The constructible universe

Constructing a simple universe

Gödel proved the relative consistency of the Axiom of Choice and the Generalized Continuum Hypothesis with Gödel-Bernays set theory by explicitly constructing a model in which both statements hold.

In the introduction to his 1940 Princeton lecture notes The Consistency of the Continuum Hypothesis Gödel describes the theory he is using:

The axioms of Σ are listed in the first chapter and are compared with those of Bernays. The translation of Gödel's results from his version of set theory to ZF is not too difficult. I will distort history in this chapter in connection with Gödel's results, as I have done throughout, harmlessly pretending that he worked in ZF.

Cardinal theorems and Cantor's paradox

The Cantor-Bernstein Theorem

At the centre of the theory of cardinal numbers is the notion of equipollence of sets and critical in the application of equipollence is the Cantor-Bernstein Theorem.

The Cantor–Bernstein Theorem is intuitively highly plausible. One might even mistakenly think that it is obvious, based on an extrapolation from the trivial finite case. However, a glance at its history reveals that the route to full understanding was not smooth; the transparent proofs known to us today make it very easy for us to forget these early difficulties.

Cantor first conjectured in 1882, and a year later proved, the theorem in the special case of subsets of ℝn, however, he assumed the Continuum Hypothesis in the proof. Over the next twelve years Cantor was to share his conjecture with several sources. The proofs Cantor outlined in this period assumed the Well-Ordering Theorem. Developments over the remaining part of the nineteenth century gradually stripped away these further assumptions.

Gödel's Incompleteness Theorems are now justly regarded both as one of the most profound discoveries in mathematical logic and as one of the gems of twentieth century mathematics.

Like all major scientific and mathematical landmarks (see also the Heisenberg Uncertainty Principle – in fact all of quantum mechanics – and the theory of complex dynamical systems (‘chaos’)), Gödel's theorems have captured the imagination of an excitable horde of pseudointellectuals who, not bothering to trouble themselves with actually studying the subject or understanding what the theorems say, instead abuse them, treating Gödel's deep and precise work as a springboard for flowery nonsense (generally hollow wordplay and embarrassingly vague analogies designed to obscure a chronic lack of content – a species of ‘mathematics envy’). For an account of how the Incompleteness Theorems have been misunderstood and mishandled see Torkel Franzen's Gödel's Incompleteness Theorem: an incomplete guide to its use and abuse. To fully appreciate Gödel's Theorems there is no better alternative than to carefully read a modern technical account which provides all the details. I recommend both Peter Smith's Gödel's Theorems and Raymond Smullyan's Gödel's Incompleteness Theorems. The latter is the main source for the condensed abstract sketch presented here.

Gödel's famous paper of 1931 begins by stating that large tracts of mathematics have become formalized, allowing proofs to be carried out by mechanical rules, the systems described by Russell and Whitehead's Principia Mathematica (PM) and Zermelo-Fraenkel set theory (ZF) being two notable and very far-ranging examples.

The vast majority of all humanly describable logical truths are, if presented without adequate preparation, either counterintuitive or beyond intuitive judgement. This is not a feature of an inherently peculiar Universe but merely a reminder of our limited cognitive ability; we have a thin grasp of large abstract structures. Fortunately we can gain access to the far reaches of such alien territory by building long strings of logical inferences, developing a new intuition as we do so. A proof of a theorem describes one such path through the darkness. Another important aspect of mathematics more closely resembles the empirical sciences, where features of a mathematical landscape are revealed experimentally, through the design of algorithms and meta-algorithms. In this book I assemble a miniature collage of a part of mathematics; an initial fragment of a huge body of work known as axiomatic set theory. The ambition of the book is a humble one – my intention is simply to present a snapshot of some of the basic themes and ideas of the theory.

Despite the impression given by the impatient practices of the media, it is not possible to faithfully condense into one convenient soundbite the details of any significant idea. One cannot hope to explain the rules of chess in six syllables, and it would be equally absurd to expect a short accessible account of set theory to be anything more than a fleeting glimpse of the whole.

Zermelo-Fraenkel set theory (ZF) is a first-order theory with one primitive binary relation ∈ and no primitive operators together with the following nonlogical axioms. Here the axioms are given in a semi-colloquial form making use of some of the notation and terminology which is discussed in more detail in the main text.

The Axiom of Pairing is redundant (i.e. it is a consequence of the other axioms).

Neither provable nor disprovable in ZF is the following, which is also assumed by most mathematicians.

Ordinal numbers

The emergence of ordinality

Cantor described an ordinal number, where M is an arbitrary well-ordered set, as ‘the general concept which results from M if we abstract from the nature of its elements while retaining their order of precedence…’. Making this precise, the classical view of ordinal numbers begins by defining the notion of an order type – the set of all ordered sets which are order isomorphic to some fixed ordered set – and then isolates the ordinal numbers as the collection of all order types of well-ordered sets.

This approach is intuitively pleasing to a certain degree, yet one might object to something as ‘simple’ as 1 being modelled by the class of all single element sets, which is, after all, a proper class. Some would provocatively adopt an extremist position, arguing that the apparent simplicity we perceive in the Platonic idea of ordinal numbers is an illusion; an unfortunate result of brainwashing in infancy. From a constructivist point of view the most immediate objection to the classical definition of ordinal numbers is that the entire set theoretical universe must be defined before we can extract any ordinals from it.

The Axiom of Choice

Strong versus weak Choice

The Axiom of Choice is one of the most interesting and most discussed axioms to have emerged in mathematics since the status of Euclid's parallel postulate was resolved in the mid-nineteenth century. The question of whether it should be adopted as one of the standard axioms of set theory was the cause of more philosophical debate among mathematicians in the twentieth century than any other question in the foundations of mathematics.

A much stronger form of Choice asserts that there is a universal choice function, that is, a function F such that, for all sets x, if x ≠ Ø then F(x) ∈ x.

Language

Building a theory of sets

The axiomatic development of set theory is among the most impressive accomplishments of modern logic. It can be used to give precise meaning to concepts which were beyond the grasp of its vague predecessors. A successful set theory describes clearly the logical and extra-logical principles of mathematics.

We want a theory of sets to be at least powerful enough to cope with the concepts of classical mathematics, in particular we need to be able to speak about the number systems discussed in the introduction. We have seen that the systems of integers, rational numbers, real numbers, complex numbers and algebraic numbers (and beyond) can be built from the natural numbers using a handful of logical constructions. Thus our theory needs to be capable of describing a model of the natural numbers, that is, a collection of sets with a successor operator satisfying Peano's Postulates, together with such notions as ordered pairs, functions and other relations of various kinds. At the same time, and this is where the creative tension comes into play, we don't want the theory to be so loose and overconfident with its assignment of sets to admit such horrors as Russell's paradox.

Primitive notions

Definitions – avoiding circularity

Some elementary observations have profound corollaries. Here is an example. Suppose finitely many points are distributed in some space and each point is joined to a number of other points by arrows, forming a complex directed network. We choose a point at random and trace a path, following the direction of the arrows, spoilt for choice at each turn. No matter how skillfully we traverse the network, and no matter how large the network is, we are forced at some stage to return to a point we have already visited. Every road eventually becomes part of a loop, in fact many loops.

A dictionary is a familiar example of such a network. Represent each word by a point and connect it via outwardly pointing arrows to each of the words used in its definition. We see that a dictionary is a dense minefield of circular definitions. In practice it is desirable to make these loops as large as possible, but this is a tactic knowingly founded on denial. The union of all such loops forms the core of the artificial language world of the dictionary, every word there in definable in terms of the loop members.

Cartesian products and relations

Functions and relations

The clarification of the concept of function, even when confined to the subject of analysis, had become a matter of desperate importance in the eighteenth and nineteenth centuries. Fourier's representation of piecewise continuous functions [a, b] → ℝ by trigonometric series and, later, Weierstrass and Riemann's examples of continuous nowhere differentiable functions and functions which are nowhere continuous yet integrable forced a generation of mathematicians (who were accustomed to treating functions of real variables as visualizable geometric entities) to reluctantly revise their understanding. A precise and general definition of function which is not tied to analysis is expressible within the set-theoretic framework.

Functions and relations saturate every part of mathematics. Even in our humble introduction we had to introduce these ideas at an early stage in order to construct models of the basic number systems. Now that the logical prerequisites have been brought into focus it is worth reviewing the formal definitions.

Russell's paradox and some of its relatives

The consequences of the Axiom of Abstraction

In general, a logical paradox is a contradiction, usually expressed in its simplest form Φ ↔ ¬Φ, which reveals a theory to be inconsistent, even though the axioms of the theory seem to be plausible and the rules of inference appear to be valid. The emphasis is on the surprise of the contradiction, the paradox arising unexpectedly in what is intuitively a perfectly sound system. Zermelo, quite early on, suggested the more precise term ‘antinomy’ to describe the phenomenon of contradictions which arise within logical systems, demoting the description ‘paradox’ to those results which are merely counterintuitive but not contradictory.

Contradictions which arise in apparently sensible systems are harsh reminders of our poor intuition regarding large abstract structures. Humans, unfortunately, are very good at cherry-picking attractive sounding ideas without questioning their total lack of empirical foundation and, more to the point, without examining their logical consequences. Even extremely convincing systems can have a sting in the tail, Frege's set theory being an important example. Contradictory systems can be embraced by an individual in the full course of a lifetime without any signs of nausea simply because the system-bearer has not lived long enough, or has not analyzed the system closely enough, to reveal the contradiction.