Natural Numbers and Natural Cardinals as Abstract Objects: A Partial Reconstruction of Frege's Grundgesetze in Object Theory

Objects: A Partial Reconstruction of Frege’s Grundgesetze in Object Theory∗ Edward N. Zalta† Center for the Study of Language and Information and Philosophy Department Stanford University In this paper, I derive a theory of numbers from a more general theory of abstract objects. The distinguishing feature of this derivation is that it involves no appeal to mathematical primitives or mathematical theories. In particular, no notions or axioms of set theory are required, nor is the notion ‘the number of F s’ taken as a primitive. Instead, entities that we may justifiably call ‘natural cardinals’ and ‘natural numbers’ are explicitly defined as species of the abstract objects axiomatized in Zalta [1983], [1988a], and [1993a]. This foundational metaphysical theory is supplemented with some plausible assumptions and the resulting system ∗This paper was published in the Journal of Philosophical Logic, 28/6 (1999): 619–660. Sections 3-5 derive from Subsections B and C in Section 15 of Principia Metaphysica, an unpublished monograph available at the URL = . †I am greatly indebted to the following people: Bernard Linsky, who pointed out that a definition analogous to Frege’s definition of predecessor was formulable in my system and who criticized the early drafts of the technical material; Karl-Georg Niebergall, whose criticisms of the original proof that every number has a successor helped me to simplify the argument; Peter Aczel, who found a model which shows that the principles added to object theory in this paper can be consistently added; and Tony Roy, who found a small error in my formulation of the Aczel model. I would also like to thank John Bacon, Johan van Benthem, Thomas Hofweber, Godehard Link, Wolfgang Malzkorn, Chris Menzel, and Gideon Rosen for our conversations about this material. Finally, I am indebted to John Perry, who has continously supported my research here at the Center for the Study of Language and Information (CSLI). Edward N. Zalta 2 yields the Dedekind/Peano axioms for number theory. The derivations of the Dedekind/Peano axioms should be of interest to those familiar with Frege’s work for they invoke patterns of reasoning that he developed in [1884] and [1893]. However, the derivation of the claim that every number has a successor does not follow Frege’s plan, but rather exploits the logic of modality that is embedded in the system. In Section 1, there is a review of the basic theory of abstract objects for those readers not familiar with it. Readers familiar with the theory should note that the simplest logic of actuality (governing the actuality operator Aφ) is now part of the theory. In Section 2, some important consequences of the theory which affect the development of number theory are described and the standard models of the theory are sketched. In Section 3, the main theorems governing natural cardinals are derived. In Section 4, the definitions and lemmas which underlie the Dedekind/Peano axioms are outlined, and in particular, the definition of ‘predecessor’ and ‘natural number’. In Section 5, the Dedekind/Peano axioms are derived. The final section consists of observations about the work in Sections 2 – 5. Although there are a myriad of philosophical issues that arise in connection with these results, space limitations constrain me to postpone the full discussion of these issues for another occasion. The issues include: how the present theory relates to the work of philosophers attempting to reconstruct Frege’s conception of numbers and logical objects; how the theory supplies an answer to Frege’s question ‘How do we apprehend numbers given that we have no intuitions of them?’; how the theory avoids ‘the Julius Caesar problem’; and how the theory fits into the philosophy of mathematics defended in Linsky and Zalta [1995]. A full discussion of these issues would help to justify the approach taken here when compared to other approaches. However, such a discussion cannot take place without a detailed development of the technical results and it will be sufficient that the present paper is devoted almost exclusively to this development. In the final section, then, there is only a limited discussion of the aforementioned philosophical issues. It includes a brief comparison of the present approach with that in Boolos [1987]. Before we begin, I should emphasize that the word ‘natural’ in the expressions ‘natural cardinal’, ‘natural number’ and ‘natural arithmetic’ 1See Parsons [1965], Wright [1983], Burgess [1984], Hazen [1985], Boolos [1986], [1987], Parsons [1987], Heck [1993], Hale [1987], Fine [1994], and Rosen [1995], and Burgess [forthcoming]. 3 Natural Numbers as Abstract Objects needs to be taken seriously. This is a theory about numbers which are abstracted from the facts about concrete objects in this and other possible worlds. As such, the numbers that we define will not count other abstract objects; for example, we cannot use them to count the natural numbers less than or equal to 2 (though we can, of course, define numerical quantifiers in the usual way and use them to assert that there are three natural numbers less than or equal to 2). This consequence will be discussed and justified in the final section. Though some reductions and systematizations of the natural numbers do identify the numbers as objects which can count the elements falling under number-theoretic properties, those reductions typically appeal to mathematical primitives and mathematical axioms. It should therefore be of interest to see a development of number theory which makes no appeal to mathematical primitives. From the present point of view, one consequence of eliminating mathematical primitives is that the resulting numbers are even more closely tied to their application in counting the objects of the natural world than Frege anticipated. This, however, would be a welcome result in those naturalist circles in which abstract objects are thought to exist immanently in the natural world, in some sense dependent on the actual pattern in which ordinary objects exemplify properties. §1: The Theory of Abstract Objects The Language: The theory of abstract objects is formulated in a syntactically second-order S5 modal predicate calculus without identity, modified only so as to include xF 1 (‘x encodes F ’) as an atomic formula along with Fx1 . . . xn. The notion of encoding derives from Mally [1912] and an informal version appears in Rapaport [1978]. Interested readers may find a full discussion of and motivation for this new form of predication in Zalta [1983] (Introduction), [1988a] (Introduction), and [1993a]. It is not too hard to show that encoding formulas of the form ‘xF ’ embody 2It should be emphasized that the following work does not constitute an attempt develop an overarching foundations for mathematics. Once the mathematicians decide which, if any, mathematical theory ought to be the foundation for mathematics, I would identify the mathematical objects and relations described by such a theory using the ideas developed in Linsky and Zalta [1995] and in Zalta [forthcoming]. 3Though, strictly speaking, on the conception developed here, reality includes modal reality, and so the identity of the abstract objects that satisfy the definition of ‘natural number’ may depend on the patterns in which possibly concrete objects exemplify properties. Edward N. Zalta 4 the same idea as Boolos’ η relation, which he uses in formulas of the form ‘Fηx’ in his papers [1986] and [1987]. The complex formulas and terms are defined simultaneously. The complex formulas include: ¬φ, φ → ψ, ∀αφ (where α is an object variable or relation variable), φ, and Aφ (‘it is actually the case that φ’). There are two kinds of complex terms, one for objects and one for n-place relations. The complex object terms are rigid definite descriptions and they have the form ıxφ, for any formula φ. The complex relation terms are λ-predicates and they have the form [λx1 . . . xn φ], where φ has no encoding subformulas. In previous work, I have included a second restriction on λ-predicates, namely, that φ not contain quantifiers binding relation variables. This restriction was included to simplify the ‘algebraic’ semantics. But since the semantics of the system will not play a role in what follows, we shall allow impredicative formulas inside λ-predicates. Models demonstrate that the theory remains consistent even in the presence of the new instances of comprehension which assert the existence of relations defined in terms of impredicative formulas. Definitions and Proper Axioms: The distinguished 1-place relation of being concrete (‘E!’) is used to partition the objects into two cells: the ordinary objects (‘O!x’) are possibly concrete, whereas abstract objects (‘A!x’) couldn’t be concrete: O!x =df E!x A!x =df ¬ E!x Thus, O!x ∨ A!x and ¬∃x(O!x & A!x) are both theorems. Though the theory asserts (see below) that ordinary objects do not encode properties, abstract objects both encode and exemplify properties (indeed, some abstract objects exemplify the very properties that they encode). Next we define a well-behaved, distinguished identity symbol =E that applies to ordinary objects as follows: x=E y =df O!x &O;!y & ∀F (Fx ≡ Fy) 4A full discussion of this would take us too far afield. I hope to discuss the connection at length in another, more appropriate context. However, I’ll say a more about this connection in the final section of the paper. 5A subformula is defined as follows: every formula is a subformula of itself. If χ is ¬φ, φ → ψ, ∀αφ, or φ, then φ (and ψ) is a subformula of χ. If φ is a subformula of ψ, and ψ is a subformula of χ, then φ is a subformula of χ. 5 Natural Numbers as Abstract Objects Given this definition, the λ-expression [λxy x=E y] is well-formed. So =E denotes a relation. The five other (proper) axioms and definitions of the theory are: 1. O!x → ¬∃FxF 2. ∃x(A!x & ∀F (xF ≡ φ)), where φ has no free xs 3. x=y =df x=E y ∨ (A!x&A;!y & ∀F (xF ≡ yF )) 4. F =G =df ∀x(xF ≡ xG) 5. α = β → [φ(α, α) ≡ φ(α, β)], where α, β are either both object variables or both relation variables and φ(α, β) is the result of replacing one or more occurrences of α by β in φ(α, α), provided β is substitutable for α in the occurrences of α that it replaces The first principle is an axiom that asserts that ordinary objects necessarily fail to encode properties. The second principle is a proper axiom schema, namely, the comprehension principle for abstract objects. This asserts the existence of an abstract object that encodes just the properties F satisfying formula φ, whenever φ is any formula with no free variables x. The third principle is a definition of a general notion of identity. Objects x and y are said to be ‘identical’ just in case they are both ordinary objects and necessarily exemplify the same properties or they are both abstract objects and necessarily encode the same properties. The fourth principle, the definition for property identity, asserts that properties are identical whenever they are necessarily encoded by the same objects. Since both the identity of objects (‘x=y’) and the identity of properties (‘F =G’) are defined notions, the fifth principle tells us that expressions for identical objects or identical relations can be substituted for one another in any context. The Logic: The logic that underlies this proper theory is essentially classical. The logical axioms of this system are the modal closures of the instances of axiom schemata of classical propositional logic, classical 6This definition can be generalized easily to yield a definition of identity for n-place relations (n ≥ 2) and propositions (n = 0). The more general formulation may be found in Zalta [1983], p. 69; Zalta [1988a], p. 52; and Zalta [1993a], footnote 21. These definitions of relation identity have been motivated and explained in the cited works. The definition allows one to consistently assert that there are distinct relations that are (necessarily) equivalent. Edward N. Zalta 6 quantification theory (modified only to admit empty descriptions), and second-order S5 modal logic with Barcan formulas (modified only to admit rigid descriptions and the actuality operator). The logical axioms for encoding are the modal closures of the following axiom: Logic of Encoding: xF → xF The logical axioms for the λ-predicates are the modal closures of the following principle of λ-conversion: λ-Conversion: [λx1 . . . xn φ]y1 . . . yn ≡ φ1n x1,...,xn , where φ has no definite descriptions and φ1n x1,...,xn is the result of substituting yi for xi (1 ≤ i ≤ n) everywhere in φ. The rules of inference (see below) will allow us to derive the following comprehension principle for n-place relations (n ≥ 0) from λ-conversion: Relations: ∃Fn ∀y1 . . .∀yn(Fny1 . . . yn ≡ φ), where φ has no free F s, no encoding subformulas and no definite descriptions 7It is a logical axiom that interchange of bound variables makes no difference to the identity of the property denoted by the λ-expression: [λx1 . . . xn φ] = [λy1 . . . yn φ′], where φ and φ′ differ only by the fact that yi is substituted for the bound occurrences of xi. The following is also a logical axiom: Fn = [λx1 . . . xn Fx1 . . . xn]. 8It is important to remember that the formulas φ in λ-expressions may not contain encoding subformulas. This restriction serves to eliminate the paradox which would otherwise arise in connection with the comprehension principle for abstract objects. Were properties of the form [λz ∃F (zF & ¬Fz)] formulable in the system, one could prove the following contradiction. By comprehension for abstract objects, the following would be an axiom: ∃x(A!x& ∀F (xF ≡ F =[λz ∃F (zF & ¬Fz)])) Call such an object ‘a’ and ask the question: [λz ∃F (zF & ¬Fz)]a? We leave it as an exercise to show that a exemplifies this property iff it does not. We remove the threat of this paradox by not allowing encoding subformulas in property comprehension. This still leaves us with a rich theory of properties, namely, all of the predicable and impredicable properties definable in standard second-order exemplification logic. 9A definite description ıyψ may appear in instances of λ-conversion whenever (it is provable that) ıyψ has a denotation. Whenever we assume or prove that ∃x(x= ıyψ), we can prove the instance of λ-conversion that asserts: