the opposite of G, namely “G can be proved in X” could then be placed in the cup on the left of the tray. However, that would mean that you could prove something that is false. That is because, as we know, “G cannot be proved in X” is true in this consistent system, and so “G can be proved in X” must be false. Therefore, we must place the opposite of G alongside G itself, in the cup of unprovable statements (because any falsity such as “G can be proved in X” must be in the cup of unprovable statements).
The fact that we can see that G is true is illustrated by putting statement G in the left cup of provable statements in system W, the table. This is allowed because, remember, G states that it cannot be proved in system X: it does not say that it cannot be proved in the richer, more complex system W. It is OK to look down from our vantage point in system W and see what can and cannot be proved in the subsystem called system X.

This illustration sets the scene for Lucy and David to build upon Gödel’s Theorem to derive increasingly remarkable conclusions about the universe, the multiverse and the ultimate key to the very existence of the multiverse and its whole family of universes.