Berto's highly readable and lucid guide introduces students and the interested reader to Gödel's celebrated Incompleteness Theorem, and discusses some of the most famous - and infamous - claims arising from Gödel's arguments.
Offers a clear understanding of this difficult subject by presenting each of the key steps of the Theorem in separate chapters
Discusses interpretations of the Theorem made by celebrated contemporary thinkers
Sheds light on the wider extra-mathematical and philosophical implications of Gödel's theories
Written in an accessible, non-technical style
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One of themselves, even a prophet of their own, said, The Cretans are always liars … This witness is true. (St Paul, Epistle to Titus, 1: 12–13)
1
Foundations and Paradoxes
In this chapter and the following, we shall learn lots of things in a short time.1 Initially, some of the things we will gain knowledge of may appear unrelated to each other, and their overall usefulness might not be clear either. However, it will turn out that they are all connected within Gödel’s symphony. Most of the work of these two chapters consists in preparing the instruments in order to play the music. We will begin by acquiring familiarity with the phenomenon of self-reference in logic–a phenomenon which, according to many, has to be grasped if one is to understand the deep meaning of Gödel’s result. Self-reference is closely connected to the famous logical paradoxes, whose understanding is also important to fully appreciate the Gödelian construction–a construction that, as we shall see, owes part of its timeless fascination to its getting quite close to a paradox without falling into it.
But what is a paradox? A common first definition has it that a paradox is the absurd or blatantly counter-intuitive conclusion of an argument, which starts with intuitively plausible premises and advances via seemingly acceptable inferences. In The Ways of Paradox, Quine claims that “a paradox is just any conclusion that at first sounds absurd but that has an argument to sustain it.”2 We shall be particularly concerned not just with sentences that are paradoxical in the sense of being implausible, or contrary to common sense (“paradox” intended as something opposed to the
, or to what is
, entren ched in pervasive and/or authoritative opinions), but with sentences that constitute authentic, full-fledged contradictions. A paradox in this strict sense is also called an antinomy.
However, sometimes the whole argument is also called a paradox.3 So we have Graham Priest maintaining that “[logical] paradoxes are all arguments starting with apparently analytic principles … and proceeding via apparently valid reasoning to a conclusion of the form ‘α and not-α’.”4
Third, at times a paradox is considered as a set of jointly inconsistent sentences, which are nevertheless credible when addressed separately.5
The logical paradoxes are usually subdivided into the semantic and set-theoretic. What is semantics, to begin with? We can understand the notion by contrasting it with that of syntax. Talking quite generally, in the study of a language (be it a natural language such as English or German, or an artificial one such as the notational systems developed by formal logicians), semantics has to do with the relationship between the linguistic signs (words, noun phrases, sentences) and their meanings, the things those signs are supposed to signify or stand for. Syntax, on the other hand, has to do with the symbols themselves, with how they can be manipulated and combined to form complex expressions, without taking into account their (intended) meanings.
Typically, such notions as truth and denotation are taken as pertaining to semantics.6 Importantly, a linguistic notion is classified as (purely) syntactic when its specification or definition does not refer to the meanings of linguistic expressions, or to the truth and falsity of sentences. The distinction between syntax and semantics is of the greatest importance: I shall refer to it quite often in the following, and the examples collected throughout the book should help us understand it better and better.
The set-theoretic paradoxes concern more technical notions, such as those of membership and cardinality. These paradoxes have cast a shadow over set theory, whose essentials are due to the great nineteenth-century mathematician Georg Cantor, and which was developed by many mathematicians and logicians in the twentieth century.
Nowadays, set theory is a well-established branch of mathematics. (One should speak of set theories, since there are many of them; but mathematicians refer mainly to one version, that due to Ernst Zermelo and Abraham Fraenkel, to which I shall refer in the following.) But the theory has also a profound philosophical importance, mainly because of the role it has had in the development of (and the debate on) the so-called foundations of mathematics. Between the end of the nineteenth century and the beginning of the twentieth, the great philosophers and logicians Gottlob Frege and Bertrand Russell attempted to provide a definitive, unassailable logical and philosophical foundation for mathematical knowledge precisely by means of set theory. When Gödel published his paper, the dispute on the foundations of mathematics was quite vigorous, because of a crisis produced by the discovery of some important paradoxes in the so-called naïve formulation of set theory.
In these initial chapters, therefore, we shall learn some history and some theory. On the one hand, we will have a look at the changes that logic and mathematics were undergoing at the beginning of the twentieth century, mainly because of the paradoxes: to know something of the logical and mathematical context Gödel was living in will help us understand why the Theorem was the extraordinary breakthrough it was. But we shall also learn some basic mathematical and set- theoretical concepts. Among the most important notions we will meet in this chapter is that of algorithm. By means of it, we should come to understand what it means for a given set to be (intuitively) decidable; what it means for a given set to be (intuitively) enumerable; and what it means for a given function to be (intuitively) computable. If this list of announcements on the subjects we shall learn sounds alarming, I can only say that the initial pain will be followed by the gain of seeing these separate pieces come together in the marvelous Gödelian jigsaw.
1 “This sentence is false”
I have claimed that the semantic paradoxes can involve different semantic concepts, such as denotation, definability, etc. We shall focus only on those employing the notions of truth and falsity, which are usually grouped under the label of the Liar. These are the most widely discussed in the literature–those for which most tentative solutions have been proposed. They are also the most classical, having been on the philosophical market for more than 2,000 years–a fact which, by itself, says something about the difficulty of dealing with them. The ancient Greek grammarian Philetas of Cos is believed to have lost sleep and health trying to solve the Liar paradox, his epitaph claiming: “It was the Liar who made me die/And the bad nights caused thereby.”
One of the most ancient versions of semantic paradox appears in St Paul’s Epistle to Titus. Paul blames a “Cretan prophet,” who was to be identified as the poet and philosopher Epimenides, and who was believed to have at one time said:
(1) All Cretans always lie.
Actually, (1) is not a real paradox in the strict sense of a sentence which, on the basis of our bona fide intuitions, would entail a violation of the Law of Non-Contradiction. It is just a sentence that, on the basis of those intuitions, cannot be true. It is self-defeating for a Cretan to say that Cretans always lie: if this were true–that is, if it were the case that all sentences uttered by any Cretan are false–then (1), being uttered by the Cretan Epimenides, would have to be false itself, against the initial hypothesis. However, there is no contradiction yet: (1) can be just false under the (quite plausible) hypothesis that some Cretan sometimes said something true.
We are dealing with a full-fledged Liar paradox (also attributed to a Greek philosopher, and probably the greatest paradoxer of Antiquity: Eubulides) when we consider the following sentence:
(2) (2) is false.
As we can see, (2) refers to itself, because it is no. 2 of the sentences highlighted in this chapter, and tells something of the very sentence no. 2. Also (1) refers to itself, but does it in a different way from (2). This is what makes (1) not strictly paradoxical. Sentence (1) claims that all the members of a set of sentences (those uttered by Cretans) are false. In addition, it belongs to that very set, due to its being uttered by a Cretan. Therefore (1) can be simply false, under the empirical hypothesis that some sentence uttered by a Cretan, and different from (1), is true. This is also what makes it look so odd: it is unsatisfactory that a logical paradox is avoided only via the empirical fact that some Cretan sometimes said something true.
Some form of self-reference can be detected in (almost) all paradoxes, so that the phenomenon of self-reference as such has been held responsible for the antinomies. Nevertheless, lots of self-referential sentences are harmless, in that we seem to be able to ascertain their truth value in an unproblematic way. For instance, you may easily observe that, among the following, (3) and (4) are true, whereas (5) is false:
(3) (3) is a grammatically well-formed sentence.
(4) (4) is a sentence contained in There’s Something About Gödel!
(5) (5) is a sentence printed with yellow ink.
In contrast, (2) is not harmless at all. Let us reason by cases. Suppose (2) is true: then what it says is the case, so it’s false. Suppose then (2) is false. This is what it claims to be, so it’s true. If we accept the Principle of Bivalence, that is, the principle according to which all sentences are either true or false, both alternatives lead to a paradox: (2) is true and false! To claim that something is both true and false is to produce a denial of the Law of Non-Contradiction. And this is how our bona fide intuitions lead us to a contradiction, via a simple reasoning by cases.
Other versions of the Liar are called strengthened Liars,7 or also revenge Liars (whereas (2) may be called the “standard” Liar):
(6) (6) is not true.
(7) (7) is false or neither true nor false.
The reason why sentences such as (6) deserve the title of strengthened Liars is the following. Some logicians (including the best one of our times, Saul Kripke) have proposed circumventing the standard Liar (2) by dispensing with the Principle of Bivalence, that is, by admitting that some sentences can be neither true nor false, and that (2) is among them. Sentence (2) is a statement such that, if it were false, it would be true, and if it were true, it would be false. But we can avoid the contradiction by granting that (2) is neither. Such a solution has some problems with sentences such as (6), which appear to deliver a contradiction even when we dismiss Bivalence. In this case, the set of sentences is divided into three subsets: the true ones, the false ones, and those which are neither. Now we can reason by cases again with (6): either (6) is true, or it is false, or neither. If it’s true, then what it says is the case, so it’s not true. If it’s false or neither true nor false, then it is not true. However, this is what it claims to be, so in the end it’s true. Whatever option we pick, (6) turns out to be both true and untrue, and we are back to contradiction. This Liar thus gains “rev...
