Through most of our tradition, due in large part to Aristotle’s influence, the notion of an actual infinity—of infinity as given, once and for all—was rejected, and only potential infinity was given currency: the infinity of a process that never ends, such as the succession of natural numbers or the division of a line.1placeholder As late as 1831, in a letter to the astronomer Heinrich Christian Schumacher, Carl Friedrich Gauss, the greatest mathematician of his generation, wrote:
“I protest right up front against the use of an infinite magnitude as something completed, which in mathematics is never permitted. The infinite is merely a façon de parler.”2placeholder
It is a strong statement; to understand it, consider an example. Suppose I say
(1) There is an infinity of prime numbers.
What I mean, according to Gauss (and Aristotle), is that, for any finite prime number, I can prove the existence of a larger, still finite one. So (1) uses the word “infinity” only in a manner of speaking, as there is in fact (Gauss and Aristotle would claim) no such thing as an infinity of primes—there is only the generation of one finite prime after another, without end. Infinity is a potential that can never be realized.
A generation later, things were changing dramatically. Starting in 1874, Georg Cantor developed the theory of transfinite sets and, despite fierce opposition that included being called a charlatan by his illustrious colleague Leopold Kronecker, the tide eventually turned in his favor, and of actual infinity. Paradoxes galore arose in his theory, but by then the profession was not willing to let it go. In his 1925 paper “On the Infinite” David Hilbert, the greatest mathematician of his generation, said, after reviewing what was called “the crisis of the foundations of mathematics”: “No one shall drive us out of the paradise which Cantor has created for us.”3placeholder Despite this proud pronouncement, actual infinity kept making trouble to this day, what with the Gödel-proved impossibility of establishing the consistency of any sufficiently powerful mathematical theory, including set theory (which bankrupted the so-called “Hilbert program”), with the Paul J. Cohen-proved impossibility of establishing the cardinality of the continuum (Hilbert’s first open problem in the famous list he presented at the 1900 second International Congress of Mathematicians, in Paris), or with the Skolem-proved paradox that the entire transfinite hierarchy of sets can be reconstructed in a domain no larger than the natural numbers. Based on these and other misgivings, I have expressed the view elsewhere that actual infinity is only the object of an unfulfilled and unfulfillable desire, more of an apt subject for theology (or literature—think of the experience of the sublime) than for serious science. Here, however, I intend to address a different issue: Did the venerable tradition of potential infinity, with all the fuss being made about its actual counterpart, get somehow lost in the shuffle? Is there still a legitimate understanding for infinity being merely a manner of speaking on the contemporary scene? Never mind choosing its side, but can we at least phrase it?
It would seem not. For consider the most elementary mathematical theory, arithmetic, which I will name PA in honor of Giuseppe Peano, the Italian mathematician who provided the axioms still used for it, and return to (1). I said that, given any prime number, I can prove the existence of a larger one. Let us look into how that is done. Say that p is prime. Consider all the prime numbers no larger than p and make the product of all of them: 2 × 3 × 5 × … × p. Call the result p* and consider q = p* + 1. q is larger than p as it is at least as large as p + 1, and, divided by any prime number no larger than p, gives 1 as a remainder. Therefore, either q is prime or one of its prime factors is larger than p. Q. E. D. For illustration, given the prime number 5, 5* = 2 × 3 × 5 = 30 and q = 31, which is prime.4placeholder This procedure is available in PA; indeed, in PA the following is a theorem (where “Px” abbreviates “x is a prime”):
(2) ∀x(Px ⊃ ∃y(y = x* + 1)).
And, if we agree that the sense in which a theory can say something to be the case is by proving it as a theorem, (2) says that there is an actual infinity of primes. For (2) can only be true if there is an actual infinity of primes: it commits us to the infinity of primes all at once; there is no way that, having proved (2), primes can only be constructed one step at a time, in a process that never ends but never even reaches infinity; that infinity is asserted by (2). PA is, I repeat, the most elementary mathematical theory; so, even at this most elementary level, it would appear there is no way we can express the view that talk of infinity be only a manner of speaking.
Enter free logic, that is: a quantification theory, with or without identity, free of existential commitments with respect to its singular terms. In standard quantification theory, the one taught in introductory symbolic logic courses,
(3) ∃x(x = a),
or, as it is often abbreviated,
(4) E!a,
that is, in English,
(5) a exists,
is provable; therefore, simply having the individual constant “a” in our language commits us to the existence of a. If this were the logic of English, simply having the name “Pegasus” would commit us to the existence of Pegasus (which explains the longstanding currency of Russell’s highly artificial theory of descriptions, and his claim that ordinary proper names are disguised definite descriptions). Not so in free logic, where (3) is not a theorem. For the sake of reference, here is an axiom system for FL, a free quantification theory with identity:
(A0) A, if A is a tautology in classical propositional logic
(A1) ∀x(A ⊃ B) ⊃ (∀xA ⊃ ∀xB)
(A2) A ⊃ ∀xA, if x is not free in A
(A3) ∀y(∀xA ⊃ A(y/x)
(A4) t = t
(A5) t = t’ ⊃ (A ⊃ A(t’//t)
(R1) B is a theorem if A and A ⊃ B are theorems
(R2) ∀xA if A is a theorem,
where A(y/x) is the result of substituting occurrences of y for all free occurrences of x in A, provided none of them becomes thereby bound, and A(t’//t) is the result of substituting occurrences of t’ for zero or more free occurrences of t in A, provided no variable free in t’ becomes thereby bound.
In an essay of 19845placeholder I proved the consistency of an arithmetic FA based on FL. These are the additional axioms and rule of FA:
(A6) ∀x∼sx = 0
(A7) st = st’ ⊃ t = t’
(A8) t + 0 = t
(A9) t + st’ = s (t + t’)
(A10) t × 0 = 0
(A11) t × st ‘ = (t × t’) + t
(A12) ∃x(x = i)
(R3) A is a theorem if A(0/x) and A ⊃ A(sx/x) are theorems,
where “s” is to be understood as the successor function, a numeral is a term of the form s…s0, and i is the numeral where “s” is repeated i times—which includes repeated 0 times. (To clarify, a numeral is the name of a number, and here for simplicity I will use the same symbol for both.)
The key to the proof of consistency is that in FL an unquantified formula like (A7) is supposed to hold for all objects (all possible references of t and t’), existent or nonexistent, whereas a quantified formula like (A6) is supposed to hold for existent objects only—quantification is only over existent objects. So (A7) says that the identity of successors, existent or nonexistent, entails the numbers of which they are successors to be identical, too (or, equivalently, that distinct numbers, existent or nonexistent, have distinct successors), whereas (A6) would allow for a nonexistent number to have successor 0.
Now consider any fragment FAj of FA which contains all and only the instances of (A12) where i is no larger than j. For example, FA2 only contains the instances
∃x(x = 0)
∃x(x = 1)
∃x(x = 2)
of (A12). The following is a finite model of Faj: the inner domain (set of existent objects) is constituted by the numbers 0, 1, 2, …, j; the outer domain (set of nonexistent objects) is constituted by the number j + 1; all numbers no larger than j have their ordinary successor, and j + 1 has successor 0. We have thus obtained a modular arithmetic with a nonexistent element, and arithmetical operations apply here as they ordinarily do in modular arithmetic. In FA2, for example, 2 + 1 = 3 but 3 + 1 = 0 (which is another way of saying that the nonexistent 3 has successor 0). It is easy to see that the model satisfies FAj.
Suppose then that FA be inconsistent, hence that there is a proof of a contradiction in it. Since a proof is a finite sequence of formulas, it can only contain finitely many instances of (A12). Be j the highest numeral for which the relevant instance of (A12) occurs in the proof. Then the proof would also be possible in FAj, and FAj would also be inconsistent, contra what we just proved: that every FAj has a (finite) model.
FA allows for all ordinary arithmetical operations.
(6) 371 + 792 = 1163
(7) 44 × 27 = 1188
are theorems in it and, if powers are introduced in the usual fashion, we can prove such additional theorems as
(8) 43 = 64
(9) 54 = 625.
Besides, we can prove the usual properties of arithmetical operations, like the commutativity and associativity of addition and multiplication, and the distributivity of multiplication over addition:
(10) t + t’ = t’ + t
(11) (t + t’) + t” = t + (t’ + t”)
(12) t × t’ = t’ × t
(13) (t × t’) × t” = t × (t’ × t”)
(14) t × (t’ + t”) = (t × t’) + (t × t”).
So FA works well for everyday counting. But, of course, it is a weaker theory than PA. (It cannot prove, for example, the unquantified variant of (A6)
(15) ∼st = 0,
where the reference of t could be existent or nonexistent.) Importantly, it cannot express its own metatheory and cannot prove the equivalent of Gödel’s second theorem. (If it could, we would have a contradiction, as that theorem states that the relevant theory cannot prove its own consistency, contra the proof of consistency I sketched.) Here I point out that this weakness can afford us a renewed sense for Gauss’s claim.
Return to (2) above. I said that its provability in PA asserts the existence of an actual infinity of prime numbers. (2) is in fact one of a family of formulas that have similar force, all provable in PA, the simplest one being
(16) ∀x∃y(y = sx),
which asserts the existence of an actual infinity of successor numbers. But none of these can be proved in FA. Suppose, for example, that there is a proof of (16) in it. Since a proof is a finite sequence of formulas, it can only contain finitely many instances of (A12). Be j the highest numeral for which the relevant instance of (A12) occurs in the proof. Then the proof would also be possible in FAj. But consider the model of FAj in which the inner domain is constituted by the numbers no greater than j and the outer domain is constituted by j + 1. In that model the (existent) number j has no existent successor, contradicting the provability of (16) in FAj and in FA. Though of course, given any specific (existent) number, we can prove the existence of its successor in FA—as much as we can apply there the usual procedure to prove the existence of larger and larger primes.
In conclusion, FA allows us to discriminate between proving formulas that assert an actual infinity of numbers—which it cannot do—and applying the procedures described by those formulas one step at a time, with no end—which it can do. For those who would adopt FA rather than PA, saying that the procedures commit us to the existence of an infinity of numbers would only be a manner of speaking. And, whether or not we want to side with potential infinity, we thus have at least a way of phrasing what that would mean.
The following are relevant passages from Aristotle: “generally the infinite has this mode of existence: one thing has always been taken after another, and each thing that is taken is always finite, but always different” (Physics, Book III, Chapter 6); “the fact that division never ceases to be possible gives the result that this actuality exists potentially, but not that it exists separately” (Metaphysics, Book IX, Chapter 6). Translations are edited by Jonathan Barnes, in The Complete Works of Aristotle (Princeton, NJ: Princeton University Press, 1984).
Gauss, Werke, vol. VIII (Leipzig: Teubner, 1900), p. 216; translation mine. The expression “façon de parler” (“manner of speaking”) is in French in Gauss’s original German text.
Translated by Erna Putnam and Gerald J. Massey in Philosophy of Mathematics: Selected Readings, edited by Paul Benacerraf and Hilary Putnam (Cambridge: Cambridge University Press, 1983), p. 191.
The essence of this proof is to be found in Proposition 20, Book IX, of Euclid’s Elements.
“Finitary Consistency of a Free Arithmetic,” Notre Dame Journal of Formal Logic 25 (1984), pp. 224-226.