Mathematics / Critical Essay

Vol. 5, NO. 1 / December 2019

The creation of numbers was the creation of things.
Thierry of Chartres1

It is a useful phrase—the director’s cut. It is useful because it sets the scene: the masterful director; his view; his vision. Mathematics has always been rich in its directors of note. In the third century BCE, Euclid subordinated two-dimensional space to five axioms. Twenty-three hundred years later, Giuseppe Peano brought the natural numbers under axiomatic control. The natural numbers are the oldest objects in our intellectual experience. Without them, we would be lost. Peano arithmetic comprises Peano’s axioms together with their logical consequences. It is a theory as rich as any in the sciences.2 The hoarse, excitable Peano is behind the lens. Peano arithmetic, PA, is what he sees; the natural numbers, , are what they are.

There it is: <PA, >, the director’s cut.

Formal Arithmetic

Kurt Gödel published his remarkable incompleteness theorems in 1931.3 With the exception of John von Neumann, mathematicians found Gödel’s work very difficult.4 It remains difficult, uncanny in its power.5 “The formulae of a formal system,” Gödel wrote, “are, looked at from outside [äusserlich betrachtet], finite series of basic signs.”6 The basic signs are what they seem, but who is looking at them from the outside? It can hardly be Peano. The symbols in Peano arithmetic are transparent: Peano is looking through them. It is what we all do. A contrived withdrawal is required before symbols can be seen as signs.

Peano arithmetic is expressed in the language of mathematics and in ordinary English.7 For all natural numbers, m = n if and only if S(m) = S(n). Two numbers are equal if and only if their successors are equal. The exuberant mathematician is handling things with an easy familiarity. And why not? He commands the scene: Peano’s axioms, Peano arithmetic, and, beyond them, the natural numbers. The logician stands apart. His is an orthopedic practice. He is concerned to see the articulated skeleton beneath Peano arithmetic, its formal system FPA.8 It is the logician who is looking at things from the outside.

Like certain objects in the physical world, FPA is atomic in nature. Its atoms are its basic signs; its molecules, their combination. Logical constants are first: ‘~’ (not), ‘’ (or), ‘’ (all), ‘0’ (zero), ‘S’ (the successor of), and, in ‘)’ and ‘(’, right and left parentheses. Although sparse, these signs cover the contingencies, ordinary numerals vanishing in favor of a stuttering ‘S’, so that ‘S S S S (0)’ in FPA does duty for ‘4’ in English. Individual variables are next: ‘x,’ ‘y,’ ‘z,’…, and the like. With arithmetical functions, predicates, and relations, it is more of the same.9 Whatever the signs, they must be combined, and this is a procedure superintended by specific rules. Elementary formulas are given explicitly. The well-formed formulas are then defined as the smallest set containing all of the elementary formulas and their negations, disjunctions, and universal quantifications.

If Peano arithmetic is an arithmetic theory, it is also an axiomatic theory. Theorems follow from the axioms in a gross cascade. It is the logician who must specify the rules of inference of a formal system, the flow of its deductions. An appeal to meaning is not allowed. A formal system is a system of signs. It might as well be a system of shapes. Some inexorable sense of rightness nevertheless prevails. Given

Mierīgie ūdēņi ir tie dziļākie Mierīgie ūdēņi ir tie dziļākie


Mierīgie ūdēņi ir tie dziļākie,

it follows that

Mierīgie ūdēņi ir tie dziļākie.

If the underlying language is inscrutable,10 this inference is crystal clear, contingent only on the antecedent identification of ‘⊃’ as a logical particle.11 The rules of inference governing a formal system are procedural; but as rules of procedure, they correspond, or express, a natural motion of the human mind, the power to move from one proposition to another.

Peano arithmetic is constructed from the Peano axioms,12 but in the context of formal arithmetic, those axioms lose some of their incidental glamor and appear as axioms in virtue of being called axioms. They have been set apart by being set apart. The inference that has just led to Mierīgie ūdēņi ir tie dziļākie is modus ponens, and modus ponens is one of the two rules of inference that figure in Gödel’s argument. The second expresses the common understanding that what holds for anything holds for something. In these cases, to go inferentially from one formula to another is go somewhere at once. There are no intermediate stops. The provable formulas comprise the smallest set containing the axioms of Peano arithmetic and closed under the release of immediate inference.

In FPA, a new mathematical object has come into being. Physical objects are described by the laws of physics. Their nature is disclosed. A formal system is determined by its formation rules, its rules of inference, and by the screening off that screens off its axioms. Its nature is induced. It is induced subject to a double standard of effectiveness. The well-formed formulas, the axioms, and the rules of inference make for three sets of formal objects. Standards for membership are strict. It must be possible to decide whether a given formula is well-formed and whether it is not. It must be possible to do both in a finite series of steps. And it must be possible to do as much for the axioms and the rules of inference.13

“Whatever withdraws us from the power of our senses,” Samuel Johnson observed, “whatever makes the past, the distant, or the future predominate over the present, advances us in the dignity of thinking beings.”14

The Recursive Scaffold

Peano arithmetic is a theory about the natural numbers, and these have remained unattended and unvoiced. They now come into their own. Gödel used the natural numbers to identify the formulas of a formal system, the identification so close that Gödel felt justified in referring to the basic signs of a formal system as natural numbers.15 To every basic sign, and then to every formula, and then to every sequence of formulas, and then to every series of such sequences, Gödel assigned a unique number. Basic signs are mapped to basic numbers:

‘0’ ↔ 1, ‘S’ ↔ 3, ‘~’ ↔ 5, ‘’ ↔ 7, ‘’ ↔ 9, ‘(’ ↔ 11, ‘)’ ↔ 13.

More complicated formulas are mapped to more complicated numbers. The basic signs appear in this list sheathed in single quotation marks. They are being mentioned, and not used. Those sheaths serve as their name, and their naming takes place from beyond FPA. It is an outside baptism.16 Gödel numbering is ingenious because of the way it reverses the natural line of sight. The director’s eye normally goes from signs to numbers. Gödel numbering encourages a reversal, one that goes from numbers to signs. The coordination achieved suggests a Cartesian coordinate system, but something stranger, too, and more difficult to grasp.

The prolegomenon to Gödel’s proof consists of forty-six definitions. Each depends on the one that has come before, and each embodies a primitive recursive function.17 These functions admit of a peculiar kind of definition. Both Richard Dedekind and Peano made use of the technique. The addition of any two natural numbers is resolved into two clauses. The first establishes that adding x to zero goes nowhere beyond x: 0 + x = x. The second defines addition in terms of succession and downward descent: x + S(y) = S(x + y). Three plus four is the successor to three plus three. Downward descent follows: three plus three is the successor to three plus two. Descent continues until it achieves its appointed apotheosis in 0. Three plus four is the sevenfold successor of 0. Multiplication? The same. “As one can easily convince oneself,” Gödel writes, “the functions x + y, x ⋅ y, xy and furthermore the relations x < y and x = y are primitive recursive.”18

Division does not figure in the Peano axioms.19 Gödel’s first definition brings it to life:

x/y =df (z)(zx & x = y. z).

There are no surprises. Whatever the number x, it is divisible by y just in case there is a number z, less than or equal to x, such that x is the product of y and z. This is, after all, what division means. The sentence ‘x/y =df (z)(zx & x = y. z)’—the whole thing—is no part of FPA. Still, it is possible to think of ‘x/y’ as a derived sign, with ‘(z)(zx & x = y. z)’ indicating a path back to the basic signs of the system.

“We will now define a sequence of functions (relations),” Gödel goes on to write, “each of which is defined from the preceding ones.”20 The definitions accumulate, one after another, each following from the one that has gone before, and each admitting the discipline of definition by primitive recursion. The expanding sequence of definitions very quickly encompasses concepts such as formula, axiom, and immediate consequence. These are not obviously arithmetical. They are not arithmetical at all. Gödel numbering shows this earthy view to be primitive. Relationships between signs may be mirrored by relationships between numbers.

Forty-five definitions having been given; the forty-sixth, and last, defines the predicate ‘Bew (x)’:

Bew (x) =df (y)(y Bw x).

The sign ‘Bew (x),’ this definition affirms, may be replaced by ‘(y)(y Bw x).’ The sign ‘y Bw x’ is, in turn, defined by the forty-fifth definition, and so back to the first. The sign ‘Bw’ is by birth arithmetical. In the strictest of strict senses, it stands for nothing beyond itself. But by virtue of its birth, it is intended to designate a relationship between numbers. At the same time, ‘Bw’ enjoys a second interpretation by virtue of its place in logical society. It stands for the German Beweis, or proof. Under this interpretation, ‘y Bw x’ coordinates two numbers y and x, where y is the Gödel number of a proof of a formula whose Gödel number is x. These definitions fix in amber or aspic the arithmetic predicate ‘Bew (x)’ and the logical predicate ‘Bew (x)’. Peering into his own film, the director is now able to see himself peering into his own film.

A chain of definitions takes ‘Bew’ to the heart of FPA and its basic signs. The definitions that precede the last are all primitive recursive. They depend on the definitions that have gone before until they empty themselves out at 0. The forty-sixth remains apart. It stakes a claim. It says that some x is Bew. Its existential quantifier is not bounded.21 It is the difference between a closed- and an open-ended search.

So far as primitive recursion goes, it is all the difference in the world.

A Step Before the Last

There is a tight connection between Gödel’s definitions and the formulas within FPA. This is the subject of Gödel’s fifth theorem. It is a theorem in two parts. Suppose that R(n1, …, nn) is a primitive recursive function or predicate controlling the numbers n1, …, nn. The numbers, note—the real things. It follows that R(n1, …, nn) may be represented within FPA by a formula in which signs stand where vernacular variables stood: R(n1, …, nn). Underlining serves to underscore the distinction between signs within the formal system and symbols about the formal system or the natural numbers. If R(n1, …, nn) is a primitive recursive function, predicate, or relation, there is always a translate R(n1, …, nn) within FPA such that

  1. if R(n1, …, nn) is true in , then R(n1, …, nn) is provable within FPA.
  2. If R(x1, …, xn) is false—then not.

As logicians say

  R(n1, …, nk) FPA  R(n1, …, nk)

  ~R(n1, …, nk) FPA  ~R(n1, …, nk).

This claim is compelling. If ‘2 + 2 = 4’ is true in N, then its translate is provable within FPA. If not, of what use FPA? The theorem does not, by itself, identify the translate.22 What can be said in its favor is that it exists. It is hard to imagine a tighter connection, but if the connection is tight, it is tight only for the primitive recursive functions. Proof is otherwise. It is not primitive recursive, satisfying 1. but not 2. The first forty-five of Gödel’s predicates and relations are strongly represented within FPA. The forty-sixth, and last, weakly represented.

And this, too, is something logicians say.


If Peano arithmetic is consistent, there is a formula within FPA such that neither it nor its negation is provable in FPA.23 The formula is undecidable. Peano arithmetic is incomplete. Gödel’s proof of his great theorem is rebarbative—there is no other word. But in the opening paragraphs of his treatise, Gödel, writing with his own matchless concision and elegance, offers an accessible, if informal, account of his argument.

Consistency gives way in favor of the assumption that every provable formula is true. Suppose all the formulas of FPA with one free variable x so arranged that the nth formula is designated R(n). A variable x is free if it does not lie within the scope of any quantifier, so that F(x) says neither that everything is F nor that something is F. The symbol [α; n] designates from beyond FPA a formula deep in the bowels of FPA. And more: it provides a recipe for its construction. It is the formula that results when x is replaced by n in α, the variable giving way to the numeral.24

With this said, consider the class K of natural numbers such that

nK ~Bew[R(n); n].

The definition of K makes use only of concepts that have already been defined. It is of a piece with the forty-six definitions on record. It follows that there is a formula G(n) in FPA that, if only it could talk, would say that n ∈ K. “There is not the slightest difficulty,” Gödel observes, “in writing out the formula G.”25

The formula designated by G is within FPA. Thus

G = R(q)

for some number q, since the formulas of FPA have been gathered into a list in which everything is included and nothing is left out.

The conclusion of the incompleteness theorem is now budding on the bunched fingertips of this argument. [R(q); q] is an instance of [αn]. The recipe enjoined by [αn] is in force. [R(q); q] is the formula within FPA that results when q replaces x in R(q). The Gödel number of R(q) is q. Whence by a chain of definitions

G(q)  [R(q); q]  q ∈ K  ~Bew[R(q); q].

[R(q); q] is a mute collocation of signs. By itself, it says nothing, but like an ancient rune, when read rightly, it says that R(q) cannot be demonstrated.26 But G = R(q). G says as much. It says it of itself. And it must exist.

The argument proceeds by contradiction. If true, then by I, it must be provable.27 But since q ∈ K it must be unprovable as well.

It cannot be both.

The assumption that G is unprovable in FPA leads again to contradiction. The argument, but not the proof, is over.

Inexpressible Truth

At much the same time as Gödel published his incompleteness theorems, Alfred Tarski published his treatise about the concept of truth in formalized languages.28 Gödel and Tarski seem to have anticipated one another, circumstances that themselves convey an air of paradox. Formal arithmetic, having served as a system’s skeleton, must now acquire the musculature of real life—an interpretation in the natural numbers. Signs become symbols. An interpreted version of FPA has among its symbols any number of individual variables, the usual sentential connectives and quantifiers, and, at most, denumerably many predicate variables of various finite ranks. Some home assembly is yet required. By a model, logicians mean an ordered pair M = < D, φ >, where D is the domain of M, and φ a function assigning to the predicate variables of FPA relations of corresponding rank on D. An assignment α is a function that maps the individual variables of FPA onto individuals in D. Whence

α satisfies S(x, …) in M,

defined by recursion on the length of S. A sentence is a formula with no free individual variables, and, as such, is either satisfied by every assignment or by none. It is True in M, or False in M. There is no wishy-washiness about it.

Tarski’s argument is scorpion shaped. Its sting is in its tail. The first few steps are obvious. A property or relation is definable within M if and only if there is a formula in FPA defining it.29 The number x is even if it is the sum of two identical numbers. The requisite formula is ‘y (x = y + y).’ A relation in M has been defined by a formula of FPA. This is mildly marvelous, but no more than that.

Let Tr be the set of Gödel numbers of the true sentences of FPA. Is Tr definable in M? If so, there must be a formula Tr(x) in FPA such that Tr(x) is satisfiable in M only for the interpretation of Tr as Tr. Suppose Tr defined in FPA so that for every sentence, there is a proof that Tr(S) ↔ S, where S is the name of S. Whereupon there is ~Tr(S) ↔ S, where S encodes the Gödel number of ‘~Tr(S).’30

There is no formula Tr. That is the sting.

The Mind as a Machine

When Gödel’s treatise first appeared in English, John Lucas concluded that men were not machines.31

Gödel’s theorem must apply to cybernetical machines, because it is of the essence of being a machine, that it should be a concrete instantiation of a formal system. It follows that given any machine which is consistent and capable of doing simple arithmetic, there is a formula which it is incapable of producing as being true—i.e. the formula is unprovable-in-the-system—but which we can see [emphasis added] to be true. It follows that no machine can be a complete or adequate model of the mind, that minds are essentially different from machines.32

Roger Penrose revived the claim and endowed it with the luster of his great reputation. “[H]uman understanding and insight,” he argued, “cannot be reduced to any set of computational rules.” This conclusion, Penrose insisted, followed from Gödel’s theorem. “[T]here must [emphasis added] be more to human thinking,” he added plaintively, “than can ever be achieved by a computer.”33 Because this thesis expressed a common human hope, or, as much, a common human fear, it was rejected by a number of philosophers and logicians.

Very early on, Hilary Putnam argued that the Lucas–Penrose argument was flawed. What can legitimately be established by the incompleteness theorem is only the conclusion that

  1. FPA is consistent if and only if G is undecidable.

What is more, 1. may itself be demonstrated within FPA. Absent a proof of the consistency of FPA, 1. does nothing to show anything.34 And by Gödel’s second incompleteness theorem, there can be no proof of the consistency of FPA within FPA.35

No one, least of all Lucas or Penrose, considering them as two heads on one body, should have argued the contrary. It is easy to see that G must be true; difficult to see that FPA is consistent; and in neither case is a proof forthcoming in FPA. What can be demonstrated is 1.; what can be seen is G. These are two different claims. They belong to two different orders of thought.36

On ne va tout de même pas pinailler pour si peu.

Nor does 1. demonstrate anything more than an incidental kinship between consistency and undecidability. They are provably equivalent within FPA. They are not are generally the same. If they were, then any consistent system would, by definition, be incomplete. Presburger arithmetic stands as a case to the contrary. Consistency and incompleteness coincide for systems rich enough to generate whole number arithmetic.

Lumbering after Putnam, Panu Raatikainen has acquired his shadow. The Lucas–Penrose argument is invalid, Raatikainen argues, because

in general, we have no idea whether the Gödel sentence of an arbitrary system is true. What we can know is only that the Gödel sentence of a system is true if and only if the system is consistent, and this much is provable in the system itself.37

The first of these claims is false, the second, incoherent. No idea? Gödel’s proof contains a strong, sustained, and powerful argument that G is not provable in FPA. This is what G says. That should give anyone some idea whether G is true. Were G false in N its negation would be true in N, and, in that case, demonstrable in FPA. This is so plainly an unwholesome conclusion that no one is disposed to endorse it, least of all Gödel: “From the remark that [R(q); q] says about itself that it is not provable, it follows at once that [R(q); q] is true, for [R(q); q] is indeed unprovable (being undecidable).”38

There remains Raatikainen’s claim that “the Gödel sentence of a system is true [emphasis added] if and only if the system is consistent.” So far as it goes, no one might scruple. That “this much is provable in the system itself” is otherwise. It provokes scruples all around. And for every good reason. There is no way to express the truth of G within FPA.39

What has been neglected in this discussion is the reason that it began: the wish, or the need, to show that men are not machines. This conclusion remains as far from the premise of any argument as it ever was. We may allow Peano to become one with FPA, and allow Lucas to see what Peano could not see and never saw. It hardly follows that Lucas may not be identified with a formal system, too, the both men, like eighteenth-century courtiers, immured in formality. There is not the slightest reason, Alonzo Church once remarked to me, that the system used to describe FPA should not itself be formalized, resulting in some still more elaborate system, FFPA, the superintendent of systems as much identified with FFPA as Peano is with FPA. In his course in mathematical logic at Princeton University, Church endeavored to do just that. It was a tedious exercise, Church, glacial and self-contained, in the end lecturing to one or two students.

The phenomenon of incompleteness led Gödel to a characteristically subtle affirmation:

Either mathematics is incompletable in this sense, that its evident axioms can never be comprised in a finite rule, that is to say, the human mind (even within the realm of pure mathematics) infinitely surpasses the powers of any finite machine, or else there exist absolutely unsolvable diophantine problems of the type specified.40

If this affirmation is subtle, it is also mysterious. A mathematician, in contemplatively considering FPA, is in a position to see something that cannot be expressed within FPA. Just how does this entail the conclusion that the human mind “infinitely [emphasis added] surpasses the power of any finite machine?” The incompleteness theorems carry something like a poisoned seed. If FFPA is a formal system, it, too, is open to incompleteness, the truth of its Gödel sentence luridly shimmering from the perspective of still some further system. Far from indicating that the human mind is not machinelike, Gödel’s argument shows only that nothing in the incompleteness theorems rules out the possibility that it is machines all the way up.

Gödel, who anticipated so much of the discussion, anticipated this as well:

Such a state of affairs would show that there is something nonmechanical in the sense that the overall plan for the historical development of machines is not mechanical. If the general plan is mechanical, then the whole [human] race can be summarized in one machine.41

Of this, all one can say is that either the general plan is mechanical or it is not, and no one knows which it is. The thesis that the human mind is a machine is returned to the same dark defile that has already entombed the study of consciousness. The defile is dark because there is nothing to be said, and it is a defile because everyone wishes to say it.

The Director’s Cut

Peano saw what he could see. We see in the light of the incompleteness theorems, and they have changed how things are seen. The truth is neither completely provable nor is it completely definable. These conclusions have retained their power to shock after ninety years. They were established at great expense. To establish these results, Gödel and Tarski needed to collapse multiple perspectives onto a single canvas, the mathematician looking at the natural numbers, the logician looking at the mathematician. But as in all great art, the incompleteness theorems demand, and as often receive, a still further perspective, the viewer beyond the canvas, the reader behind the proof, his vision, his point of view. The incompleteness theorems could not otherwise live in the minds of future generations. The process of withdrawal and encompassing reflection, if indefinite, is also endless. The incompleteness theorems provoke an intellectual experience only against the ever-receding background of a common language, a way of life, a culture of contemplation. No matter the degree to which things have been formalized, the formal objects that result are always described, and so discerned, amidst the inconvenience and distraction of the world in which we find ourselves explaining things to one another, specifying inferences, drawing conclusions, signing contracts, writing love letters, and, in general, getting on with life in ways that are inseparably a part of life.

No formal system explains itself. It cannot say anything and we cannot say everything.42

If this is not a fact of logic, it is a fact of life, and so a feature of life. Every description of voluntary action is as incomplete as Peano arithmetic. No matter the network of causal influences acting on a human being, its description inevitably comes to seem incomplete, the strings in plain sight, the puppet master hidden. There can be no proof of this, of course, but it is something we sense, and it is true. I do not know that we can expect anything more from life, or from logic.

There it is: <PA>, the director’s cut.

So it is. So we are. Even so.


  1. Original: “Creatio numerorum, rerum est creatio.” Thierry of Chartres, The Commentary on the De arithmetica of Boethius, ed. Irene Caiazzo (Toronto: Pontifical Institute of Mediaeval Studies, 2015), ix. 
  2. Colin McLarty has even suggested that it may be possible to derive Fermat’s last theorem from within Peano arithmetic. If true, this would be remarkable. See Colin McLarty, “The Large Structures of Grothendieck Founded on Finite Order Arithmetic,” (2014), arXiv:1102.1773v4: 16. 
  3. Kurt Gödel, “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I,” Monatshefte für Mathematik und Physik 38 (1931): 173–98. The English translation of this masterpiece appeared almost 30 years after its publication in German: On Formally Undecidable Statements of Principia Mathematica and Related Systems, trans. Bernard Meltzer (New York: Basic Books, 1962). 
  4. Ernst Zermelo thought it contained a mistake, and could not be persuaded otherwise. Paul Finsler insisted that he had anticipated Gödel’s result, vexing Gödel endlessly with claims of priority. Emil Post, a very great logician, had anticipated Gödel’s result and by 10 years—Gödel’s result, but not his methods. See John Dawson, Jr., “The Reception of Gödel’s Incompleteness Theorems,” PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association, Vol. II: Symposia and Invited Papers (1984): 253–71. 
  5. Logicians now say that they take the theorem and its proof in stride. No doubt. They have had ninety years to study it. But, while Gödel’s proof is no longer mystifying, it remains what it always was and that is rebarbative. 
  6. Kurt Gödel, On Formally Undecidable Statements of Principia Mathematica and Related Systems, trans. Bernard Meltzer (New York: Basic Books, 1962), 38. The translation in the more recent edition published by Solomon Feferman et al. has “in outward appearance,” which is both unidiomatic (as opposed to “to all outward appearances”), and pointlessly passive. The German quite plainly means “considered (betrachet) from the outside (äusserlich).” Someone is considering something. Feferman et al. remark that Gödel approved their translation. This is no very great recommendation. Gödel was not a native English speaker. Stefan Bauer-Mengelberg, who worked closely with Gödel as his translator, once remarked to me that Gödel was infuriating in his obsessive commitment to his own idiosyncratic version of English grammar. See Kurt Gödel, Collected Works, Vol. I, ed. Solomon Feferman et al. (New York: Oxford University Press, 1989), 147. 
  7. German, of course, in the original. 
  8. Clement Greenberg remarks that
    The Old Masters had sensed that it was necessary to preserve what is called the integrity of the picture plane: that is, to signify the enduring presence of flatness underneath and above the most vivid illusion of three-dimensional space. The apparent contradiction involved was essential to the success of their art, as it is indeed to the success of all pictorial art. The Modernists have neither avoided nor resolved this contradiction; rather, they have reversed its terms. One is made aware of the flatness of their pictures before, instead of after, being made aware of what the flatness contains [emphasis added]. Whereas one tends to see what is in an Old Master before one sees the picture itself, one sees a Modernist picture as a picture first. This is, of course, the best way of seeing any kind of picture, Old Master or Modernist, but Modernism imposes it as the only and necessary way, and Modernism’s success in doing so is a success of self-criticism.
    Clement Greenberg, “Modernist Painting.” 
  9. More of the same in the sense that ‘F’ in ‘F(x)’ is itself designated a variable, although not one open to quantification. Were it open, the system that results would be second order. 
  10. Latvian, as it happens. 
  11. It is crystal clear as an inference from P ⊃ P and P to P. Trivial? Of course. But crystal clear anyway. 
  12. And the logical axioms of Alfred North Whitehead and Bertrand Russell’s Principia Mathematica
  13. Feferman remarks of formal systems that their
    formulas are built up from basic arithmetical (and possibly other) relations by means of the propositional connectives (such as ¬, , , →) and quantifiers (such as , ) and whose provable formulas are obtained from a given set of axioms (both logical and non-logical) by closing under certain rules of inference. Moreover, F is assumed to be effectively given, i.e. the set of axioms of F and its rules of inference are supposed to be effectively decidable, so that its set of provable formulas is effectively enumerable.
    Nice. Solomon Feferman, “Penrose’s Gödelian Argument,” 4. 
  14. Samuel Johnson, A Journey to the Western Islands of Scotland (1775), quoted in “Samuel Johnson,” in Oxford Essential Quotations, ed. Susan Ratcliffe (Oxford University Press, 2017). 
  15. Kurt Gödel, On Formally Undecidable Statements of Principia Mathematica and Related Systems, trans. Bernard Meltzer (New York: Basic Books, 1962), 38. 
  16. Whenever it is clear in context whether a formula is being named or used, quotation marks are dropped. 
  17. This is Newspeak; Gödel’s original term was rekursiv. The distinction between primitive recursion and recursion is important. Hence Newspeak. 
  18. Kurt Gödel, On Formally Undecidable Statements of Principia Mathematica and Related Systems, trans. Bernard Meltzer (New York: Basic Books, 1962), 46. 
  19. Division, no, but addition and multiplication, yes. It is possible to define addition and multiplication recursively within the Peano axioms, but the definitions require the recursion theorem, and this, in turn, requires a certain amount of set theory. 
  20. Kurt Gödel, On Formally Undecidable Statements of Principia Mathematica and Related Systems, trans. Bernard Meltzer (New York: Basic Books, 1962), 49. 
  21. Proofs are recursively enumerable, but not recursive; truths are neither. 
  22. Gödel did not prove the fifth theorem, leaving his remarks as a sketch. See Kurt Gödel, On Formally Undecidable Statements of Principia Mathematica and Related Systems, trans. Bernard Meltzer (New York: Basic Books, 1962), 55–56. 
  23. Simple consistency suffices for the first claim; ω-consistency for the second. Shortly after Gödel published his work, J. Barkley Rosser showed how to demote ω-consistency in favor of simple consistency all over again. 
  24. The variable signx’ giving way to the numeral. 
  25. Kurt Gödel, On Formally Undecidable Statements of Principia Mathematica and Related Systems, trans. Bernard Meltzer (New York: Basic Books, 1962), n. 10. 
  26. The essential argument is now called the fixed-point lemma, or the diagonalization lemma, and may be expressed with less by way of elaboration. Let A(x) designate a formula in FPA with one free variable in x. It is always possible to construct a formula G, designating some formula in FPA such that FPA  G ↔ A(n), where n is the Gödel number of G, and n the numeral naming it in FPA. On the level of the underlying assembly language, n is written as S, …, S(0). With the identification of A in A(x) as ~Bew, it follows that FPA  G ↔ G(n), with G saying of itself that it is unprovable. This is, perhaps, a simpler way of presenting things. Gödel’s introduction is rhetorically more effective than contemporary accounts because it suggests, if only indirectly, some of the heavy lifting that goes into the full formal proof. The claim that [αn] is the formula that results when x is replaced by n in α at once invites two questions: Just what results? And how is the replacement carried out? The requisite operation is one of substitution, and it is no easy business to define it properly. No matter the rhetorical ease afforded by Gödel’s introduction, some logicians considered it a mistake because it replaces consistency with soundness. The development of model theory has made this argument anachronistic. 
  27. It is here that Gödel’s introduction, in appealing to soundness rather than consistency, parts company with his proof. 
  28. Alfred Tarski, “The Concept of Truth in Formalized Languages (1935),” in Logic, Semantics, Metamathematics, trans. Joseph Woodger, ed. John Corcoran (Indianapolis: Hackett, 1983), 152–278, whence the concept of truth. See Alfred Tarski and Robert Vaught, “Arithmetical Extensions of Relational Systems,” Compositio Mathematica 13 (1956): 81–102, for the definition of the far more elegant concept of truth in a model. 
  29. A relation R of n arguments on D is said to be M-definable with parameters if it is the extension in M of some formula of FPA containing m + n free variables, m of which have been fixed in D. 
  30. S is a sentence and, by itself, it cannot code for anything. Nonetheless, it is easy to show that it is equivalent to some formula A(x) with one free variable whose translate within the system takes over the coding. 
  31. John Lucas, “Mind, Machines and Gödel,” Philosophy 36 (April–July 1961): 112–27. 
  32. John Lucas, “Mind, Machines and Gödel,” Philosophy 36 (April–July 1961): 113. 
  33. Roger Penrose, Shadows of the Mind (New York: Oxford University Press, 1994), 65. 
  34. Hilary Putnam, “Minds and Machines,” in Dimensions of Mind, ed. Sidney Hook (New York: New York University Press, 1960), 138–64. 
  35. Lucas, in rebuttal, has not covered himself in glory. Addressing an imaginary opponent, he remarks that
    [i]f, however, he acknowledges that the system cannot prove its Gödelian formula, then we know it is consistent, since it cannot prove every well-formed formula, and knowing that it is consistent, know also that its Gödelian formula is true.
    But this is to collapse the discussion into the frank fallacy in which P ⊃ Q and Q give rise to P. 
  36. It is perfectly possible to see that still waters run deep; and to see nothing on seeing mierīgie ūdēņi ir tie dziļākie; and yet to know perfectly well that they are in some sense the same. 
  37. Panu Raatikainen in a review of Torkel Franzen’s Gödel’s Theorem: An Incomplete Guide to Its Use and Abuse, in Notices of the American Mathematical Society 54, no. 3 (March 2007): 382. 
  38. Kurt Gödel, Collected Works, Vol. I: Publications 1929–1936, ed. Solomon Feferman et al. (New York: Oxford University Press, 1989), 151. The German is richtig
  39. FPA says nothing about truth either as a relation or a formal sign, and, in any case, Tarski’s theorem makes it plain that this claim is untrue as stated. Other systems may admit a partial definition of Tr, or a set of axioms incorporating Tr, but they, of course, are not this system, our very own. 
  40. Kurt Gödel, “Some Basic Theorems on the Foundations of Mathematics and Their Implications (1951),” in Collected Works, Vol. III: Unpublished Essays and Lectures, ed. Solomon Feferman et al. (New York: Oxford University Press, 1995), 310. 
  41. Gödel quoted in Jack Copeland, “The Mathematical Objection: Turing, Gödel, and Penrose on the Mind,” July 2008, 14. 
  42. For the moment, we possess no real theories about human nature or human life: we are immersed in the flow. Whatever the theories we may in the future have, there is no obvious reason to suppose that these theories and our theories of the physical world should be consistent, and every reason to suppose the contrary. This means only that they cannot be simultaneously upheld, reason enough not to uphold them simultaneously. 

David Berlinski is an American writer.

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