Predicative Arithmetic. (MN-32):

Predicative Arithmetic. (MN-32):

Edward Nelson
Copyright Date: 1986
Pages: 198
  • Cite this Item
  • Book Info
    Predicative Arithmetic. (MN-32):
    Book Description:

    This book develops arithmetic without the induction principle, working in theories that are interpretable in Raphael Robinson's theory Q. Certain inductive formulas, the bounded ones, are interpretable in Q. A mathematically strong, but logically very weak, predicative arithmetic is constructed.

    Originally published in 1986.

    ThePrinceton Legacy Libraryuses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

    eISBN: 978-1-4008-5892-7
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-vi)
  2. Table of Contents
    (pp. vii-viii)
  3. Chapter 1 The impredicativity of induction
    (pp. 1-2)

    The induction principle is this: if a property holds for 0, and if whenever it holds for a numbernit also holds forn+ 1, then the property holds for all numbers. For example, let$\theta (n)$be the property that there exists a numbermsuch that$2 \cdot m = n \cdot (n + 1)$. Then$\theta (0)$(letm= 0). Suppose$2 \cdot m = n \cdot (n + 1)$. Then$2 \cdot (m + n + 1) = (n + 1).((n + 1) + 1)$, and thus if$\theta (n)$then$\theta (n+1)$. The induction principle allows us to conclude$\theta (n)$for all numbersn. As a second example, let$\pi (n)$be the property that there exists a non-zero numbermthat is divisible by all numbers...

  4. Chapter 2 Logical terminology
    (pp. 3-7)

    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 [Sh], especially the first four chapters. Our logical terminology and notation are those of [Sh] 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, a1, ... of the variables is calledalphabetical order.Roman letters are used...

  5. Chapter 3 The axioms of arithmetic
    (pp. 8-9)

    ByPeano Arithmeticwe mean the theoryIwhose nonlogical symbols are the constant 0, the unary function symbol S (successor), and the binary function symbols + and ·, and whose nonlogical axioms are

    3.1$Ax.Sx \ne 0$,

    3.2$Ax.Sx = Sy \to x = y$,

    3.3$Ax \cdot x + 0 = x$,

    3.4$Ax.x + Sy = S(x + y)$,

    3.5$Ax \cdot x \cdot 0 = 0$,

    3.6$Ax\cdotx\cdot{S_y} = x\cdoty + x$,

    and all induction formulas in the language ofI. We have adopted the usual convention that · takes predecence over + in restoring parentheses, so that (3.6) is an abbreviation for$x\cdot{S_y} = (x\cdot{y}) + x$.

    If we simply drop the induction formulas as axioms, then the resulting theory is too weak to be of much arithmetical interest;...

  6. Chapter 4 Order
    (pp. 10-11)

    The following formula is the defining axiom of a binary predicate symbol that we adjoin to Q1:

    4.1Def.$x \le y \leftrightarrow \exists zx + z = y$.

    Call the resulting theory Q’1. In this chapter we prove a few simple theorems in Q’1.

    4.2Thm.$0 \le x \le x \le Sx$.

    Proof.We have$0 + x = x$by (3.3) and (3.11), so$0 \leqslant x.$We have$x + 0 = x$by (3.3), so$x \le x,$and$x + S0 = S(x + 0) = Sx$by (3.4) and (3.3), so$x \le Sx$.

    4.3Thm.$x \le 0 \to x = 0$.

    Proof.Suppose$x \le 0$. Then there existszsuch that$x + z = 0$. Suppose$z = 0$. Then$x = 0$by (3.3) and thus$z = 0 \to x = 0$, so suppose$z \ne 0$. Then$SPz = z$by (3.7). so that$z = 0$so that$x + SPz = 0,$and...

  7. Chapter 5 Induction by relativization
    (pp. 12-15)

    Let C be a unary formula. We use the following abbreviations:

    ${C^1}$for$\forall y(y \le x \to C\left[ y \right])$,

    ${C^2}$for$\forall y({C^1}\left[ y \right] \to {C^1}\left[ {y + x} \right])$,

    ${C^3}$for$\forall y({C^2}\left[ y \right] \to {C^2}\left[ {y \cdot x} \right])$.

    Then C¹, C², and C³ are unary formulas with free variablex. We will show that if C is inductive, then C³ is stronger than C, is hereditary, and is not only inductive but respects P, +, ·, and the defining axiom of$ \le $. This relativization scheme is due to R. Solovay; see [PD] and [Pu].

    Metatheorem 5.1 LetTbe an extension of${{Q'}_1}^$(possibly${{Q'}_1}^$itself), and letCbe a unary formula ofT.Then the following...

  8. Chapter 6 Interpretability in Robinson’s theory
    (pp. 16-22)

    For the pleasure of working from minimal assumptions, let us show that we can drop the axioms (3.8)–(3.12). We will extend the relativization scheme of the preceding chapter by building into the construction the necessary associativity, etc., and then use this to show that${Q_1}$is interpretable in${Q_0}$(and so in Robinson’s theoryQ). The reader who wishes to skip this chapter can simply substitute${Q_1}$forQin later statements about interpretability inQ.

    For this chapter, and this chapter only, we make the abbreviations:

    $\alpha $for$\forall x\forall y$(x + y) + z= x + (y +z),...

  9. Chapter 7 Bounded induction
    (pp. 23-28)

    In this chapter we define the notion of bounded formulas and show that induction can be used on them.

    The initial occurrence of$\exists x$in a part$\exists xB$of A is calledmanifestly boundedin case B is of the form$x \le a$& C where a is a term not containing x. The formula A is calledmanifestly boundedin case each occurrence of an existential quantifier in A is manifestly bounded. (Recall that inside each universal quantifier there lurks an existential quantifier.) Formulas are built up from atomic formulas by means of ¬, V, and existential quantifiers$\exists x$(see [Sh,§2.4])....

  10. Chapter 8 The bounded least number principle
    (pp. 29-31)

    The least number principle is a form of the induction principle that is useful in many proofs. Used indirectly, it is equivalent to proof by infinite descent or complete induction. Here we formulate a predicative version of the least number principle.

    For a formula A and variables${x_1}, \ldots ,{x_\nu },$let${y_1}, \ldots ,{y_\nu }$be in alphabetical order the first ν variables not occurring in A and distinct from${x_1}, \ldots ,{x_\nu };$then we write

    ${\min _{{x_1}..{x_\upsilon }}}A$


    $A & \neg {\exists _{{y_\upsilon }}}({y_1} \le {x_1}\& \cdots \& {y_\upsilon } \le {x_\upsilon }\& ({y_1} \ne {x_1} \cdots V{y_\upsilon } \ne {x_\upsilon })\& {A_{{x_{1 \cdots {x_\upsilon }}}}}[{y_1} \cdots {y_\upsilon }])$.

    Metatheorem 8.1 LetUbe an extension of Q2and letAbe a formula ofUthat is bounded over Q2. Then

    BLNP.$\exists {x_1} \cdots \exists {x_v}A \to \exists {x_1} \cdots \exists {x_v}{\min _{{x_1} \cdots {x_v}}}A$

    is a theorem...

  11. Chapter 9 The Euclidean algorithm
    (pp. 32-35)

    Until further notice we will work in bounded extensions ofQ2.At one point in the proof of REL we quoted five axioms to show that one timesxequalsx.This sort of thing might become tiresome if continued much longer, so let’s stop doing it. The development picks up from where we left off at the end of Chapter 4, but now we have BI and BLNP available.

    9.1 Thm.${z_1} + y = {z_2} + y \to {z_2}$.

    Proof.Clearly (9.1)y[0]. Suppose (9.1)${z_1} + {S_y} = {z_2} + {S_y}$. Then${z_1} + y = {z_2} + y$and so${z_1} = {z_2}$. Thusindy(9.1), so by BI we have (9.1).

    9.2Def.$x - y = z \leftrightarrow z - y = x$otherwisez= 0....

  12. Chapter 10 Encoding
    (pp. 36-42)

    In the last chapter we copied the usual proofs, observing that only bounded inductions are involved. But now we come to a fork in the road. Arithmetic is too limited unless it can express notions such as finite sums and products, exponentiation, etc. The usual semi-formal treatment of such notions is based on a pun on the word “number”, confounding the formal notion of number as a term of a theory with the genetic notion of number used in counting. For example, when one writes

    $\sum\limits_{i = 1}^n {f(i)} = f(1) + \cdots + f(n)$,

    one is trying simultaneously to usenas a term of a theory and...

  13. Chapter 11 Bounded separation and minimum
    (pp. 43-45)

    Let Q’2be the current theory; that is, the extension of Q2obtained by adjoining the defining axioms up to the present.

    Metatheorem 11.1 LetTbe an extension ofQ’2,letAbe a bounded formula ofT,and let x, y, and z be distinct variables such that z does not occur inA. Then

    BSD.$\{ x \in y:A\} = z \leftrightarrow {\min _z}\forall x(x \in y\& A \to x \in z)$

    is the defining axiom of a bounded function symbol. (The variables in the term$x \in y:A$are y and the variables distinct from x that occur free inA.) The following is a theorem ofT[(BSD)]:

    BS.$\{ x \in y:A\}$is a set &...

  14. Chapter 12 Sets and functions
    (pp. 46-50)

    Although conceptually simple, the encoding procedure is laborious. In this chapter we will develop an elementary theory (in the mathematical sense of a collection of theorems) of certain finite sets of numbers, and afterwards we can hopefully forget the details of the encoding procedure.

    12.1Thm. 0 is a set &$\neg (x \in 0)$.

    Proof. Clearly 0 is a set. By(10.21),$\neg (x \in 0)$.

    12.2Thm.aandbare sets &$\forall x(x \in a \leftrightarrow x \in b) \to a = b$.

    Proof· Supposehyp(12.2). Then$a \le b \le a$Thus (12.2).

    12.3Def.$\{ x\} = \{ y \in 2 \cdot {\left| {Encx} \right|_4} \cdot 4 \cdot 4 - Encx \cdot 4 + 2:y = x\} $.

    12.4Thm. {x} is a set &$(y \in \{ x\} \leftrightarrow y = x).$

    proof. By BS, {x} is a set &$(y \in \{ x\} \to y = x).$. We have enc(1,...

  15. Chapter 13 Exponential functions
    (pp. 51-53)

    In this chapter we prove some familiar properties of exponentiation by assuming as given a function I that is exponentiation ($i \mapsto {x^i}$) on the domain of all$i \le k$

    13.1Def.$\exp (x,k,f) \mapsto f$is a function &f(0) = 1 &$\forall i(i \in Domf \leftrightarrow i \le k)$&$\forall i(i < k \to f(i + 1) = x \cdot f(i)).$.

    We haverhs(13.1):$i \le Max(f,k)$).

    13.2Thm. exp(x, k, f) & exp(x, k, g)$ \to f = g$.

    Proof. By (12.17).

    13.3Thm. exp(x, k, f) & exp(y, k, g) &$ exp(x · y, k, h)\to h(k) = f(k) \cdot g(k) $

    Proof. Supposehyp(13.3). We will show that in fact$1:i \le k \to h(i) = f(i) \cdot g(i)$. Suppose$\exists i\neg (1)$. By BLNP there exists a minimal suchi. Clearly$i \ne 0$. Therefore...

  16. Chapter 14 Exponentiation
    (pp. 54-59)

    Let${Q'''}_{_2}$be the current theory. Notice: in this chapter we will work in unbounded extensions of${Q'''}_{_2}$. (When I say that something is unbounded, I mean merely that I do not claim that it is bounded.) We mark with “!” the defining axiom of any unbounded symbol. The “!” serves as a warning that we may not use BI (or BLNP, BSD, BS, MIND, MIN, MAXD, MAX, or FS) on formulas containing the symbol.

    14.1Def!$\varepsilon (k) \leftrightarrow \forall x\exists f\exp (x,k,f)$.

    14.2Thm.$in{d_k}\varepsilon (k)$.

    Proof.We have$\forall x\exp (x,0,\{ \langle 0,1\rangle \} )$and so$\varepsilon (0)$. Suppose exp(x, k, f) and let g = f U {(Sk, x...

  17. Chapter 15 A stronger relativization scheme
    (pp. 60-63)

    We concluded Chapter 14 by proving some properties of #0, but some of these held only conditionally, subject to λ(x) for certainx. What we can’t prove, we can postulate. Let${Q_3}$be the theory obtained from${{Q'''}_2}$(this is the bounded extension of ${Q_2}$ defined at the beginning of Chapter 14) by adjoining a binary function symbol # and the following nonlogical axioms:

    15.1$Ax.x\# y = |x\# y{|_2} = |x{|_2}\# |y{|_2} = |x{|_2}\# |y| = x\# |y{|_2}$

    15.2$Ax.x\# 1 = 1$,

    15.3$Ax.x\# 2 = 2$,

    15.4$Ax.x\# y = y\# x$,

    15.5$Ax.(x\# y)\# z = x\# (y\# z)$

    15.6$Ax.x\# (|y{|_2} \cdot |z{|_2}) = (x\# |y{|_2}) \cdot (x\# |z{|_2})$

    15.7$Ax.y \le z \to x\# y \le x\# z$

    Metatheorem 15.1${Q_3}$is interpretable in${Q_2}$.

    Demonstration.Let T be the (unbounded) extension by definitions of${Q_2}$obtained by adjoining all...

  18. Chapter 16 Bounds on exponential functions
    (pp. 64-69)

    Now we take up the question of how bigfis when exp(x, k, f) holds.

    16.1Thm.$\{ x\} \le 146\cdot{(SPx)^2}$.

    Proof.From the defining axioms of the function symbols involved, we find that$|x{|_2} \le SPx$and$Encx \le SPx \cdot SPx \cdot 4$, and so we have$\{ x\} \le 2 \cdot (SPx \cdot SPx \cdot 4) \cdot 4 \cdot 4 + (SPx \cdot SPx \cdot 4) \cdot 4 + 2 \le 146\cdot{(SPx)^2}$

    16.2Thm.$a \cup b \le 5 \cdot SPa \cdot SPb$

    Proof.We have$a \cup b \le a \cdot SPb \cdot 4 + b \le 5 \cdot SPa \cdot SPb$.

    16.3Thm.$\langle x,y\rangle \le 5 \cdot {(Max(x,y))^2}$.

    Proof.We have$\langle x,y\rangle = {(x + y)^2} + y \le 4 \cdot {(Max(x,y))^2} + 5 \cdot {(Max(x,y))^2}$.

    16.4Def.K = 18250.

    16.5Thm.$f \cup \{ \langle x,y\rangle \} \le K.SPf \cdot {(SPMax(x,y))^4}$

    Proof.By (16.1), (16.2), and (16.3).

    16.6Thm.$2 \le x$&$\exp (x,k,f)$&$i \le k \to i \le f(i)$.

    Proof.Suppose$\exists i\neg (16.6)$. By BLNP there exists a minimal suchi. Clearly$i \ne 0$. Therefore$i - 1 \le f(i - 1)$, and$i \ne 0$by (13.9), so$i \le x \cdot f(i - 1) = f(i)$, a contradiction. Thus (16.6)....

  19. Chapter 17 Bounded replacement
    (pp. 70-72)

    Let${{Q'}_4}$be the current theory.

    Metatheorem 17.1 LetTbe an extension of${{Q'}_4}$,and letDbe such that$\exists yD$is a bounded formula ofTand${x_1}, \ldots ,{x_\nu }$are the variables distinct from x and y occurring free inD.Then

    BR. a is a set &$\forall x(x \in a \to (\exists yD) \to \exists f$(f is a function & Domf=a&$\forall x(x \in a \to {D_y}[f(x)])$)

    is a theorem of T, and

    BRD.$\{ \langle x,y\rangle :x \in a{\rm{\& }}{\min _y}D\} $=$f \leftrightarrow f$is a function & Domf=a&$\forall x(x \in a \to {({\min _y}D)_y}[f(x)])$, otherwisef= 1...

  20. Chapter 18 An impassable barrier
    (pp. 73-81)

    Let us pause to examine from an impredicative point of view what we are doing. Take a strong theory T containing 0 and S, say an extension by definitions of Peano ArithmeticIor even of ZFC (Zermelo-Fraenkel set theory with the axiom of choice). Let${\rm{\hat T}}$be the theory obtained by adjoining a unary predicate symbol ø and the axiom

    Fin. ø(0) & (ø(x) → ø(Sx)).

    This adjunction does not increase the power of the theory in any way; we can for example interpret ø(x) byx=x.If T is an axiomatization of arithmetic, say an extension...

  21. Chapter 19 Sequences
    (pp. 82-89)

    In this chapter we will predicatively arithmetize the basic syntactical notions of juxtaposition, substitution, and occurrence.

    19.1Thm.$\exists z\forall wf(w) \le f(z)$

    Proof.We have (19.1):$z \le f$,$w \le f$. Suppose$\exists f\neg (19.1)$. By BLNP there exists a minimal such f. Suppose$\neg $(fis a function) V f = 0, and let z = 0. Then (19.1), a contradiction, and thusJis a function &$f \ne 0$. By (12.2) there existstsuch that$t \in f$, and there existxandysuch that$\langle x,y\rangle = t$....

  22. Chapter 20 Cardinality
    (pp. 90-94)

    A set has to be small in three ways: the formula describing it must be bounded, there must be a bound on its elements, and there must be a logarithmic bound on its cardinality. We elaborate on this statement in this chapter and the next.

    20.1Def. fis injective ↔$\forall x\forall y(x \in Domf\& y \in Domf\& x \ne y \to f(x) \ne f(y))$.

    We use the abbreviations “a is an injective sequence” for “a is injective & a is a sequence”, and “a is an injective sequence of ***” for “a is injective & a is a sequence of ***”.

    20.2Def. uis an injection into a ↔ u is...

  23. Chapter 21 Existence of sets
    (pp. 95-97)

    Metatheorem 21.1 Let${{Q''}_4}$be the current theory, letU be an extension of${{Q''}_4}$,letAbe a bounded formula ofU, and letaandbbe bounded terms ofUnot containingxory.Then the following is a theorem ofU:

    SET.$\forall \times (A \to x \le a)$&$\forall y(\forall \times (x \in y \to A) \to Cardy \le Logb) \to \exists y$(y is a set &$\forall \times (x \in y \leftrightarrow A)$).

    Demonstration.We prove (SET) in U as follows. Supposehyp(SET).

    Write α for

    $y \le Exp\log (730 \cdot {(SPa)^2},b)$& y is a set &$\forall \times (x \in y \to A)$,

    and let y = Max y α. Clearly${\alpha _y}[0]$, so by MAX we have α. Suppose${\rm{A \& x }} \notin y$and let${y_1} = y \cup \{ x\} $(Here y1is...

  24. Chapter 22 Semibounded replacement
    (pp. 98-100)

    This chapter is a digression, and I do not intend to use it in the sequel except in occasional remarks. The semibounded replacement principle differs from the bounded replacement principle of Chapter 17 in that 3y D is no longer required to be bounded, though D itself is.

    Metatheorem 22.1 LetDbe a bounded formula of${{Q''}_4}$such that${x_1}, \ldots ,{x_v}$are the variables distinct from x and y occurring free inD.Consider the formula

    SBR. a is a set &$\forall x(x \in a \to \exists yD) \to \exists f$(fis a function & Domf=a&$\forall x(x \in a \to Dy[f(x)])$).

    Then${{Q''}_4}$[(SBR)]is interpretable in${{Q''}_4}$.


  25. Chapter 23 Formulas
    (pp. 101-110)

    Now we are ready to begin to investigate which results of finitary math ematical logic can be established predicatively. We will follow the presentation in [Sh] very closely. See [Sh,§2.4] in connection with this chapter.

    It will be convenient to enlarge our stock of variables. We also letA, B, C,andD-possibly with 0, 1, 2, ... as a subscript—be variables, and if x is a variable we let x’ be a variable. The notion of alphabetical order is understood to be suitably redefined, with the relative order of the old variables being unchanged. Generally speaking, our...

  26. Chapter 24 Proofs
    (pp. 111-114)

    We give a predicative arithmetization of the predicate calculus. We modify the treatment in [Sh, §2.6] by adopting tautological consequence as a rule of inference; see the conclusion of [Sh,§3.1].

    24.1Def. Bis a substitution$axiom \leftrightarrow \exists \wedge \exists x\exists a$(ais substitutable forxin A &$B = {A_x}[a]\tilde \to \tilde \exists xA)$.

    24.2Def. Bis an identity$axiom \leftrightarrow \exists x$(xis a variable &$B = x\tilde = x$.

    24.3Def.Equals(x', y') = {(i, x'(z')"= y'(i)): i E Domx'}.

    24.4Def. Equals(x', y') = {(i, x'(z')"= y'(i)): i E Domx'}.

    24.5Def. B. is a logical$axiom \leftrightarrow \exists x$Def. Bis an equality axiom +-4 3x' 3y'(x' and y' are sequences of...

  27. Chapter 25 Derived rules of inference
    (pp. 115-133)

    The material in [Sh,§§3.2–3.5] is concerned with derived rules of inference. Straightforward arithmetizations of all of these results are theorems of our theory. All of these derived rules of inference can be expressed by bounded function symbols. All of the induction arguments in these sections of [Sh] are bounded, with one exception—and an alternate predicative proof can be given for it. The reader who is willing to accept these conclusions should read on for a few paragraphs, where some notational conventions are introduced, and then skip the remainder of this lengthy chapter.

    Sometimes in the course of the...

  28. Chapter 26 Special constants
    (pp. 134-135)

    There is another important derived rule of inference. When we have proved$\exists \times A$it is very useful to have a name for such an x. In our proofs we have been saying “there exists x such that A”. More formally, for a closed instantiation$\exists \times A$one can adjoin a new constant r, called aspecial constant,and thespecial axiom$\exists \times A \to {A_x}[r]$, and this device can be iterated. When$\exists \times A$is not closed, it is necessary to treat its free variables as constants; this is a tacit use of the Theorem on Constants. In this chapter we will construct a predicative...

  29. Chapter 27 Extensions by definition
    (pp. 136-151)

    In this chapter we will predicatively arithmetize the notions of extension by definition of a predicate symbol and extension by definition of a function symbol, and show how to construct bounded function symbols that translate proofs into the original theory. We will follow [Sh, §4.6] except for the proof that an extension by definition of a function symbol is conservative. We begin with some general properties of translation functions.

    27.1Def. gis a translation function onsgis a function &sis a set of formulas &$Domg = s$&$\forall B(B \in s \to g(B)$is a formula & Free g(B)...

  30. Chapter 28 Interpretations
    (pp. 152-156)

    The proof in [Sh,§4.7] of the Interpretation Theorem has a straightforward predicative arithmetization. We begin with the notion of an interpretationi,with universe u0, of one language in another.

    28.1Def.inter${p_0}(i,{u_0},{l_1},{l_2}) \leftrightarrow {l_1}$and${l_2}$are languages &$& {u_0}$is a predicate symbol & Index${u_0} = 1$&${u_0} \in {l_2}$& Dom$i = {l_1}$& Ran$i \subseteq {l_2}$&$\forall \upsilon (\upsilon \in {l_1} \to Indexi(\upsilon ) = Index\upsilon \& (i(\upsilon )$is a predicate symbol ↔υ is a predicate symbol)).

    28.2Def.${u_{(1)}} = \{ \langle j,v\rangle :j \in Domu\& (u(j) \in Domi \to \upsilon = i(u(j)))\& (u(j) \notin Dom\iota \to \upsilon = u(j))\} $.

    This construction is used to interpret a term or an atomic formulau: replace each nonlogical symbol byiof it. We construct the interpretation${A^{(\iota ,{u_0})}}$of a general formulaAin...

  31. Chapter 29 The arithmetization of arithmetic
    (pp. 157-161)

    Now we can begin to prove results about predicative arithmetic within predicative arithmetic. In this chapter we will arithmetize Robinson’s theory and show it to be tautologically consistent (and we do this within a theory that is interpretable in Robinson’s theory).

    29.1$Def.\tilde 0 = {F_{0,0}}$.

    29.2$Def.\bar S = {F_{1,0}}$

    29.3$Def.\bar P = {F_{1,1}}$

    29.4$Def. \mp = {F_{2,0}}$

    29.5$Def.\bar \cdot = {F_{2,1}}$

    29.6$Def.\tilde Sa = \bar S * a$

    29.7$Def.\tilde Pa = \bar P * a$

    29.8$Def.a\tilde + b = \mp * a * b$

    29.9$Def.a\tilde \cdot b = \bar \cdot * a * b$

    If x is the νthvariable in alphabetical order, we use${\tilde x}$as an abbreviation for${X_{\bar \nu }}$. (Recall that${\bar \nu }$is$S \cdots {S_0}$with ν occurences of S.) Also, we use$\{ {a_1}, \ldots {a_\nu }\} $as an abbreviaion of$\{ {a_1}\} U \cdots U\{ {a_\nu }\} $.

    29.10$Def.{{\bar Q}_0} = (\{ \tilde 0,\bar S,\bar P, \mp ,\bar \cdot \} ,\{ \tilde S\tilde x\tilde \ne \tilde 0,\tilde S\tilde x\tilde = \tilde S\tilde y\tilde \to \tilde x\tilde = \tilde y,\tilde x\tilde + \tilde 0\tilde = \tilde x,\tilde x\tilde + \tilde S\tilde y\tilde = \tilde S(\tilde x\tilde + \tilde y),\tilde x\tilde \cdot \tilde 0\tilde = \tilde 0,\tilde x\tilde \cdot \tilde S\tilde y\tilde = \tilde x\tilde \cdot \tilde y\tilde + \tilde x,\tilde P\tilde x\tilde = \tilde y\tilde \leftrightarrow \tilde S\tilde y\tilde = \tilde x\tilde V(\tilde x\tilde = \tilde 0\tilde \& \tilde y\tilde = \tilde 0)\} )$

    29.11Def. a...

  32. Chapter 30 The consistency theorem
    (pp. 162-172)

    We have already discussed the Hilbert-Ackermann Consistency Theorem of [Sh, §4.3] in connection with Assertion 18.1. This theorem is basically an algorithm for eliminating quantifiers from proofs. The first step is to eliminate each use of$\exists $-introduction; this is done by introucing special constants and special axioms. Then the formulas belonging to special constants are eliminated, beginning with special constants of maximal rank. We follow [Sh,§4.3] closely.

    30.1Def.spconstseq$({r^1},{x^1},{C^1},t) \leftrightarrow {r^1}$is an injective sequence of constants &${x^1}$is a sequence of variables &$Ln{r^1} = Ln{x^1} = Ln{C^1}\& \{ \left\langle {i,\widetilde\exists {x^1}\left( i \right){C^1}\left( i \right)} \right\rangle :i \in Dom{r^1}\} $is a sequence of closed formulas &$\forall i\forall j(1 \leqslant i < j \leqslant Ln{r^1} \to \neg ({r^1}(i)occur\sin {C^1}(j)))$&tis a theory & Lang...

  33. Chapter 31 Is exponentiation total?
    (pp. 173-177)

    Why are mathematicians so convinced that exponentiation is total (everywhere defined)? Because they believe in the existence of abstract objects called numbers. What is a number? Originally, sequences of tally marks were used to count things. Then positional notation—the most powerful achievement of mathematics—was invented. Decimals (i.e., numbers written in positional notation) are simply canonical forms for variable-free terms of arithmetic. It has been universally assumed, on the basis of scant evidence, that decimals are the same kind of thing as sequences of tally marks, only expressed in a more practical and efficient notation. This assumption is based...

  34. Chapter 32 A modified Hilbert program
    (pp. 178-180)

    Hilbert’s program was to secure the foundations of classical mathematics by giving a finitary consistency proof for it. This formulation of the program was undoubtedly influenced by his controversy with Brouwer—finitary methods are those (or perhaps a subset of those) that are acccepted by the intuitionists. As far at least as arithmetic is concerned, Hilbert’s aim of demonstrating that classical mathematics is no less secure than is intuitionistic mathematics was achieved by Gödel’s five page paper [Gö2], published in 1933, in which he gave an interpretation of classical arithmetic within intuitionistic arithmetic. But by then the problem had been...

  35. Bibliography
    (pp. 181-182)
  36. General index
    (pp. 183-185)
  37. Index of defining axioms
    (pp. 186-189)
  38. Back Matter
    (pp. 190-190)