Basic arithmetic

It seems to be child's play to grasp the fundamental notions involved in the arithmetic of addition and multiplication. Starting from zero, there is a sequence of ‘counting’ numbers, each having just one immediate successor. This sequence of numbers – officially, the natural numbers – continues without end, never circling back on itself; and there are no ‘stray’ numbers, lurking outside this sequence. Adding n to m is the operation of starting from m in the number sequence and moving n places along. Multiplying m by n is the operation of (starting from zero and) repeatedly adding m, n times. And that's about it.

Once these fundamental notions are in place, we can readily define many more arithmetical notions in terms of them. Thus, for any natural numbers m and n, m < n iff there is a number k ≠ 0 such that m + k = n. m is a factor of n iff 0 < m and there is some number k such that 0 < k and m × k = n. m is even iff it has 2 as a factor. m is prime iff 1 < m and m's only factors are 1 and itself. And so on.

Using our basic and/or defined concepts, we can then make various general claims about the arithmetic of addition and multiplication. There are familiar truths like ‘addition is commutative’, i.e. for any numbers m and n, we have m + n = n + m.

At the end of this short chapter, we introduce the pivotal idea of a p.r. adequate theory of arithmetic, i.e. one that can appropriately capture all p.r. functions, properties and relations. Then, in the next chapter, we will show that Q and hence PA are p.r. adequate.

However, we haven't yet explained the idea of capturing a function as opposed to capturing a property or relation. So we must start with that.

Capturing a function

Suppose f is a one-place numerical function. Now define the relation Rf by saying that m has the relation Rf to n just in case f(m) = n. We'll say Rf is f's corresponding relation. Functions and their corresponding relations match up pairs of things in exactly the same way: f and Rf have the same extension, namely the set of ordered pairs 〈m, f(m)〉.

And just as the characteristic function trick (Section 11.6) allows us to take ideas defined for functions and apply them to properties and relations, this very simple tie between functions and their corresponding relations allows us to carry over ideas defined for relations and apply them to functions (total functions, as always in this book.)

For a start, consider how we can use this tie to define the idea of expressing a function using an open wff.

As we noted in Section 10.1, the intuitive principle of mathematical induction looks to be a second-order principle that quantifies over numerical properties, and which can't be directly expressed in a first-order theory that only quantifies over numbers. So you might well be wondering: why not work with a second-order arithmetic, rather than hobble ourselves by artificially forcing our formal arithmetic into a first-order straightjacket? True, we now know that – so long as it stays consistent and properly axiomatized – a richer theory won't entirely escape the reach of the Gödel-Rosser Theorem, any more than a first-order theory can. But still, we ought to say at least a little about second-order arithmetics.

Indeed, there is a pressing issue about such theories which really needs to be addressed head on at this point. For if you have done a standard university algebra course, you might very well be feeling pretty puzzled by now. Such a course typically introduces axioms for some version of ‘Second-order Peano Arithmetic’, and there is an elementary textbook proof that these axioms pin down a unique type of structure. But if this second-order arithmetic does pin down the structure of the natural numbers, then – given that any arithmetic sentence makes a determinate claim about this structure – it apparently follows that this theory does enough to settle the truth-value of every arithmetic sentence. Which makes it sound as if there can after all be a (consistent) negation-complete axiomatic theory of arithmetic richer than first-order PA, flatly contradicting the Gödel-Rosser Theorem.

The pieces we need are finally all in place. So in this chapter we at long last learn how to construct ‘Gödel sentences’ and use them to prove that PA is incomplete. Then in the next chapter, we show how our arguments can be generalized to prove that PA – and any other formal arithmetic satisfying very modest constraints – is not only incomplete but incompletable. Our initial discussion in these two chapters uses ideas from Gödel's own treatment in 1931. Then in Chapter 19, we start extending Gödel's work.

The beautiful proofs now come thick and fast: savour them slowly!

Reminders

We start with some quick reminders – or bits of headline news, if you have impatiently been skipping forward in order to get to the exciting stuff.

Fix on some acceptable scheme for coding up wffs of PA by using Gödel numbers (‘g.n.’), and coding up sequences of wffs by super Gödel numbers. Then,

1. i. The diagonalization of ϕ is ∃y(y = ⌜ϕdlcorn ∧ ϕ), where ‘⌜ϕdlcorn’ here stands in for the numeral for ϕ's g.n. – the diagonalization of ϕ(y) is thus equivalent to ϕ(⌜ϕdlcorn). (Section 15.5)

2. ii. diag(n) is a p.r. function which, when applied to a number n which is the g.n. of some wff ϕ, yields the g.n. of ϕ's diagonalization. (Section 15.6)

3. […]

Q is Σ1-complete, a fact which will turn out to be very important. But, as we saw, in other ways Q is an extremely weak theory. To derive elementary general truths like ∀x(0 + x = x) that are beyond Q's reach, we obviously will have to use a formal arithmetic that incorporates some stronger axiom(s) for proving quantified wffs. This chapter explains the induction axioms we need to add, working up to the key theory PA, first-order Peano Arithmetic.

Induction and the Induction Schema

(a) In informal argumentation, we frequently appeal to the following principle of mathematical induction in order to prove general claims:

In fact, we used informal inductions in the last chapter. For example, to prove that Q correctly decides all Σ1 wffs, we in effect began: let n have the property P if Q-correctly-decides-Σ1-wffs-of-degree-no-more-than n. Then we argued (i) 0 has property P, and (ii) for any number n, if it has P, then n + 1 also has P. So we concluded (iii) every number has P, i.e. Q correctly decides any Σ1 wff, whatever its degree.

The previous chapter concerned axiomatized formal theories in general. This chapter introduces some key concepts we need in describing formal arithmetics in particular, notably the concepts of expressing and capturing numerical properties. But we need to start with two quick preliminary sections, about notation and about the very idea of a property.

Three remarks on notation

(a) Gödel's First Incompleteness Theorem is about the limitations of axiomatized formal theories of arithmetic: if a theory T is consistent and satisfies some other fairly minimal constraints, we can find arithmetical truths that can't be derived in T. Evidently, in discussing Gödel's result, it will be very important to be clear about when we are working ‘inside’ some specified formal theory T and when we are talking informally ‘outside’ that particular theory (e.g. in order to establish truths that T can't prove).

However, we do want our informal talk to be compact and perspicuous. Hence we will tend to borrow the standard logical notation from our formal languages for use in augmenting mathematical English (so, for example, we might write ‘∀x∀y(x + y = y + x)’ as a compact way of expressing the ‘ordinary’ arithmetic truth that the order in which you sum numbers doesn't matter).

An algorithm, we said, is a sequential step-by-step procedure which can be fully specified in advance of being applied to any particular input. Every minimal step is to be ‘small’ in the sense that it is readily executable by a calculator with limited cognitive resources. The rules for moving from one step to the next must be entirely determinate and self-contained. And an algorithmic procedure is to deliver its output after a finite number of computational steps. The Church–Turing Thesis, as we are interpreting it, is then the claim that a numerical function is effectively computable by such an algorithm iff it is µ-recursive/ Turing-computable (note, we continue to focus throughout on total functions).

The Thesis, to repeat, is not a claim about what computing ‘machines’ can or can't do. Perhaps there can, at least in principle, be ‘machines’ that out-compute Turing machines – but if so, such hypercomputing set-ups will not be finitely executing algorithms (see Section 34.3).

And as we also stressed, it is enough for our wider purposes that we accept the Thesis's link between effective computability by an algorithm and µ-recursiveness/Turing computability; we don't have to take a particular stance on the status of the Thesis.

But all the same, it is very instructive to see how we might go about following Turing (and perhaps Gödel) in defending a bolder stance by trying to give an informal proof that the intuitive and formal concepts are indeed coextensive.

In this chapter, we introduce Turing's classic analysis of algorithmic computability. And then – in the next chapter – we will establish the crucial result that the Turing-computable total functions are exactly the µ-recursive functions. This result is fascinating in its own right; it is hugely important historically; and it enables us later to establish some further results about recursiveness and incompleteness in a particularly neat way. So let's dive in without more ado.

The basic conception

Think of executing an algorithmic computation ‘by hand’, using pen and paper. We follow strict rules for writing down symbols in various patterns. To keep things tidy, let's write the symbols neatly one-by-one in the squares of some suitable square-ruled paper. Eventually – assuming that we don't find ourselves carrying on generating output for ever – the computation process stops and the result of the computation is left written down in some block of squares on the paper.

Now, Turing suggests, using a two-dimensional grid for writing down the computation is not of the essence. Imagine cutting up the paper into horizontal strips a square deep, and pasting these together into one long tape. We could use that as an equivalent workspace.

Using a rich repertoire of symbols is not of the essence either. Suppose some computational system uses 27 symbols. Number these off using a five-binarydigit code (so the 14th symbol, for example, gets the code ‘01110’).

Back in Chapter 8, we introduced the weak arithmetic Q, and soon saw that it is boringly incomplete. Then in Chapter 10 we introduced the much stronger first-order theory PA, and remarked that we couldn't in the same easy way show that it fails to decide some elementary arithmetical claims. However, in the last chapter it has turned out that PA also remains incomplete.

Still, that result in itself isn't yet hugely exciting, even if it is a bit surprising. After all, just saying that a particular theory T is incomplete leaves wide open the possibility that we can patch things up by adding an axiom or two more, to get a complete theory T+. As we said at the very outset, the real force of Gödel's arguments is that they illustrate general methods which can be applied to any theory satisfying modest conditions in order to show that it is incomplete. This reveals that a theory like PA is not only incomplete but in a good sense incompletable.

The present chapter explains these crucial points.

Generalizing the semantic argument

In Section 16.3, we showed that PA is incomplete on the semantic assumption that its axioms are true (and its logic is truth-preserving). In this section, we are going to extend this first ‘semantic’ argument for incompleteness to other theories.

In Chapter 3, we proved that the theorems of any properly axiomatized theory – and hence, in particular, the theorems of any properly axiomatized arithmetic – can be effectively enumerated. In this chapter, we prove by contrast that the truths of any sufficiently expressive arithmetic language can't be effectively enumerated (we will explain in just a moment what ‘sufficiently expressive’ means).

Suppose then that T is a properly axiomatized theory with a sufficiently expressive language. Since T is axiomatized, its theorems can be effectively enumerated. Since T's language is sufficiently expressive, the truths of its language can't be effectively enumerated. Hence the theorems and the truths can't be the same: either some T-theorems aren't truths, or some truths aren't T-theorems. Let's concentrate on sound theories whose theorems are all true. Then for any sound axiomatized theory T which is sufficiently expressive, there will be truths which aren't T-theorems. Let ϕ be such an unprovable truth: then ¬ϕ will be false, so that too will be unprovable in our sound theory T. Hence T must be negation incomplete.

So much for the headline news. The rest of this chapter fills in the details.

Sufficiently expressive languages

Recall: a two-place relation R is effectively decidable iff there is an algorithmic procedure that decides whether Rmn, for any given m and n (Section 2.2). And a relation R can be expressed in language L iff there is an open L-wff ϕ such that is true iff Rmn (Section 4.5).

Theorem 5.7, our first shot at an incompleteness theorem, applies to sound theories. But we have already remarked in Section 1.2 that Gödel's arguments show that we don't need to assume soundness to prove incompleteness. In this chapter we see how to argue from consistency to incompleteness.

But if we are going to weaken one assumption (from soundness to mere consistency) we'll need to strengthen another assumption: we'll now consider theories that don't just express enough but which can capture, i.e. prove, enough.

Starting in Chapter 8, we'll begin examining various formal theories of arithmetic ‘from the bottom up’, in the sense of first setting down the axioms of the theories and then exploring what the different theories are capable of proving. For the moment, however, we are continuing to proceed the other way about. In the previous chapter, we considered theories that have sufficiently expressive languages, and so can express what we'd like any arithmetic to be able to express. Now we introduce the companion concept of a sufficiently strong theory, which is one that by definition can prove what we'd like any moderately competent theory of arithmetic to be able to prove about decidable properties of numbers. We then establish some easy but quite deep results about such theories.

The idea of a ‘sufficiently strong’ theory

Suppose that P is some effectively decidable property of numbers, i.e. one for which there is a mechanical procedure for deciding, given a natural number n, whether n has property P or not.

We have achieved our first main goal, namely to prove Gödel's First Incompleteness Theorem. And it will do no harm to pause for breath and quickly survey what we've established and how we established it. Equally importantly, we should make it clear what we have not proved. The Theorem attracts serious misunderstandings. We will briefly block a few of these.

What we've proved

To begin with the headlines about what we have proved (we are going to be repeating ourselves, but – let's hope! – in a good way). Suppose we are trying to regiment the truths of basic arithmetic – i.e. the truths expressible in terms of successor, addition, multiplication, and the apparatus of first-order logic. Ideally, we'd like to construct a consistent theory T whose language includes LA and which proves all the truths of LA (and no falsehoods). So we'd like T to be negation complete, at least for sentences of LA. But, given some entirely natural assumptions, there can't be such a negation-complete theory.

The first natural assumption is that T should be set up so that it is effectively decidable whether a putative T-proof really is a well-constructed derivation from T's axioms. So, in short, we want T to be a properly axiomatized theory. Indeed, we surely want more: we want it to be decidable what counts as a T-proof without needing open-ended search procedures (if would be a very odd kind of theory where, e.g., checking whether some wff is an axiom takes an unbounded search).

This chapter falls into three parts. We first introduce Gödel's simple but wonderfully powerful idea of associating the expressions of a formal theory with code numbers. In particular, we'll fix on a scheme for assigning code numbers first to expressions of LA and then to proof-like sequences of expressions. This double coding scheme will correlate various syntactic properties with purely numerical properties.

For example, take the syntactic property of being a term of LA. We can define a corresponding numerical property Term, where Term(n) holds just when n codes for a term. Likewise, we can define Atom(n), Wff(n), and Sent(n) which hold just when n codes for an atomic wff, a wff, or a closed wff (sentence) respectively. It will be easy to see that these numerical properties are primitive recursive ones.

More excitingly, we can define the numerical relation Prf (m, n) which holds just when m is the code number in our scheme of a PA-derivation of the sentence with number n. It will also be easy to see – at least in an informal way – that this relation too is primitive recursive.

The second part of the chapter introduces the idea of the diagonalization of a wff. This is the idea of taking a wff ϕ(y), and substituting (the numeral for) its own code number in place of the free variable.

The title of Gödel's great paper is ‘On formally undecidable propositions of Principia Mathematica and related systems I’. And as we noted in Section 18.5, his First Incompleteness Theorem does indeed undermine Principia's logicist ambitions. But logicism wasn't really Gödel's main target. For, by 1931, much of the steam had already gone out of the logicist project. Instead, the dominant project for showing that classical infinitary mathematics is in good order was Hilbert's Programme, which we have already mentioned a few times. This provided the real impetus for Gödel's early work; it is time we filled out just a bit more of the story.

However, this book certainly isn't the place for a detailed treatment of the changing ideas of Hilbert and his followers as their ideas developed pre- and post-Gödel; nor is it the place for an extended discussion of the later fate of Hilbertian ideas. So our necessarily brief remarks will do no more than sketch the logical geography of some broadly Hilbertian territory: those with more of a bent for the history of logic can be left to fight over e.g. the question of Hilbert's precise path through the landscape.

Another topic we'll take up in this Interlude is the vexed one of the impact of the incompleteness theorems, and in particular the Second Theorem, on the issue of mechanism: do Gödelian results show that minds cannot be machines?

In the last chapter, we gave some headline news about the Second Theorem for PA, and about how it rests on the Formalized First Theorem. In this actionpacked chapter, we'll say something about how the Formalized First Theorem is proved. More exactly, we state the so-called Hilbert-Bernays-Löb derivability conditions on the provability predicate for theory T and show that these suffice for proving the Formalized First Theorem for T, and hence the Second Theorem. We'll also prove a very nice theorem due to Löb.

More notation

To improve readability, let's introduce some neat notation. We will henceforth abbreviate ProvT(⌜ϕ⌝) simply by ▪Tϕ. So ConT can now alternatively be abbreviated as ¬▪T⊢.

Two comments. First, note that our box symbol actually does a double job: it further abbreviates the long predicative expression already abbreviated by ProvT, and it absorbs the corner quotes that turn a wff into the standard numeral for that wff's Gödel number. If you are logically pernickety, then you might be rather upset about introducing a notation which in this way rather disguises the complex logical character of what is going on. But my line is that abbreviatory convenience here trumps notational perfectionism.

Second, we will very often drop the explicit subscript from the box symbol, and let context supply it. We'll also drop other subscripts in obvious ways. For example, here's the Formalized First Theorem for theory T in our new notation:

The turnstile signifies the existence of a proof in T's deductive system.

In the last Interlude, we gave a five-stage map of our route to Gödel's First Incompleteness Theorem. The first two stages we mentioned are now behind us. They involved (1) introducing the standard theories Q and PA, then (2) defining the p.r. functions and – the hard bit! – proving Q's p.r. adequacy. In order to do the hard bit, we have already used one elegant idea from Gödel's epoch-making 1931 paper, namely the β-function trick. But most of his proof is still ahead of us: at the end of this Interlude, we'll review the stages that remain.

But first, let's relax for a moment after our labours, and take a very short look at some of the scene-setting background. We'll say more about the historical context in a later Interlude (Chapter 28). But for now, we ought at least to say enough to explain the title of Gödel's great paper, ‘On formally undecidable propositions of Principia Mathematica and related systems I’.

Principia's logicism

As we noted in Section 10.8, Frege aimed in Grundgesetze der Arithmetik to reconstruct arithmetic on a secure footing by deducing it from logic plus definitions. But – in its original form – his overall logicist project flounders on Frege's fifth Basic Law, which leads to contradiction. And the fatal flaw that Russell exposed in Frege's system was not the only paradox to bedevil early treatments of the theory of classes.