- •Contents
- •Preface to the Fifth Edition
- •1 Enumerability
- •1.1 Enumerability
- •1.2 Enumerable Sets
- •Problems
- •2 Diagonalization
- •Problems
- •3 Turing Computability
- •Problems
- •4 Uncomputability
- •4.1 The Halting Problem
- •4.2* The Productivity Function
- •Problems
- •5 Abacus Computability
- •5.1 Abacus Machines
- •5.2 Simulating Abacus Machines by Turing Machines
- •5.3 The Scope of Abacus Computability
- •Problems
- •6 Recursive Functions
- •6.1 Primitive Recursive Functions
- •6.2 Minimization
- •Problems
- •7 Recursive Sets and Relations
- •7.1 Recursive Relations
- •7.2 Semirecursive Relations
- •7.3* Further Examples
- •Problems
- •8.1 Coding Turing Computations
- •8.2 Universal Turing Machines
- •8.3∗ Recursively Enumerable Sets
- •Problems
- •9.1 First-Order Logic
- •9.2 Syntax
- •Problems
- •10.1 Semantics
- •10.2 Metalogical Notions
- •Problems
- •11 The Undecidability of First-Order Logic
- •11.1 Logic and Turing Machines
- •11.2 Logic and Primitive Recursive Functions
- •11.3 Lemma
- •Problems
- •12 Models
- •12.1 The Size and Number of Models
- •12.2 Equivalence Relations
- •Problems
- •13 The Existence of Models
- •13.1 Outline of the Proof
- •13.2 The First Stage of the Proof
- •13.3 The Second Stage of the Proof
- •13.4 The Third Stage of the Proof
- •13.5* Nonenumerable Languages
- •Problems
- •14 Proofs and Completeness
- •14.1 Sequent Calculus
- •14.2 Soundness and Completeness
- •14.3* Other Proof Procedures and Hilbert’s Thesis
- •Problems
- •15 Arithmetization
- •15.1 Arithmetization of Syntax
- •Problems
- •16 Representability of Recursive Functions
- •16.2 Minimal Arithmetic and Representability
- •16.3 Mathematical Induction
- •16.4* Robinson Arithmetic
- •Problems
- •17.1 The Diagonal Lemma and the Limitative Theorems
- •17.2 Undecidable Sentences
- •17.3* Undecidable Sentences without the Diagonal Lemma
- •Problems
- •18 The Unprovability of Consistency
- •Historical Remarks
- •19 Normal Forms
- •19.1 Disjunctive and Prenex Normal Forms
- •19.2 Skolem Normal Form
- •19.3 Herbrand’s Theorem
- •19.4 Eliminating Function Symbols and Identity
- •Problems
- •20 The Craig Interpolation Theorem
- •20.1 Craig’s Theorem and Its Proof
- •20.2 Robinson’s Joint Consistency Theorem
- •20.3 Beth’s Definability Theorem
- •Problems
- •21 Monadic and Dyadic Logic
- •21.1 Solvable and Unsolvable Decision Problems
- •21.2 Monadic Logic
- •21.3 Dyadic Logic
- •Problems
- •22 Second-Order Logic
- •Problems
- •23.2 Arithmetical Definability and Forcing
- •Problems
- •24 Decidability of Arithmetic without Multiplication
- •Problems
- •25 Nonstandard Models
- •25.1 Order in Nonstandard Models
- •25.2 Operations in Nonstandard Models
- •25.3 Nonstandard Models of Analysis
- •Problems
- •26 Ramsey’s Theorem
- •Problems
- •27 Modal Logic and Provability
- •27.1 Modal Logic
- •27.2 The Logic of Provability
- •27.3 The Fixed Point and Normal Form Theorems
- •Problems
- •Annotated Bibliography
- •General Reference Works
- •Textbooks and Monographs
- •By the Authors
- •Index
25
Nonstandard Models
By a model of (true) arithmetic is meant any model of the set of all sentences of the language L of arithmetic that are true in the standard interpretation N. By a nonstandard model is meant one that is not isomorphic to N. The proof of the existence of an (enumerable) nonstandard model of arithmetic is as an easy application of the compactness theorem (and the Lowenheim¨–Skolem theorem). Every enumerable nonstandard model is isomorphic to a nonstandard model M whose domain is the same as that of N, namely, the set of natural numbers; though of course such an M cannot assign the same denotations as N to the nonlogical symbols of L. In section 25.1 we analyze the structure of the order relation in such a nonstandard model. A consequence of this analysis is that, though the order relation cannot be the standard one, it at least can be a recursive relation. By contrast, Tennenbaum’s theorem tells us that it cannot happen that the addition and multiplication relations are recursive. This theorem and related results will be taken up in section 25.2. Section 25.3 is a sort of appendix (independent of the other sections, but alluding to results from several earlier chapters) concerning nonstandard models of an expansion of arithmetic called analysis.
25.1 Order in Nonstandard Models
Let M be a model of (true) arithmetic not isomorphic to the standard model N. (The existence of such models was established in the problems at the end of Chapter 12, as an application of the compactness theorem.) What does such a model look like? We’ll call the objects in the domain |M| NUMBERS. M assigns as denotation to the symbol 0 some NUMBER O we’ll call ZERO, and to the symbol some function † on NUMBERS we’ll call SUCCESSOR. It assigns to < some relation we’ll call LESS THAN, and to + and · some functions and we’ll call ADDITION and MULTIPLICATION. Our main concern in this section will be to understand the LESS THAN relation.
First of all, no NUMBER is LESS THAN itself. For no (natural) number is less than itself. So x x <x is true in N, so it is true in M, and so as asserted no NUMBER is LESS THAN itself. This argument illustrates our main technique for obtaining information about the ‘appearance’ of M: observe that the natural numbers have a certain property, conclude that a certain sentence of L is true in N, infer that it must also be true in M (since the same sentences of L are true in M as in N), and decipher the sentence ‘over’ M. In this way we can conclude that exactly one of any two NUMBERS is LESS THAN the other, and that if one NUMBER is LESS THAN another, which is LESS
302
25.1. ORDER IN NONSTANDARD MODELS |
303 |
THAN a third, then the first is LESS THAN the third. LESS THAN is a linear ordering of the NUMBERS, just as less than is a linear ordering of the numbers.
Zero is the least number, so ZERO is the LEAST NUMBER. Any number is less than its successor, and there is no number between a given number and its successor (in the sense of being greater than the former and less than the latter), so any NUMBER is LESS THAN its SUCCESSOR, and there is no NUMBER between a given NUMBER and its SUCCESSOR. In particular, 0 (that is, 1 or one) is the next-to-least number, and O†
(which we may call I or ONE) is the next-to-LEAST NUMBER; 0 is next-to-next-to-least and O†† is next-to-next-to-LEAST; and so on. So there is an initial segment O, O†,
O††, . . . of the relation LESS THAN that is isomorphic to the series 0, 0 , 0 , . . . of the (natural) numbers.
We call O, O†, O††, . . . the standard NUMBERS. Any others are nonstandard. The standard NUMBERS are precisely those that can be obtained from ZERO by applying
the SUCCESSOR operation a finite number of times. For any (natural) number n, let us write h(n) for O††...†(n times), which is the denotation of the numeral n or 0 ... (n times)
in M. Then the standard NUMBERS are precisely the h(n) for n a natural number. Any others are nonstandard. Any standard NUMBER h(n) is LESS THAN any nonstandard NUMBER m. This is because, being true in N, the sentence
z((z = 0 & . . . & z = n) → n <z)
must be true in M, so any NUMBER other than h(0), . . . , h(n) must be GREATER THAN h(n).
[It is not quite trivial to show that there must be some nonstandard NUMBERS in any nonstandard model M. If there were not, then h would be a function from (natural) numbers onto the domain of M. We claim that in that case, h would be an isomorphism between N and M, which it cannot be if M is nonstandard. First, h would be one-to-one, because when m = n, m = n is true in N and so in M, so the denotations of m and n in M are distinct, that is, h(m) = h(n). Further, when m + n = p, m + n = p is true in N and so in M, so h(m + n) = h(m) h(n). Finally, h(m · n) = h(m) h(n) by a similar argument.]
Any number other than zero is the successor of some unique NUMBER, so any NUMBER other than ZERO is the SUCCESSOR of some unique number. So we can define a function ‡ from NUMBERS to NUMBERS by letting O‡ = O and otherwise letting m‡ be the unique NUMBER of which m is the SUCCESSOR. If n is standard, then n† and n‡ are standard, too, and if m is nonstandard, then m† and m‡ are nonstandard. Moreover, if n is standard and m nonstandard, then n is LESS THAN m‡.
We’ll now define an equivalence relation ≈ on NUMBERS. If a and b are NUMBERS, we’ll say that a ≈ b if for some standard(!) NUMBER c, either a c = b or b c = a. Intuitively speaking, a ≈ b if a and b are a finite distance away from each other, or in other words, if one can get from a to b by applying † or ‡ a finite number of times. Every standard NUMBER bears the relation ≈ to all and only the standard NUMBERS. We call the equivalence class under ≈ of any NUMBER a the block of a. Thus a’s block is
{. . . , a‡‡‡, a‡‡, a‡, a, a†, a††, a†††, . . .}.
304 |
NONSTANDARD MODELS |
Note that a’s block is infinite in both directions if a is nonstandard, and is ordered like the integers (negative, zero, and positive).
Suppose that a is LESS THAN b and that a and b are in different blocks. Then since a† is LESS THAN or equal to b, and a and a† are in the same block, a† is LESS THAN b. Similarly, a is LESS THAN b‡. It follows that if there is even one member of a block A that is LESS THAN some member of a block B, then every member of A is LESS THAN every member of B. If this is the case, we’ll say that block A is LESS THAN block B. A block is nonstandard if and only if it contains some nonstandard number. The standard block is the LEAST block.
There is no LEAST nonstandard block, however. For suppose that b is a nonstandard NUMBER. Then there is an a LESS THAN b such that either a a = b or a a I = b. [Why? Because for any (natural) number b greater than zero there is an a less than b such that either a + a = b or a + a + 1 = b.] Let’s suppose a a = b. (The other case is similar.) If a is standard, so is a a. So a is nonstandard. And a is not in the same block as b: for if a c = b for some standard c, then a c = a a, whence c = a, contradicting the fact that a is nonstandard. (The laws of addition that hold in N hold in M.) So a’s block is LESS THAN b’s block. Similarly, there is no GREATEST block.
Finally, if one block A is LESS THAN another block C, then there is a third block B that A is LESS THAN, and that is LESS THAN C. For suppose a is in A and c is in C, and a is LESS THAN c. There there is an b such that a is LESS THAN b, b is LESS THAN c, and either a c = b b or a c I = b b. (Averages, to within a margin of error of one-half, always exist in N; b is the AVERAGE in M of a and c.) Suppose a c = b b. (The argument is similar in the other case.) If b is in A, then b = a d for some standard d, and so a c = a d a d, and so c = a d d (laws of addition), from which it follows, as d d is standard, that c is in A. So b is not in A, and, similarly not in C either. We may thus take as the desired B the block of b.
To sum up: the elements of the domain of any nonstandard model M of arithmetic are going to be linearly ordered by LESS THAN. This ordering will have an initial segment that is isomorphic to the usual ordering of natural numbers, followed by a sequence of blocks, each of which is isomorphic to the usual ordering of the integers (negative, zero, and positive). There is neither an earliest nor a latest block, and between any two blocks there lies a third. Thus the ordering of the blocks is what was called in the problems at the end of Chapter 12 a dense linear ordering without endpoints, and so, as shown there, it is isomorphic to the usual ordering of the rational numbers. This analysis gives us the following result.
25.1a Theorem. The order relations on any two enumerable nonstandard models of arithmetic are isomorphic.
Proof: Let K be the set consisting of all natural numbers together with all pairs (q, a) where q is a rational number and a and integer. Let <K be the order on K in which the natural numbers come first, in their usual order, and the pairs afterward, ordered as follows: (q, a) <K (r, b) if and only if q < r in the usual order on rational numbers, or (q = r and a < b in the usual order on integers). Then what we have shown above is that the order relation in any enumerable nonstandard model of
25.1. ORDER IN NONSTANDARD MODELS |
305 |
arithmetic is isomorphic to the ordering <K of K. Hence the order relations in any two such models are isomorphic to each other.
This result can be extended from models of (true) arithmetic to models of the theory P (introduced in Chapter 16).
25.1b Theorem. The order relations on any two enumerable nonstandard models of P are isomorphic.
Proof: We indicate the proof in outline. What one needs to do in order to extend Theorem 25.1a from models of arithmetic to models of P is to replace every argument ‘S must be true in M because S is true in N’ that occurs above, by the argument ‘S must be true in M because S is a theorem of P’. To show that S is indeed a theorem of P, one needs to ‘formalize’ in P the ordinary, unformalized mathematical proof that S is true in N. In some cases (for instance, laws of arithmetic) this has been done already in Chapter 16; in the other cases (for instance, the existence of averages) what needs to be done is quite similar to what was done in Chapter 16. Details are left to the reader.
Any enumerable model of arithmetic or P (or indeed any theory) is isomorphic to one whose domain is the set of natural numbers. Our interest in the remainder of this chapter will be in the nature of the relations and functions that such a model assigns as denotations to the nonlogical symbols of the language. A first result on this question is a direct consequence of Theorem 25.1a.
25.2 Corollary. There is a nonstandard model of arithmetic with domain the natural numbers in which the order relation is a recursive relation (and the successor function a recursive function).
Proof: We know the order relation on any nonstandard model of arithmetic is isomorphic to the order <K on the set K defined in the proof of Theorem 25.1a. The main step in the proof of the corollary will be relegated to the problems at the end of the chapter. It is to show that there is a recursive relation on the natural numbers that is also isomorphic to the order <K on the set K . Now, given any enumerable nonstandard model M of arithmetic, there is a function h from the natural numbers to |M| that is an isomorphism between the ordering on the natural numbers and the ordering <M on M. Much as in the proof of the canonical-domains lemma (Corollary 12.6), define an operation † on natural numbers by letting n† be the (unique) m such that h(m) = h(n) M; and define functions and similarly. Then the interpretation with domain the natural numbers and with ,† , , as the denotation of <, , +, · will be isomorphic to M. It will thus be a model of arithmetic, with the order relation recursive. (If one is careful, one can get the successor function † to be recursive as well.)
The Lowenheim¨ –Skolem theorem tells us that any theory that has an infinite model has a model with domain the natural numbers. The arithmetical Lowenheim¨–Skolem theorem asserts that any axiomatizable theory that has an infinite model has a model with domain the natural numbers and the denotation of every nonlogical symbol an arithmetical relation or function. The proof of this result requires careful review of