Predicative Arithmetic. (MN-32) [Course Book ed.] 9781400858927

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PREDICATIVE ARITHMETIC

by Edward Nelson

Mathematical Notes 32

Princeton University Press Princeton, New Jersey 1986

Copyright (c) 1986

by Princeton University Press

All Rights Reserved

Printed in the United States of America by Princeton University Press, 41 William Street, Princeton, New Jersey 08540

ISBN 0-691-081^55-6

The Princeton Mathematical Notes are edited by William Browder Robert Langlands, John Milnor, and Elias M. Stein

Library of Congress Cataloging in Publication Data will be found on the last printed page of this book

To Nancy I put myself into this work, so there is also something of you in it.

Acknowledgments This work was begun in December, 1979, at the Institute for Advanced Study. The next year I lectured on some of this material at Princeton Uni­ versity, and benefited from the comments of David Anderson, Jay Hook, Patrick Smith, Hale Trotter, Brian White, and Mitsuru Yasuhara. I also had helpful conversations with Laurence Kirby and Simon Kochen, and especially with Sam Buss, who found many errors and made many sugges­ tions for improvements. The work was partially supported by the National Science Foundation. I am grateful to these people and institutions.

Table of contents 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.

The impredicativity of induction Logical terminology The axioms of arithmetic Order Induction by relativization Interpretability in Robinson's theory Bounded induction The bounded least number principle The Euclidean algorithm Encoding Bounded separation and minimum Sets and functions Exponential functions Exponentiation A stronger relativization scheme Bounds on exponential functions Bounded replacement An impassable barrier Sequences Cardinality Existence of sets Semibounded replacement Formulas Proofs Derived rules of inference Special constants Extensions by definition

1 3 8 10 12 16 23 29 32 36 43 46 51 54 60 64 70 73 82 90 95 98 101 111 115 134 136

viii 28. Interpretations 29. The arithmetization of arithmetic 30. The consistency theorem 31. Is exponentiation total? 32. A modified Hilbert program Bibliography General index Index of defining axioms

TABLE OF CONTENTS 152 157 162 173 178 181 183 186

Chapter 1 The impredicativity of induction The induction principle is this: if a property holds for 0, and if whenever it holds for a number η it also holds for η + 1, then the property holds for all numbers. For example, let β(η) be the property that there exists a number m such that 2 · m = η • (η + 1). Then 0(0) (let m = 0). Suppose 2 · m — η • (η + 1). Then 2 • (m + η + l) = (η + 1) · ((η + 1) + 1), and thus if θ(η) then θ(τι + 1). The induction principle allows us to conclude θ(η) for all numbers n. As a second example, let π(η) be the property that there exists a non-zero number m that is divisible by all numbers from 1 to n. Then 7Γ (0) (let m = 1). Suppose m is a non-zero number that is divisible by all numbers from 1 to n. Then m • (n + l) is a non-zero number that is d i v i s i b l e b y a l l n u m b e r s f r o m 1 t o η + 1 , a n d t h u s if π ( η ) t h e n π { η + l ) . The induction principle would allow us to conclude π(η) for all numbers n. The reason for mistrusting the induction principle is that it involves an impredicative concept of number. It is not correct to argue that induction only involves the numbers from 0 to n; the property of η being established may be a formula with bound variables that are thought of as ranging over all numbers. That is, the induction principle assumes that the natural number system is given. A number is conceived to be an object satisfy­ ing every inductive formula; for a particular inductive formula, therefore, the bound variables are conceived to range over objects satisfying every

2

1. THE IMPREDICATIVITY OF INDUCTION

inductive formula, including the one in question. In the first example, at least one can say in advance how big is the number m whose existence is asserted by θ{η): it is no bigger than re-(n+l). This induction is bounded, and one can hope that a predicative treatment of numbers can be constructed that yields the result θ(η). In the second example, the number m whose existence is asserted by π(η) cannot be bounded in terms of the data of the problem. It appears to be universally taken for granted by mathematicians, what­ ever their views on foundational questions may be, that the impredicativity inherent in the induction principle is harmless—that there is a concept of number given in advance of all mathematical constructions, that discourse within the domain of numbers is meaningful. But numbers are symbolic constructions; a construction does not exist until it is made; when some­ thing new is made, it is something new and not a selection from a pre­ existing collection. There is no map of the world because the world is coming into being. Let us explore the possibility of developing arithmetic predicatively.

Chapter 2 Logical terminology I tried several times to write a brief, clear summary of the logical terminology that will be used in this investigation, but it always came out long and muddy. Instead, I refer the reader to the beautiful exposition in Shoenfield's book iSh], especially the first four chapters. Our logical terminology and notation are those of iSh] except for some departures and ildditions that will be indicated. Lower case italic letters, possibly with 0, 1, ... as a subscript, are variables. The order a, b, .. . , Z, ao, bo, ... , Zo, al, ... of the variables is called alphabetical order. Roman letters are used as in iSh] as syntactical variables when talking about expressions. We define Ax[al as follows: if no variable occurring in the term a occurs bound in the formula A, substitute a for each free occurrence of the variable x in Ai otherwise let Xl, ... ,xv be in alphabetical order the \'ariables that occur in a and occur bound in A, let YI, ... ,Yv be in alphabetical order the first v variabJes distinct from all variabJes occurring i:-. A or a, substitute y I" for each bound occurrence of xI" in A for all J1 from 1 to v, and then substitute a for each free occurrence of x. For example, if A is

+ y) + Z = x + (y + Z), is \:fx\:fy (x + y) + 0 = x + (y + 0)

'Vx'Vy (x

then Az[Ol

a + (y + x). We write Ax, xvial".av] for Ax,[al] ... Xv[a v]. 3

and Az[x] is \:fa\:fy (a + y)

+x =

4

2. LOGICAL TERMINOLOGY We make the following abbreviations, in addition to those of [Shl. If

PI> •.• ,Pv are binary predicate symbols, we write

as in x = y = z for x = y & Y = z. We follow [Shl is associating V, &, and -+ from right to left, and letting V and & take precedence over -+ and +-+ in restoring parentheses, so that

is equivalent to Al & ... & A v -

-+

I

Av ,

but we adopt the convention that Al

+-+ ••• +-+

Av - I

Av

+-+

is an abbreviation for

If A is Al -+ A 2 , we write hypA for Al and conA for A 2 • If A is Al we write IhsA for Al and rhsA for A 2 • We write

3!xA for 3x(A & VY(Ax[Yl

-+

+-+

A2 ,

y = x)),

where y is the first variable in alphabetical order distinct from x and all variables occurring in A. If all bound occurrences of x in A occur in the part 3xB, we write scope3xA for B, and similarly with 3 replaced by 3! or v. We write the defining axiom of a function symbol f as fXI ... Xv = Y -+ D (instead of y = fXI ... Xv form fXI ... Xv

= Y +-+ A,

+-+

D as in [Sh,§4.6]). Sometimes we write it in the

otherwise y = e

where e is a constant. This is an abbreviation for fXI··

.Xv

= Y +-+ A V (-,3yA & y = e).

2. LOGICAL TERMINOLOGY

5

Then the existence condition is trivial, but the uniqueness condition still has to be verified. Also, we sometimes adjoin a new function symbol f by writing fxi... x„ = a where a is a term containing no variable other than xi,... ,x„; the defining axiom is understood to be

fxi... X 1 , = y ^ y = a where y is the first variable in alphabetical order distinct from xi,... ,x„. A formula will be called u n a r y in case one and only one variable occurs free in it. If C is unary and χ is the variable occurring free in it, we write C[a] for C x [a]. Let C be a unary formula. Then we write Ac for the formula obtained by replacing each part of A of the form 3yB by 3y(C[y] k B). (We follow [Sh] in regarding Vy as an abbreviation for ^3y-i; if one chooses not to eliminate the defined symbol V, then an equivalent formula is obtained if in addition one replaces each part of A of the form VyB by Vy(C[y] —» B).) Let X 1 , . . . , x„ be in alphabetical order the variables occurring free in A; then we write C ( f r e e A ) for Cfx 1 ! & ··· & C[x „j. If A is closed, then C(/reeA) is the empty expression; we make the conven­ tion that if A is closed, then all occurrences of "C(/reeA)" together with attendant logical connectives are to be deleted. (In general, I will not worry about the distinction between use and mention, but in this case I use quo­ tation marks lest the reader be puzzled as to how to go about deleting the empty expression.) We write A c for C(/reeA) —> A c and call A c the r e l a t i v i z a t i o n of A by C. Let C be a unary formula of a theory T. For f a ^-ary function symbol, we say that C respects f in case Ι~τ Cfx 1 ] - •

• C [ x „ ] -• Cffx 1 ... *„].

2. LOGICAL TERMINOLOGY

6

We say that C respects A in case rT A C. We say that C is inductive in case C respects 0 and S (assuming T to contain the constant 0 and the unary function symbol S), so that C is inductive if and only if

rT C[O]

& (C[x]

---7

C[Sx])

(or, equivalently, if and only if rT C[O] & \fx(C[x]---7 C[Sxl)). We say that C is hereditary in case

rT C[x]

& y