Is every theorem provable?

“Take any definite unsolved problem, such as … the existence of an infinite number of prime numbers of the form \(2^n + 1\). However unapproachable these problems may seem to us and however helpless we stand before them, we have, nevertheless, the firm conviction that their solution must follow by a finite number of purely logical processes…”
“…This conviction of the solvability of every mathematical problem is a powerful incentive to the worker. We hear within us the perpetual call: There is the problem. Seek its solution. You can find it by pure reason, for in mathematics there is no ignorabimus.”, David Hilbert, 1900.

“The meaning of a statement is its method of verification.”, Moritz Schlick, 1938 (aka “The verification principle” of logical positivism)

The problems shown uncomputable in Chapter 9, while natural and important, still intimately involved NAND-TM programs or other computing mechanisms in their definitions. One could perhaps hope that as long as we steer clear of functions whose inputs are themselves programs, we can avoid the “curse of uncomputability”. Alas, we have no such luck.

In this chapter we will see an example of a natural and seemingly “computation free” problem that nevertheless turns out to be uncomputable: solving Diophantine equations. As a corollary, we will see one of the most striking results of 20th century mathematics: Gödel’s Incompleteness Theorem, which showed that there are some mathematical statements (in fact, in number theory) that are inherently unprovable. We will actually start with the latter result, and then show the former.

Hilbert’s Program and Gödel’s Incompleteness Theorem

“And what are these …vanishing increments? They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?”, George Berkeley, Bishop of Cloyne, 1734.

The 1700’s and 1800’s were a time of great discoveries in mathematics but also of several crises. The discovery of calculus by Newton and Leibnitz in the late 1600’s ushered a golden age of problem solving. Many longstanding challenges succumbed to the new tools that were discovered, and mathematicians got ever better at doing some truly impressive calculations. However, the rigorous foundations behind these calculations left much to be desired. Mathematicians manipulated infinitesimal quantities and infinite series cavalierly, and while most of the time they ended up with the correct results, there were a few strange examples (such as trying to calculate the value of the infinite series \(1-1+1-1+1+\ldots\)) which seemed to give out different answers depending on the method of calculation. This led to a growing sense of unease in the foundations of the subject which was addressed in the works of mathematicians such as Cauchy, Weierstrass, and Riemann, who eventually placed analysis on firmer foundations, giving rise to the \(\epsilon\)’s and \(\delta\)’s that students taking honors calculus grapple with to this day.

In the beginning of the 20th century, there was an effort to replicate this effort, in greater rigor, to all parts of mathematics. The hope was to show that all the true results of mathematics can be obtained by starting with a number of axioms, and deriving theorems from them using logical rules of inference. This effort was known as the Hilbert program, named after the influential mathematician David Hilbert.

Alas, it turns out the results we’ve seen dealt a devastating blow to this program, as was shown by Kurt Gödel in 1931:

Defining “Proof Systems”

Before proving Theorem 11.1, we need to define “proof systems” and even formally define the notion of a “mathematical statement”. In geometry and other areas of mathematics, proof systems are often defined by starting with some basic assumptions or axioms and then deriving more statements by using inference rules such as the famous Modus Ponens, but what axioms shall we use? What rules? We will use an extremely general notion of proof systems, not even restricting ourselves to ones that have the form of axioms and inference.

Mathematical statements. At the highest level, a mathematical statement is simply a piece of text, which we can think of as a string \(x\in \{0,1\}^*\). Mathematical statements contain assertions whose truth does not depend on any empirical fact, but rather only on properties of abstract objects. For example, the following is a mathematical statement:¹

“The number \(2\),\(696\),\(635\),\(869\),\(504\),\(783\),\(333\),\(238\),\(805\),\(675\),\(613\), \(588\),\(278\),\(597\),\(832\),\(162\),\(617\),\(892\),\(474\),\(670\),\(798\),\(113\) is prime”.

Mathematical statements do not have to involve numbers. They can assert properties of any other mathematical object including sets, strings, functions, graphs and yes, even programs. Thus, another example of a mathematical statement is the following:²

The following Python function halts on every positive integer n

def f(n):
    if n==1: return 1
    return f(3*n+1) if n % 2 else f(n//2)

Proof systems. A proof for a statement \(x\in \{0,1\}^*\) is another piece of text \(w\in \{0,1\}^*\) that certifies the truth of the statement asserted in \(x\). The conditions for a valid proof system are:

1. (Effectiveness) Given a statement \(x\) and a proof \(w\), there is an algorithm to verify whether or not \(w\) is a valid proof for \(x\). (For example, by going line by line and checking that each line follows from the preceding ones using one of the allowed inference rules.)

2. (Soundness) If there is a valid proof \(w\) for \(x\) then \(x\) is true.

These are quite minimal requirements for a proof system. Requirement 2 (soundness) is the very definition of a proof system: you shouldn’t be able to prove things that are not true. Requirement 1 is also essential. If there is no set of rules (i.e., an algorithm) to check that a proof is valid then in what sense is it a proof system? We could replace it with a system where the “proof” for a statement \(x\) is “trust me: it’s true”.

We formally define proof systems as an algorithm \(V\) where \(V(x,w)=1\) holds if the string \(w\) is a valid proof for the statement \(x\). Even if \(x\) is true, the string \(w\) does not have to be a valid proof for it (there are plenty of wrong proofs for true statements such as 4=2+2) but if \(w\) is a valid proof for \(x\) then \(x\) must be true.

Gödel’s Incompleteness Theorem: Computational variant

Our first formalization of Theorem 11.1 involves statements about Turing machines. We let \(\mathcal{H}\) be the set of strings \(x\in \{0,1\}^*\) that have the form “Turing machine \(M\) does not halt on the zero input”.

Quantified integer statements

There is something “unsatisfying” about Theorem 11.3. Sure, it shows there are statements that are unprovable, but they don’t feel like “real” statements about math. After all, they talk about programs rather than numbers, matrices, or derivatives, or whatever it is they teach in math courses. It turns out that we can get an analogous result for statements such as “there are no positive integers \(x\) and \(y\) such that \(x^2 - 2 = y^7\)”, or “there are positive integers \(x,y,z\) such that \(x^2 + y^6 = z^{11}\)” that only talk about natural numbers. It doesn’t get much more “real math” than this. Indeed, the 19th century mathematician Leopold Kronecker famously said that “God made the integers, all else is the work of man.” (By the way, the status of the above two statements is unknown.)

To make this more precise, let us define the notion of quantified integer statements:

We often care deeply about determining the truth of quantified integer statements. For example, the statement that Fermat’s Last Theorem is true for \(n=3\) can be phrased as the quantified integer statement

The twin prime conjecture, that states that there is an infinite number of numbers \(p\) such that both \(p\) and \(p+2\) are primes can be phrased as the quantified integer statement

where we replace an instance of \(\ensuremath{\mathit{PRIME}}(q)\) with the statement \((q>1) \wedge \forall_{a\in \N} \forall_{b\in\N} (a=1) \vee (a=q) \vee \neg (a\times b =q)\).

The claim (mentioned in Hilbert’s quote above) that are infinitely many primes of the form \(p=2^n+1\) can be phrased as follows:

where \(\ensuremath{\mathit{DIVIDES}}(a,b)\) is the statement \(\exists_{c\in\N} b\times c = a\). In English, this corresponds to the claim that for every \(n\) there is some \(p>n\) such that all of \(p-1\)’s prime factors are equal to \(2\).

Much of number theory is concerned with determining the truth of quantified integer statements. Since our experience has been that, given enough time (which could sometimes be several centuries) humanity has managed to do so for the statements that it cared enough about, one could (as Hilbert did) hope that eventually we would be able to prove or disprove all such statements. Alas, this turns out to be impossible:

Theorem 11.8 is a direct corollary of the following result, just as Theorem 11.3 was a direct corollary of the uncomputability of \(\ensuremath{\mathit{HALTONZERO}}\):

Since a quantified integer statement is simply a sequence of symbols, we can easily represent it as a string. For simplicity we will assume that every string represents some quantified integer statement, by mapping strings that do not correspond to such a statement to an arbitrary statement such as \(\exists_{x\in \N} x=1\).

In the rest of this chapter, we will show the proof of Theorem 11.8, following the outline illustrated in Figure 11.1.

Diophantine equations and the MRDP Theorem

Many of the functions people wanted to compute over the years involved solving equations. These have a much longer history than mechanical computers. The Babylonians already knew how to solve some quadratic equations in 2000BC, and the formula for all quadratics appears in the Bakhshali Manuscript that was composed in India around the 3rd century. During the Renaissance, Italian mathematicians discovered generalization of these formulas for cubic and quartic (degrees \(3\) and \(4\)) equations. Many of the greatest minds of the 17th and 18th century, including Euler, Lagrange, Leibniz and Gauss worked on the problem of finding such a formula for quintic equations to no avail, until in the 19th century Ruffini, Abel and Galois showed that no such formula exists, along the way giving birth to group theory.

However, the fact that there is no closed-form formula does not mean we can not solve such equations. People have been solving higher degree equations numerically for ages. The Chinese manuscript Jiuzhang Suanshu from the first century mentions such approaches. Solving polynomial equations is by no means restricted only to ancient history or to students’ homework. The gradient descent method is the workhorse powering many of the machine learning tools that have revolutionized Computer Science over the last several years.

But there are some equations that we simply do not know how to solve by any means. For example, it took more than 200 years until people succeeded in proving that the equation \(a^{11} + b^{11} = c^{11}\) has no solution in integers.³ The notorious difficulty of so called Diophantine equations (i.e., finding integer roots of a polynomial) motivated the mathematician David Hilbert in 1900 to include the question of finding a general procedure for solving such equations in his famous list of twenty-three open problems for mathematics of the 20th century. I don’t think Hilbert doubted that such a procedure exists. After all, the whole history of mathematics up to this point involved the discovery of ever more powerful methods, and even impossibility results such as the inability to trisect an angle with a straightedge and compass, or the non-existence of an algebraic formula for quintic equations, merely pointed out to the need to use more general methods.

Alas, this turned out not to be the case for Diophantine equations. In 1970, Yuri Matiyasevich, building on a decades long line of work by Martin Davis, Hilary Putnam and Julia Robinson, showed that there is simply no method to solve such equations in general:

As usual, we assume some standard way to express numbers and text as binary strings. The constant \(100\) is of course arbitrary; the problem is known to be uncomputable even for polynomials of degree four and at most 58 variables. In fact the number of variables can be reduced to nine, at the expense of the polynomial having a larger (but still constant) degree. See Jones’s paper for more about this issue.

Hardness of quantified integer statements

We will not prove the MRDP Theorem (Theorem 11.10). However, as we mentioned, we will prove the uncomputability of \(\ensuremath{\mathit{QIS}}\) (i.e., Theorem 11.9), which is a special case of the MRDP Theorem. The reason is that a Diophantine equation is a special case of a quantified integer statement where the only quantifier is \(\exists\). This means that deciding the truth of quantified integer statements is a potentially harder problem than solving Diophantine equations, and so it is potentially easier to prove that \(\ensuremath{\mathit{QIS}}\) is uncomputable.

Our proof of the uncomputability of \(\ensuremath{\mathit{QIS}}\) (i.e. Theorem 11.9) will, as usual, go by reduction from the Halting problem, but we will do so in two steps:

1. We will first use a reduction from the Halting problem to show that deciding the truth of quantified mixed statements is uncomputable. Quantified mixed statements involve both strings and integers. Since quantified mixed statements are a more general concept than quantified integer statements, it is easier to prove the uncomputability of deciding their truth.

2. We will then reduce the problem of quantified mixed statements to quantified integer statements.

Step 1: Quantified mixed statements and computation histories

We define quantified mixed statements as statements involving not just integers and the usual arithmetic operators, but also string variables as well.

For example, the true statement that for every string \(a\) there is a string \(b\) that corresponds to \(a\) in reverse order can be phrased as the following quantified mixed statement

Quantified mixed statements are more general than quantified integer statements, and so the following theorem is potentially easier to prove than Theorem 11.9:

Step 2: Reducing mixed statements to integer statements

We now show how to prove Theorem 11.9 using Theorem 11.13. The idea is again a proof by reduction. We will show a transformation of every quantified mixed statement \(\varphi\) into a quantified integer statement \(\xi\) that does not use string-valued variables such that \(\varphi\) is true if and only if \(\xi\) is true.

To remove string-valued variables from a statement, we encode every string by a pair integer. We will show that we can encode a string \(x\in \{0,1\}^*\) by a pair of numbers \((X,n)\in \N\) s.t.

• \(n=|x|\)

• There is a quantified integer statement \(\ensuremath{\mathit{COORD}}(X,i)\) that for every \(i<n\), will be true if \(x_i=1\) and will be false otherwise.

This will mean that we can replace a “for all” quantifier over strings such as \(\forall_{x\in \{0,1\}^*}\) with a pair of quantifiers over integers of the form \(\forall_{X\in \N}\forall_{n\in\N}\) (and similarly replace an existential quantifier of the form \(\exists_{x\in \{0,1\}^*}\) with a pair of quantifiers \(\exists_{X\in \N}\exists_{n\in\N}\)) . We can then replace all calls to \(|x|\) by \(n\) and all calls to \(x_i\) by \(\ensuremath{\mathit{COORD}}(X,i)\). This means that if we are able to define \(\ensuremath{\mathit{COORD}}\) via a quantified integer statement, then we obtain a proof of Theorem 11.9, since we can use it to map every mixed quantified statement \(\varphi\) to an equivalent quantified integer statement \(\xi\) such that \(\xi\) is true if and only if \(\varphi\) is true, and hence \(\ensuremath{\mathit{QMS}}(\varphi)=\ensuremath{\mathit{QIS}}(\xi)\). Such a procedure implies that the task of computing \(\ensuremath{\mathit{QMS}}\) reduces to the task of computing \(\ensuremath{\mathit{QIS}}\), which means that the uncomputability of \(\ensuremath{\mathit{QMS}}\) implies the uncomputability of \(\ensuremath{\mathit{QIS}}\).

The above shows that proof of Theorem 11.9 all boils down to finding the right encoding of strings as integers, and the right way to implement \(\ensuremath{\mathit{COORD}}\) as a quantified integer statement. To achieve this we use the following technical result :

Using Lemma 11.15 we can encode a \(x\in\{0,1\}^*\) by the numbers \((X,n)\) where \(X = \prod_{x_i=1} p_i\) and \(n=|x|\). We can then define the statement \(\ensuremath{\mathit{COORD}}(X,i)\) as

where \(\ensuremath{\mathit{DIVIDES}}(a,b)\), as before, is defined as \(\exists_{c\in\N} a\times c = b\). Note that indeed if \(X,n\) encodes the string \(x\in \{0,1\}^*\), then for every \(i<n\), \(\ensuremath{\mathit{COORD}}(X,i)=x_i\), since \(p_i\) divides \(X\) if and only if \(x_i=1\).

Thus all that is left to conclude the proof of Theorem 11.9 is to prove Lemma 11.15, which we now proceed to do.

To sum up we have shown that for every quantified mixed statement \(\varphi\), we can compute a quantified integer statement \(\xi\) such that \(\ensuremath{\mathit{QMS}}(\varphi)=1\) if and only if \(\ensuremath{\mathit{QIS}}(\xi)=1\). Hence the uncomputability of \(\ensuremath{\mathit{QMS}}\) (Theorem 11.13) implies the uncomputability of \(\ensuremath{\mathit{QIS}}\), completing the proof of Theorem 11.9, and so also the proof of Gödel’s Incompleteness Theorem for quantified integer statements (Theorem 11.8).

Exercises

Bibliographical notes

As mentioned before, Gödel, Escher, Bach (Hofstadter, 1999) is a highly recommended book covering Gödel’s Theorem. A classic popular science book about Fermat’s Last Theorem is (Singh, 1997) .

Cantor’s are used for both Turing and Gödel’s theorems. In a twist of fate, using techniques originating from the works of Gödel and Turing, Paul Cohen showed in 1963 that Cantor’s Continuum Hypothesis is independent of the axioms of set theory, which means that neither it nor its negation is provable from these axioms and hence in some sense can be considered as “neither true nor false” (see (Cohen, 2008) ). The Continuum Hypothesis is the conjecture that for every subset \(S\) of \(\mathbb{R}\), either there is a one-to-one and onto map between \(S\) and \(\N\) or there is a one-to-one and onto map between \(S\) and \(\mathbb{R}\). It was conjectured by Cantor and listed by Hilbert in 1900 as one of the most important problems in mathematics. See also the non-conventional survey of Shelah (Shelah, 2003) . See here for recent progress on a related question.

Thanks to Alex Lombardi for pointing out an embarrassing mistake in the description of Fermat’s Last Theorem. (I said that it was open for exponent 11 before Wiles’ work.)

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Compiled on 12/06/2023 00:07:25

Copyright 2023, Boaz Barak.
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