Appendix: Bar Induction in the Proof 1 of Termination of Gentzens Reduction 2 Procedure 3 Annika Siders and Jan von Plato 4 1 Introduction 5 We shall give an explicit formulation to the use of bar induction in Gentzen s 6 original proof of consistency, as a continuation of the analysis in the preceding essay 7 about the Hilfssatz, referred to here as HH. 8 The article Bernays (1970) was the first one to explain in print the ideas in 9 Gentzen s original proof of consistency, and it also made clear that the proof 10 was in the end based on bar induction. There is a review of Bernays article by 11 Joseph Shoenfield in which the latter writes that the progress made in formalizing 12 intuitionistic systems in recent years should make it possible to formalize this 13 proof and thus see exactly what intuitionistic principles are needed to carry it out 14 (Mathematical Reviews, MR0276062). 15 2 Bar Induction in the 1935 Proof 16 We prove that derivable sequents reduce to endform. As the basic predicate B in the 17 induction, the property is used that the succedent of a derivable sequent is an atomic 18 formula, here an equation. For the inductive predicate I, we use the property that a 19 derivable sequent with an atomic formula as a succedent reduces to endform. For the 20 proof, we show first that reduction steps in the succedent preserve the derivability 21 of a sequent: 22 Lemma If! C is a derivable sequent and an S-move is applied to it, a derivable 23 sequent is obtained. 24 A. Siders J. von Plato Department of Philosophy, University of Helsinki, Helsinki, Finland e-mail: annika.siders@helsinki.fi; jan.vonplato@helsinki.fi 127
128 A. Siders and J. von Plato We go through the possible S-moves in turn: 25 SVar. If! C has free variables, numbers are chosen at will to instantiate these 26 until there are no free variables left. Derivability is maintained under substitution 27 so that the reduced sequent is derivable. 28 S&. The sequent is! A & B, and both of the reduced sequents! A and 29! B are derivable by rule &E. 30 S:. The sequent is!:a. The following derivation by the rules of the calculus 31 NLK shows that A;!0 D 1 is derivable, with Wk, Ref,andDN standing for the 32 rules of weakening, refutation, and elimination of double negation, respectively: 33 A! A!:A : 0 D 1; A! A Wk : 0 D 1;!:A Wk Ref A;!::0 D 1 A;! 0 D 1 DN S8. The sequent is!8xa.x/, and any instance! A.n/ is derivable by rule 34 8E. QED. 35 Theorem Derivable sequents reduce to endform. 36 For a proof, we go through the four conditions for bar induction: 37 1. B has to be decidable. This is the case. 38 2. For any given derivable sequent! C and any sequence of reduction steps, 39 there is a step in the sequence by which the succedent formula has turned into an 40 equality. To show this, consider the reductions steps: If there are free variables 41 in! C, move Svar must be applied first, to substitute them by constants. 42 Thereafter the other S-moves must be applied, each producing a shorter formula 43 in the succedent until it is an equation. 44 3. Given a derivable sequent such that each applicable reduction step produces a 45 sequent that reduces to endform, to show that the sequent before the reduction 46 reduces to endform. This is immediate. 47 4. Finally, it has to be shown that if a derivable sequent has been reduced so that it 48 has the property B, i.e., is of the form, it is a derivable sequent that 49 reduces to endform. The derivability part follows by the lemma. The rest is an 50 induction on the last rule in the derivation of. Ifm D n is true, the 51 sequent is in endform. Therefore we may assume m D n to be false. 52 The possible cases are: 53 4.1. is an initial sequent. Then the antecedent is the false equation 54 m D n and the sequent in endform. 55 4.2. is a mathematical groundsequent, for which we take the 56 formulation with free parameters, as in HH, Section IV.4, with all free 57 variables removed by steps of Svar: 58! m D m; n D m! m D n; m D k; k D n! m D n; 59 k C 1 D k! 0 D 1;! h C k D k C h;!.h C k/c l D h C.k C l/: 60
Appendix: Bar Induction in the Proof of Termination of Gentzens Reduction Procedure 129 The reflexivity groundsequent is in endform and symmetry has a false 61 antecedent n D m whenever the succedent m D n is false. With transitivity, 62 if m D n is false, if m D k in the antecedent is true, then k D n in the 63 antecedent is false and similarly if k D n is true. With k C 1 D k! 0 D 1, 64 the antecedent is false, and for the rest, the succedent is true. 65 4.3. The last rule is a logical one. There are the cases &E;8E,andDN. 66 4.3.1. The last rule is &E: 67! A & m D n The premiss reduces to endform by assumption, and therefore also the 68 conclusion. The reductionis similar if the secondformofrule &E is applied. 69 4.3.2. The last rule is 8E: 70!8x:x D n The premiss reduces to endform by assumption, and therefore also the 71 conclusion. The reduction is similar if the right member of the equation was 72 quantified. 73 4.3.3. The last rule is DN: 74!::m D n The first step of reduction for the premiss gives : m D n;! 0 D 1.Ifstep 75 A: is applied to : m D n, the reduced sequent is : m D n; 76 with a false equation in the succedent. Therefore some other reduction step 77 must be applied, and if A: is applied at some later stage to : m D n, a 78 similar useless loop is produced. Therefore : m D n in the antecedent can 79 be left intact and reduces to endform by the same steps as : m D 80 n;! 0 D 1. 81 4.4. The last rule is CI with 0 ; 00 and the conclusion 0 ; 00! m D n: 82 &E 8E DN 0! m D 0 m D x; 00! m D x C 1 0 ; 00! m D n If m D 0 is false, the conclusion reduces to endform by the same steps as 83 0! m D 0.Ifm D 0 is true, Svar gives in particular for the second premiss 84 the reducible sequent m D 0; 00! m D 0 C 1 with a false succedent. The 85 steps of reduction leave the true equation m D 0 intact and apply as well for 86 the reduction of 0 ; 00! m D n. 87 By 1 4, the conditions for bar induction are satisfied and all derivable 88 sequents have the property I, i.e., reduce to endform. QED. 89 CI
130 A. Siders and J. von Plato References 90 AQ1 Bernays, P. (1970) On the original Gentzen consistency proof for number theory. In J. Myhill et al., 91 eds, Intuitionism and Proof Theory, pp. 409 417, North-Holland. 92 Shoenfield, J. (1972) Review of Bernays (1970). Mathematical Reviews, MR0276062. 93
AUTHOR QUERY AQ1. Please note that reference Shoenfield (1972) has not been cited in the text but present in the list. Please provide in-text citation for this reference or delete this from the reference list.