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On the Original Gentzen Consistency Proof for Number Theory
ON THE ORIGINAL GENTZEN CONSISTENCY PROOF FOR NUMBER THEORY +
PAUL BERNAYS The first published Gentzen consistency proof for the formal system of first order number theory, including standard logic, the Peano axioms and recursive definitions, was given in Gentzen’s paper ‘Die Widerspruchsfreiheit der reinen Zahlentheorie’ (Math. Ann. 112 (1936)). It was however not his original proof but a revised version of it. The revision was motivated by a criticism, in which I myself for some time concurred of the original proof on the grounds that it implicitly included an application of the fan theorem. Gentzen did not expressly oppose this opinion; he took care of the criticism by modifying his consistency proof before it was published. Fortunately the text of the original proof is preserved in galley proof*. On rereading the original text, I came to doubt the mentioned opiniont. It seems worth while reconsidering the original proof, because 1. it is certainly easier to follow than the first published proof and at least as easy to follow as the second Gentzen consistency proof, 2. it does not require the generalized form of induction (ordinal induction up to E 0 ) . I shall give in this paper a description of Gentzen’s original proof in a way sufficiently detailed to make apparent the kind of constructive methods here used. At the same time I shall make a slight simplification in this proof, making use of some remarks in Gentzen’s Annalen proof. I begin by describing the formal system to be considered. The logical calculus used is the calculus of sequents’ r + A where r, the antecedent, is a finite (possibly empty) list of formulas separated by commas, and Since Prof. Bernays was unfortunately unable to be present, this paper was read by William Howard. - Eds. An Englishtranslationofit will soon appear in an edition of Gentzen’s works in English by Manfred Szabo. t Georg Kreisel had previously stated in conversation that he dissented from this opinion. t ‘Sequent’as a technical term should be distinguished from ‘sequence’ in its usual sense. 409 +
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where A, the succedent, is a single formula. The logical symbols v (or), 3 (existence) and 3 (implication) are eliminated, and only & (and), --, (not), V (universality) are used. As Gentzen admits arithmetical function symbols (with the restriction that for numerical arguments the value of the function must be computable), there is no loss of generality in assuming that all prime formulas are equations between terms. (Terms are built up in the usual way from free number variables*, numerals and function symbols.) The rules of the calculus, after the mentioned elimination, are the following: Initial sequents are: (1). Logical initial sequents, i.e. sequents of one of the forms A & B +A,
A & B + B,
A,TA+1=2,
A, B + A & B ,
TTA+A,
(Vx)F(x)
--f
F(t),
where x is a bound variable, and t a term. (2). Arithmetic initial sequents, i.e. sequents whose formulas are equations and which have the property that by replacing each free variable by a numeral (of course equal variables by equal numerals) and by computing the function values, either the succedent formula gets the truth value ‘true’ or one of the antecedent formulas gets the truth value ‘false’, according to the usual valuation of numerical equations. The rules of inference are: (a). Rules of structural change in a sequent, permitting one (al). to interchange the order of the formulas of the antecedent, (a2). to add an arbitrary formula to the antecedent, (a3). to delete a repetition of a formula in the antecedent, (a4). to change a bound variable of a universal quantifier, everywhere in its scope, into another bound variable. (b). Logical inference schemata +A
A, A + B r,A+B
*
(cut)+
Gentzen makes a notational distinction between free and bound variables. + This schema does not occur in the original text as a fundamental schema, but it is derivable from the schemata for negation there stated, which Gentren later changed in 8 14 of the Annalen paper.
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ON THE ORIGINAL GENTZEN CONSISTENCY PROOF
T,A-+1=2 r + i A
r
r -+ F(a) -+
(c). Induction
r
-+
(vX)F(X)
(negation introduction) (where a is a free variable not occurring in any formula of r nor in F(x), and where x is a bound variable) (universality introduction)
F(l) F(a), A + F(a+l) r, A -+ F(t)
where t is a term and a a free variable not occurring in or F(t).
r, A,
F(l)
In this calculus from any two formulas A, A we can derive, using the initial sequent A, A -P 1 = 2 and cut, the sequent + 1 = 2; hence, in order to prove consistency of the considered formal system, it is sufficient to show that the sequent -+ 1 = 2 is not derivable. A concept of ‘reduction’ of a sequent is introduced. The following are the possible ‘reduction steps’ on a sequent: ( ~ 1 ~ ) . Replacing a free variable, wherever it occurs in the sequent, by the the same numeral, which can be arbitrarily chosen. ( ~ 1 ~ ) . Replacing a function symbol all of whose arguments are constants by its value. (PI). When the succedent has the form (Vx)F(x), replacing it by F(k), where k is an arbitrarily chosen numeral. (&). When the succedent has the form A & B, replacing it by A or by B, according to an arbitrary choice. (P3). When the sequent has the form r -, A, replacing it by ~,r-+i=2. (y). When the succedent is a false numerical equation: (yl). Replacing an antecedent formula (Vx)F(x) by F(k), or adding F(k) to it in the antecedent, where k is a numeral. (y2). Replacing an antecedent formula A & B by one of the formulas A, B or adding one of these in the antecedent. (y3). If an antecedent formula A occurs, replacing the succedent A in the formula by A and possibly cancelling the formula antecedent.
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The description of these reduction steps must be completed by the following rule of preference: The step (a1)has preference over all other reduction steps, and (a1),(a2)have preference over the other reduction steps. That means: the steps ( a 2 )are admitted only for a sequent which contains no free variable, and the steps (pl), (p2), (p3), (yl), (y2), (y3) are admitted only for sequents which contain neither a free variable nor a term which is not a numeral. Related to the concept of a reduction step is that of a ‘reductionprocess’. (Reduziervorschrift). By a reduction process for a sequent is meant a procedure consisting of a terminating sequence of successive reduction steps by which the sequent is brought into a ‘final form’ (Endform), i.e. to a sequent satisfying at least one of the two conditions: (1) that the succedent is a true numerical equation, or (2) that some antecedent formula is a false numerical equation. This defining property of a reduction process is to be understood in the strong sense that, whenever reduction steps of the kinds (al),(pl), (p2) occur in the procedure, the final form will be attained for every decision on the arbitrary choices, by suitably making choices in connection with steps of the kind (yl), (y2), (y3). A consequence of the rule of preference is that whenever for a sequent S there is a reduction process, there is a reduction process for any sequent resulting from S by replacing some or all of the free variables of S by numerals. As an illustration of a reduction process may serve the sequents of the form A + A. First, by applications of (a1)the free variables are removed. In the resulting sequent A* + A*, by applications of (p,), (p2) and (a2) the succedent is brought into either the form of an equation r = s where r and s are numerals, or into the form C, each term in C being a numeral. In the first case the equation r = s is a numerical equation. If this equation is true (i.e. of the form m = m) then a final form has been reached. Otherwise we have a sequent A* + m = n with m = n false. Now we can apply the reduction steps (a2),(yl), (yz) to the antecedent A*; and this can be done in an exactly parallel fashion to the reduction steps (a2),(PI), (p,) previously applied to the succedent A*. In this way the antecedent turns into an equation m = n, so that we get a false numerical equation in the antecedent, and so the sequent is again brought into a final form. In the second case, where we have, after the first reduction steps, the sequent A* + C, we get, by applying (p3), the sequent C, A* + 1 = 2.
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Applying here again to the antecedent formula A* the reduction steps (yl), (y,), (a2)in an exactly parallel fashion to the steps (pl), (p2), (a,) previously applied to the succedent A*, we obtain the sequent C, C + 1 = 2, then by applying (y3) we obtain C -+ C. Here the formula C contains at least one logical symbol less than A*. Proceeding with the sequent C + C in the same way as we did before with A* -+ A*, we either obtain a final form or we get a sequent D -+ D, where D contains fewer logical symbols than C. Thus, continuing in the same way, we come, if we have not already obtained a final form, to a sequent Q -+ Q, where Q is a numerical equation m = n. The sequent m = n -+ m = n has in any case a final form. It is to be observed that the course of the reduction following the given process depends on the choices to be made at the reduction steps (a1), (PI), (pz), which are arbitrary. The succession of these choices constitutes a free choice sequence in Brouwer’s sense*. And the given reduction process consists in assigning to any such free choice sequence and to any sequent A + A a succession of sequents, which in a finitist way can be seen to end with a sequent in final form. Using the method of the reduction process for sequents A + A, and also that process itself, reduction processes can be given for each of the logical initial sequents. The arithmetical initial sequents can be brought into final form by reduction steps (al),(a,). If it can be shown that for every derivable sequent there exists a reduction process, then it follows that the formal system under consideration is consistent. For there is no reduction process for the sequent -+ 1 = 2. Hence, in order to prove consistency, it is sufficient to show for each of the inference rules, that if we have a reduction process for the premise, or in the many-premise case for each of the premises, we get a reduction process for the conclusion. Let us try to state this for the various rules of inference. For the rules (al), (a,), (a4) the statement is trivial. For (a3) the statement follows from the fact that in the reduction steps (yl), (y,), (y3) one is allowed to leave unchanged the antecedent formula to which the step applies. For the schemata of negation and universality introduction the statement is obvious.
v2)
* The choice of a member of a conjunction A & B at a step can be transformed to the choice of a number, by agreeing that the first or the second conjunction member will be chosen, according as the number is odd or even.
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In the schema of induction the conclusion r, A + F(t), by the use of steps (aI),(az)can be turned into a sequent r*,A* --f F*(n) where n is a numeral. By applying (a1),(az)with the same replacements of the variables of r, A, F(l) to the premises, these will be turned into the sequents r*+ F*(l) and F*(k), A* + F*(k+ l), where k is an arbitrary numeral. (Possibly not yet all permitted applications of (a2)are made here.) From the assumption that a reduction process exists for the sequents r -+ F(l) and F(a), A + F(a+ l), it follows that we have also reduction processes for r*-+ F*(l) and also for F*(k), A* + F*(k+ l), where k is any numeral. (It is assumed that the variable a does not occur in r, A, F(I), F(t).) If now n is 1, then the sequent r*,A* + F(n) is either identical with r*+ F(l) or obtainable from it by rule (az). If n is the successor of m, then r*,A* + F*(n) is obtainable from r*-+ F*(l) and the m sequents F*(k), A* + F*(k+ I), wherein k is successively 1,2, . . ., m by applications of cut, together with rule (a3). Thus the only thing still to be shown is that also cut has the property that if for both premises -+ A and A, A + B we have a reduction process, we can get from them a reduction process for the conclusion r, A + B. To show this, we begin with a remark about free variables. A reduction of the sequent r, A + B must begin by replacing the free variables by numerals. By such replacements (arbitrarily chosen) r, A, B becomes r, A, B, and by making these replacements for those free variables in A which occur in the sequent r, A --t B, and adding replacementsfor the other free variables (if any) in A, the formula A becomes A. Since by assumption we have reduction processes for the sequents r + A and A, A + B, we also have such processes for the sequents P -+ A and A, 2 -+ 6 resulting from the former by reduction steps (a1).The proof will be complete if from these reduction processes we can obtain a process for the sequent F, -+ B. There is no loss of generality in assuming from the beginning that the sequents r + A and A, A -+ B (therefore also r, A + B) contain no free variables. Likewise there is no loss of generality in assuming that all terms in the sequents are computed so that no other terms occur than numerals. First consider the case in which the formula A contains no logical symbols and hence is a numerical equation. Notice two things: 1. From a reduction process for a sequent S we can immediately obtain a reduction process for any sequent obtained from S by adding other formulas to the antecedent. 2. From a reduction process for a sequent S which has a true numerical -
a
-
I
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equation Q in the antecedent, we can immediately get a reduction process for the sequent obtained from S by deleting the formula Q from the antecedent. If now in the sequent A, A -+ B for which we have a reduction process, A is a true numerical equation, it follows from the second remark that we can also obtain a process for the sequent A -+ B, and, by the first remark, also for the sequent r, A -+ B. If A is a false numerical equation, then by applying reduction steps (PI), (p2), (p3) to the sequent r, A -+ B we come either to a final form; or to a sequent r, A -+ Q, where Q is a false numerical equation; or to a sequent C, r, A -+ 1 = 2. By hypothesis we have a reduction process for -+ A; and by the first remark above also for r, A -+ A and C, r, A -+ A. But since Q and 1 = 2, like A, are false numerical equations, the process for r, A -+ A is also a process for r, A + Q, and the process for C, r, A -+ A is also a process for C, r, A -+ 1 = 2. Hence, if A is a numerical equation, we can get from reduction processes for the sequents r -+ A, A, A -+ B a reduction process for r, A -+ B. In order to prove the same thing for a cut with an arbitrary A (supposing only that in the sequents of the cut all terms are numerals), we proceed by reducing successively the number of logical symbols in the formula A, which Gentzen calls the mix-formula (Mischformel) of the cut. So we have to show that we have a method for getting from reduction processes for the premises of a cut schema (with no other terms than numerals) a reduction process for the conclusion, provided that we know already how to do this for any cut schema whose mix-formula has fewer logical symbols than the one of the cut schema under consideration. This proviso we shall refer as to 'the inductive hypothesis'. We begin by paralleling the reduction of the sequent A, A -+ B. As long as the reduction steps do not involve the antecedent formula A, each such step can likewise be applied to the sequent I?, A + B. If by these steps the sequent A, A -+ B is reduced to a final form, the same holds for r, A + B. Otherwise there is in the reduction of A, A -+ B a first step which involves the mix-formula A. This step must be of one of the kinds (yl), (y2), (y3), and the sequent to which it applies has the form A, A* -+ Q, where Q is a false numerical equation. The mix-formula A has one of the forms C. (Vx)F(x), C & D, In case A is a formula (Vx)F(x), the reduction step under consideration is of the kind (yl) and it yields either a sequent F(k), A* -+ Q, where k is a numeral, or a sequent A, F(k), A* -+ Q. If the numeral k in F(k)
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occurs as a part of a term which contains a function symbol, we have to apply one or more steps (a2), by which F(k) becomes a formula F,, all terms of which are numerals. Hence in the reduction of A, A -+ B we come either to F,, A* -+ Q (Case 1) or to A, F,,A* -+ Q (Case 2), whereas in the reduction of I?, A -+ B we have at this stage arrived at r, A* -+ Q. In case 1 we are almost at our goal. For in the reduction of the sequent J? + A the first steps can be chosen so as to turn it into r + F, . And the reduction process for r -+ A gives a process for r -+ F,. Likewise the process for A, A -+ B contains a process for F, , A* -+ Q. And since in the formula F, the number of logical symbols is smaller than in A, we can apply the inductive hypothesis according to which we know already how to obtain from the reduction process for r -+ F, and F, ,A* -+ Q a reduction process for r, A* -+ Q which, combined with the reduction steps already used to get from r, A + B to r, A* -+ Q, yields a reduction process for the sequent r, A -,B. In case 2 we have a reduction process for the sequent A, F,, A* -+ Q, and the reduction of r, A -+ B has again arrived at the sequent r, A* -+ Q, Let us now provisionally assume that from the reduction process for r -+ A and A, F,, A* -+ Q we have obtained a process for the sequent r, F,, A* -+ Q which results from the two of them by cut: then we have likewise a process for F, ,r, A* -+ Q; and from this together with the process for r -+ F, (already used in case 1) we obtain according to the inductive hypothesis a process for r, r, A* -+ Q, and hence also for r, A* -+ Q. which again together with the performed reduction of r, A -+ B to r, A* -+ Q yields a reduction process for r, A -+ B. So in case 2 our goal is attained upon the provisional assumption. Before discussing this assumption, let us consider the other two possible forms of the mix-formula A. If A is a conjunction C & D, almost nothing is changed, except the formula F, is replaced by one of the conjunction C, the reduction step to be applied to members. When A is a formula A, A* -+ Q makes this sequence either into A* -+ C (case 1) or into A, A* -+ C (case 2). And in the reduction of the sequent r + A the first step makes it into C, r -+ 1 = 2. Now in the first case, since C has fewer logical symbols than A, from the reduction process for the sequents A* + C and C, r -+ 1 = 2 we obtain by the inductive hypothesis a process for the sequent A*, r -+ 1 = 2, and hence also a process for r, A* -+ Q, which, together with the reduction steps leading from r, A -+ B to r, A* -+ Q yields a process for r, A B. -+
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In case 2 the provisional assumption is that from the reduction processes for r + A and A, A* + C we get a process for the sequent r, A* + C (which results from the two of them by the cut schema). This process together with the one for C, r + 1 = 2, yields, according to the inductive hypothesis, a process for r, A*, r + 1 = 2, and hence for r, A* + Q, and so again for r, A + B. Now it remains to consider the provisional assumption which, for all forms of the mix-formula syas that from the reduction processes we have for r -+ A and for a sequent A, A, + B,, we can get a process for r, A, -+ B, . It might first seem that nothing has been gained by replacing, in the assertion to be proved, the sequent A, A -+ B by A, A1 -+ B,. But the last sequent occurs in the reduction of A, A -+ B, and its reduction is the part of the reduction of A, A -+ B which follows the steps leading from A, A -+ B to A, A, -+ B,. Thus the replacement means progressing in the reduction of A, A -+ B, and therefore it can take place only a limited number of times. This argument concludes the consistency proof. Concerning this proof of Gentzen’s one might ask in what respect it transgresses the methods formalizable in the formal system under consideration, as must be the case by the Godel incompleteness theorem. Gentzen himself gave the answer by stating that it is in the concept of a reduction process that the transgression comes about. Indeed this concept involves universal quantiflcation over free choice sequences, and this quantification occurs not only in assertions, but also in hypotheses. The concept of a reduction process is also introduced in Gentzen’s Annalen paper, but there it is not properly used; it is rather replaced by the more elementary concept of a reduction step applied to a derivation (‘Reduktionsschritt an einer Herleitung’). Whereas in the concept of a reduction process the requirement of terminating after finitely many steps is involved, in the Annalen paper the ending of the procedure is proved, as you know, by assigning ordinals below e0 to the derivations, next showing that every reduction step on a delivation lowers its ordinal, and finally proving ordinal induction up to eo.
PAUL BERNAYS The first published Gentzen consistency proof for the formal system of first order number theory, including standard logic, the Peano axioms and recursive definitions, was given in Gentzen’s paper ‘Die Widerspruchsfreiheit der reinen Zahlentheorie’ (Math. Ann. 112 (1936)). It was however not his original proof but a revised version of it. The revision was motivated by a criticism, in which I myself for some time concurred of the original proof on the grounds that it implicitly included an application of the fan theorem. Gentzen did not expressly oppose this opinion; he took care of the criticism by modifying his consistency proof before it was published. Fortunately the text of the original proof is preserved in galley proof*. On rereading the original text, I came to doubt the mentioned opiniont. It seems worth while reconsidering the original proof, because 1. it is certainly easier to follow than the first published proof and at least as easy to follow as the second Gentzen consistency proof, 2. it does not require the generalized form of induction (ordinal induction up to E 0 ) . I shall give in this paper a description of Gentzen’s original proof in a way sufficiently detailed to make apparent the kind of constructive methods here used. At the same time I shall make a slight simplification in this proof, making use of some remarks in Gentzen’s Annalen proof. I begin by describing the formal system to be considered. The logical calculus used is the calculus of sequents’ r + A where r, the antecedent, is a finite (possibly empty) list of formulas separated by commas, and Since Prof. Bernays was unfortunately unable to be present, this paper was read by William Howard. - Eds. An Englishtranslationofit will soon appear in an edition of Gentzen’s works in English by Manfred Szabo. t Georg Kreisel had previously stated in conversation that he dissented from this opinion. t ‘Sequent’as a technical term should be distinguished from ‘sequence’ in its usual sense. 409 +
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where A, the succedent, is a single formula. The logical symbols v (or), 3 (existence) and 3 (implication) are eliminated, and only & (and), --, (not), V (universality) are used. As Gentzen admits arithmetical function symbols (with the restriction that for numerical arguments the value of the function must be computable), there is no loss of generality in assuming that all prime formulas are equations between terms. (Terms are built up in the usual way from free number variables*, numerals and function symbols.) The rules of the calculus, after the mentioned elimination, are the following: Initial sequents are: (1). Logical initial sequents, i.e. sequents of one of the forms A & B +A,
A & B + B,
A,TA+1=2,
A, B + A & B ,
TTA+A,
(Vx)F(x)
--f
F(t),
where x is a bound variable, and t a term. (2). Arithmetic initial sequents, i.e. sequents whose formulas are equations and which have the property that by replacing each free variable by a numeral (of course equal variables by equal numerals) and by computing the function values, either the succedent formula gets the truth value ‘true’ or one of the antecedent formulas gets the truth value ‘false’, according to the usual valuation of numerical equations. The rules of inference are: (a). Rules of structural change in a sequent, permitting one (al). to interchange the order of the formulas of the antecedent, (a2). to add an arbitrary formula to the antecedent, (a3). to delete a repetition of a formula in the antecedent, (a4). to change a bound variable of a universal quantifier, everywhere in its scope, into another bound variable. (b). Logical inference schemata +A
A, A + B r,A+B
*
(cut)+
Gentzen makes a notational distinction between free and bound variables. + This schema does not occur in the original text as a fundamental schema, but it is derivable from the schemata for negation there stated, which Gentren later changed in 8 14 of the Annalen paper.
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T,A-+1=2 r + i A
r
r -+ F(a) -+
(c). Induction
r
-+
(vX)F(X)
(negation introduction) (where a is a free variable not occurring in any formula of r nor in F(x), and where x is a bound variable) (universality introduction)
F(l) F(a), A + F(a+l) r, A -+ F(t)
where t is a term and a a free variable not occurring in or F(t).
r, A,
F(l)
In this calculus from any two formulas A, A we can derive, using the initial sequent A, A -P 1 = 2 and cut, the sequent + 1 = 2; hence, in order to prove consistency of the considered formal system, it is sufficient to show that the sequent -+ 1 = 2 is not derivable. A concept of ‘reduction’ of a sequent is introduced. The following are the possible ‘reduction steps’ on a sequent: ( ~ 1 ~ ) . Replacing a free variable, wherever it occurs in the sequent, by the the same numeral, which can be arbitrarily chosen. ( ~ 1 ~ ) . Replacing a function symbol all of whose arguments are constants by its value. (PI). When the succedent has the form (Vx)F(x), replacing it by F(k), where k is an arbitrarily chosen numeral. (&). When the succedent has the form A & B, replacing it by A or by B, according to an arbitrary choice. (P3). When the sequent has the form r -, A, replacing it by ~,r-+i=2. (y). When the succedent is a false numerical equation: (yl). Replacing an antecedent formula (Vx)F(x) by F(k), or adding F(k) to it in the antecedent, where k is a numeral. (y2). Replacing an antecedent formula A & B by one of the formulas A, B or adding one of these in the antecedent. (y3). If an antecedent formula A occurs, replacing the succedent A in the formula by A and possibly cancelling the formula antecedent.
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The description of these reduction steps must be completed by the following rule of preference: The step (a1)has preference over all other reduction steps, and (a1),(a2)have preference over the other reduction steps. That means: the steps ( a 2 )are admitted only for a sequent which contains no free variable, and the steps (pl), (p2), (p3), (yl), (y2), (y3) are admitted only for sequents which contain neither a free variable nor a term which is not a numeral. Related to the concept of a reduction step is that of a ‘reductionprocess’. (Reduziervorschrift). By a reduction process for a sequent is meant a procedure consisting of a terminating sequence of successive reduction steps by which the sequent is brought into a ‘final form’ (Endform), i.e. to a sequent satisfying at least one of the two conditions: (1) that the succedent is a true numerical equation, or (2) that some antecedent formula is a false numerical equation. This defining property of a reduction process is to be understood in the strong sense that, whenever reduction steps of the kinds (al),(pl), (p2) occur in the procedure, the final form will be attained for every decision on the arbitrary choices, by suitably making choices in connection with steps of the kind (yl), (y2), (y3). A consequence of the rule of preference is that whenever for a sequent S there is a reduction process, there is a reduction process for any sequent resulting from S by replacing some or all of the free variables of S by numerals. As an illustration of a reduction process may serve the sequents of the form A + A. First, by applications of (a1)the free variables are removed. In the resulting sequent A* + A*, by applications of (p,), (p2) and (a2) the succedent is brought into either the form of an equation r = s where r and s are numerals, or into the form C, each term in C being a numeral. In the first case the equation r = s is a numerical equation. If this equation is true (i.e. of the form m = m) then a final form has been reached. Otherwise we have a sequent A* + m = n with m = n false. Now we can apply the reduction steps (a2),(yl), (yz) to the antecedent A*; and this can be done in an exactly parallel fashion to the reduction steps (a2),(PI), (p,) previously applied to the succedent A*. In this way the antecedent turns into an equation m = n, so that we get a false numerical equation in the antecedent, and so the sequent is again brought into a final form. In the second case, where we have, after the first reduction steps, the sequent A* + C, we get, by applying (p3), the sequent C, A* + 1 = 2.
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Applying here again to the antecedent formula A* the reduction steps (yl), (y,), (a2)in an exactly parallel fashion to the steps (pl), (p2), (a,) previously applied to the succedent A*, we obtain the sequent C, C + 1 = 2, then by applying (y3) we obtain C -+ C. Here the formula C contains at least one logical symbol less than A*. Proceeding with the sequent C + C in the same way as we did before with A* -+ A*, we either obtain a final form or we get a sequent D -+ D, where D contains fewer logical symbols than C. Thus, continuing in the same way, we come, if we have not already obtained a final form, to a sequent Q -+ Q, where Q is a numerical equation m = n. The sequent m = n -+ m = n has in any case a final form. It is to be observed that the course of the reduction following the given process depends on the choices to be made at the reduction steps (a1), (PI), (pz), which are arbitrary. The succession of these choices constitutes a free choice sequence in Brouwer’s sense*. And the given reduction process consists in assigning to any such free choice sequence and to any sequent A + A a succession of sequents, which in a finitist way can be seen to end with a sequent in final form. Using the method of the reduction process for sequents A + A, and also that process itself, reduction processes can be given for each of the logical initial sequents. The arithmetical initial sequents can be brought into final form by reduction steps (al),(a,). If it can be shown that for every derivable sequent there exists a reduction process, then it follows that the formal system under consideration is consistent. For there is no reduction process for the sequent -+ 1 = 2. Hence, in order to prove consistency, it is sufficient to show for each of the inference rules, that if we have a reduction process for the premise, or in the many-premise case for each of the premises, we get a reduction process for the conclusion. Let us try to state this for the various rules of inference. For the rules (al), (a,), (a4) the statement is trivial. For (a3) the statement follows from the fact that in the reduction steps (yl), (y,), (y3) one is allowed to leave unchanged the antecedent formula to which the step applies. For the schemata of negation and universality introduction the statement is obvious.
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* The choice of a member of a conjunction A & B at a step can be transformed to the choice of a number, by agreeing that the first or the second conjunction member will be chosen, according as the number is odd or even.
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In the schema of induction the conclusion r, A + F(t), by the use of steps (aI),(az)can be turned into a sequent r*,A* --f F*(n) where n is a numeral. By applying (a1),(az)with the same replacements of the variables of r, A, F(l) to the premises, these will be turned into the sequents r*+ F*(l) and F*(k), A* + F*(k+ l), where k is an arbitrary numeral. (Possibly not yet all permitted applications of (a2)are made here.) From the assumption that a reduction process exists for the sequents r -+ F(l) and F(a), A + F(a+ l), it follows that we have also reduction processes for r*-+ F*(l) and also for F*(k), A* + F*(k+ l), where k is any numeral. (It is assumed that the variable a does not occur in r, A, F(I), F(t).) If now n is 1, then the sequent r*,A* + F(n) is either identical with r*+ F(l) or obtainable from it by rule (az). If n is the successor of m, then r*,A* + F*(n) is obtainable from r*-+ F*(l) and the m sequents F*(k), A* + F*(k+ I), wherein k is successively 1,2, . . ., m by applications of cut, together with rule (a3). Thus the only thing still to be shown is that also cut has the property that if for both premises -+ A and A, A + B we have a reduction process, we can get from them a reduction process for the conclusion r, A + B. To show this, we begin with a remark about free variables. A reduction of the sequent r, A + B must begin by replacing the free variables by numerals. By such replacements (arbitrarily chosen) r, A, B becomes r, A, B, and by making these replacements for those free variables in A which occur in the sequent r, A --t B, and adding replacementsfor the other free variables (if any) in A, the formula A becomes A. Since by assumption we have reduction processes for the sequents r + A and A, A + B, we also have such processes for the sequents P -+ A and A, 2 -+ 6 resulting from the former by reduction steps (a1).The proof will be complete if from these reduction processes we can obtain a process for the sequent F, -+ B. There is no loss of generality in assuming from the beginning that the sequents r + A and A, A -+ B (therefore also r, A + B) contain no free variables. Likewise there is no loss of generality in assuming that all terms in the sequents are computed so that no other terms occur than numerals. First consider the case in which the formula A contains no logical symbols and hence is a numerical equation. Notice two things: 1. From a reduction process for a sequent S we can immediately obtain a reduction process for any sequent obtained from S by adding other formulas to the antecedent. 2. From a reduction process for a sequent S which has a true numerical -
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equation Q in the antecedent, we can immediately get a reduction process for the sequent obtained from S by deleting the formula Q from the antecedent. If now in the sequent A, A -+ B for which we have a reduction process, A is a true numerical equation, it follows from the second remark that we can also obtain a process for the sequent A -+ B, and, by the first remark, also for the sequent r, A -+ B. If A is a false numerical equation, then by applying reduction steps (PI), (p2), (p3) to the sequent r, A -+ B we come either to a final form; or to a sequent r, A -+ Q, where Q is a false numerical equation; or to a sequent C, r, A -+ 1 = 2. By hypothesis we have a reduction process for -+ A; and by the first remark above also for r, A -+ A and C, r, A -+ A. But since Q and 1 = 2, like A, are false numerical equations, the process for r, A -+ A is also a process for r, A + Q, and the process for C, r, A -+ A is also a process for C, r, A -+ 1 = 2. Hence, if A is a numerical equation, we can get from reduction processes for the sequents r -+ A, A, A -+ B a reduction process for r, A -+ B. In order to prove the same thing for a cut with an arbitrary A (supposing only that in the sequents of the cut all terms are numerals), we proceed by reducing successively the number of logical symbols in the formula A, which Gentzen calls the mix-formula (Mischformel) of the cut. So we have to show that we have a method for getting from reduction processes for the premises of a cut schema (with no other terms than numerals) a reduction process for the conclusion, provided that we know already how to do this for any cut schema whose mix-formula has fewer logical symbols than the one of the cut schema under consideration. This proviso we shall refer as to 'the inductive hypothesis'. We begin by paralleling the reduction of the sequent A, A -+ B. As long as the reduction steps do not involve the antecedent formula A, each such step can likewise be applied to the sequent I?, A + B. If by these steps the sequent A, A -+ B is reduced to a final form, the same holds for r, A + B. Otherwise there is in the reduction of A, A -+ B a first step which involves the mix-formula A. This step must be of one of the kinds (yl), (y2), (y3), and the sequent to which it applies has the form A, A* -+ Q, where Q is a false numerical equation. The mix-formula A has one of the forms C. (Vx)F(x), C & D, In case A is a formula (Vx)F(x), the reduction step under consideration is of the kind (yl) and it yields either a sequent F(k), A* -+ Q, where k is a numeral, or a sequent A, F(k), A* -+ Q. If the numeral k in F(k)
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occurs as a part of a term which contains a function symbol, we have to apply one or more steps (a2), by which F(k) becomes a formula F,, all terms of which are numerals. Hence in the reduction of A, A -+ B we come either to F,, A* -+ Q (Case 1) or to A, F,,A* -+ Q (Case 2), whereas in the reduction of I?, A -+ B we have at this stage arrived at r, A* -+ Q. In case 1 we are almost at our goal. For in the reduction of the sequent J? + A the first steps can be chosen so as to turn it into r + F, . And the reduction process for r -+ A gives a process for r -+ F,. Likewise the process for A, A -+ B contains a process for F, , A* -+ Q. And since in the formula F, the number of logical symbols is smaller than in A, we can apply the inductive hypothesis according to which we know already how to obtain from the reduction process for r -+ F, and F, ,A* -+ Q a reduction process for r, A* -+ Q which, combined with the reduction steps already used to get from r, A + B to r, A* -+ Q, yields a reduction process for the sequent r, A -,B. In case 2 we have a reduction process for the sequent A, F,, A* -+ Q, and the reduction of r, A -+ B has again arrived at the sequent r, A* -+ Q, Let us now provisionally assume that from the reduction process for r -+ A and A, F,, A* -+ Q we have obtained a process for the sequent r, F,, A* -+ Q which results from the two of them by cut: then we have likewise a process for F, ,r, A* -+ Q; and from this together with the process for r -+ F, (already used in case 1) we obtain according to the inductive hypothesis a process for r, r, A* -+ Q, and hence also for r, A* -+ Q. which again together with the performed reduction of r, A -+ B to r, A* -+ Q yields a reduction process for r, A -+ B. So in case 2 our goal is attained upon the provisional assumption. Before discussing this assumption, let us consider the other two possible forms of the mix-formula A. If A is a conjunction C & D, almost nothing is changed, except the formula F, is replaced by one of the conjunction C, the reduction step to be applied to members. When A is a formula A, A* -+ Q makes this sequence either into A* -+ C (case 1) or into A, A* -+ C (case 2). And in the reduction of the sequent r + A the first step makes it into C, r -+ 1 = 2. Now in the first case, since C has fewer logical symbols than A, from the reduction process for the sequents A* + C and C, r -+ 1 = 2 we obtain by the inductive hypothesis a process for the sequent A*, r -+ 1 = 2, and hence also a process for r, A* -+ Q, which, together with the reduction steps leading from r, A -+ B to r, A* -+ Q yields a process for r, A B. -+
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In case 2 the provisional assumption is that from the reduction processes for r + A and A, A* + C we get a process for the sequent r, A* + C (which results from the two of them by the cut schema). This process together with the one for C, r + 1 = 2, yields, according to the inductive hypothesis, a process for r, A*, r + 1 = 2, and hence for r, A* + Q, and so again for r, A + B. Now it remains to consider the provisional assumption which, for all forms of the mix-formula syas that from the reduction processes we have for r -+ A and for a sequent A, A, + B,, we can get a process for r, A, -+ B, . It might first seem that nothing has been gained by replacing, in the assertion to be proved, the sequent A, A -+ B by A, A1 -+ B,. But the last sequent occurs in the reduction of A, A -+ B, and its reduction is the part of the reduction of A, A -+ B which follows the steps leading from A, A -+ B to A, A, -+ B,. Thus the replacement means progressing in the reduction of A, A -+ B, and therefore it can take place only a limited number of times. This argument concludes the consistency proof. Concerning this proof of Gentzen’s one might ask in what respect it transgresses the methods formalizable in the formal system under consideration, as must be the case by the Godel incompleteness theorem. Gentzen himself gave the answer by stating that it is in the concept of a reduction process that the transgression comes about. Indeed this concept involves universal quantiflcation over free choice sequences, and this quantification occurs not only in assertions, but also in hypotheses. The concept of a reduction process is also introduced in Gentzen’s Annalen paper, but there it is not properly used; it is rather replaced by the more elementary concept of a reduction step applied to a derivation (‘Reduktionsschritt an einer Herleitung’). Whereas in the concept of a reduction process the requirement of terminating after finitely many steps is involved, in the Annalen paper the ending of the procedure is proved, as you know, by assigning ordinals below e0 to the derivations, next showing that every reduction step on a delivation lowers its ordinal, and finally proving ordinal induction up to eo.