On the Formal Theory of the Ordinal Diagrams

Transcription

1 On the Formal Theory of the Ordinal Diagrams By GAISI TAKEUTI Department of Mathematics, Tokyo University of Education, Tokyo In the former paper [4], the author developed the theory of ordinal diagrams and by the use of transfinite induction for the system of ordinal diagrams he proved the consistency of the logical system called RNN in the former paper [5]. RNN is obtained from GLC (See [6]) modifying it as follows: 1. Every beginning sequence of RNN is of the form D D or of the form a= b) A(a) A(b) or the "mathematische Grundsequenz" in Gentzen [2]. 2. The following inference-schema called 'induction' is added: where a is contained neither in A(O), nor i C nor in ƒ, and t is an arbitrary term. A(a) and A(a+1) are called the chief formulas of this induction. 3. The inference _??_ left on an f-variable is restricted by the condition that _??_ƒõf(ƒõ) is regular (See [5] for ' regular'). In this paper, we shall show that the formal theory of the ordinal diagrams can be developed in RNN. From this, we see easily that the formal theory of the ordinal diagrams is consistent

2 2 T he similar treatment on a formal theory of Ackermann's ordinal numbers (See Ackermann [1].) was carried out by Kino [3]. 1. The beginning sequences in RNN The "mathematische Grundsequenzen " used in this paper are those on the concepts a=b, a<b, a b, 0, 1, a+b, a-b, a Eb, ab and the follow ing ones: where O(n; a) is the formal expression of the concept ` a is an ordinal diagram of order n' in [4], where a denotes the ordinal diagram in [4] expressed by a in this paper. Thus in intuitive explanation of notions, the notions in [4] corresponding to those in this paper we cite using wave where 2a means the formal expression of the positive integer 'a' in [4]. Thus _??_ is a where 3i 5r 7a neabs the formal expression of '(i;r,a)'in [4]

3 [4] where 11a 13b means the formal expression of 'a#b' in [4] where o(a) is the formal expression of the minimum number m in [4].such that a is an ordinal diagram of order m where ¼ (i; a, b) is the formal expression of the concept 'a ¼i b' in where = (a, b) is the formal expression of the concept 'a=b' in [4]

4 where 1(a) is the formal expression of the number of the components of a in [4] where comp (i; a) is the formal expression of the i-th component of a in [4] where cor(i; a, b) is the formal expression of 'li' in [4], such that a has k components a1 c c, ak, b has the same number of them b1, c c, bkr, and there exsists a permutation (l1 c c, lk) of (1, c c, ď) satisfying ali= bi where <(i; a, b) is the formal expression of 'a< i_??_' in [4]

5 where cof (i; j; a) is the formal expression of the integer k in [4],.such that the k-th component of d is the i-th in order of magunitude for.the order <i In the following we write simply comp (i; j; a) for comp (cof (i; j; a); a) where cos(i; a, b) is the formal expression of the minimum number m in [4], such that comp(i; m, b) is greater than comp(i; m, a) for the -order<i

6 6 where sect (i; a, b) is the formal expression of the i-section of is being not less than b for the order <i provided that there exists such one where part(i; j, k; a) is the formal expression of the ordinal diagram ' comp(i; j; a)# c c #comp(i; k; a)' in [4] The provable sequences in RNN We see that the following sequences are provable in RNN which follows from induction, and the right side of which is sometimes called the axiom of course-of-values induction

7 which is proved by course-of-values induction on 1(a) and induction on i which follows from which is proved by course-of-values induction on a which is proved by course-of-values induction on a which is proved by course-of-values induction on 1(a) which is proved by course-of-values induction on b which is proved by course-of-values induction on a by use of

8 _??_ƒõ which follows from 2.17, which follows from which is proved by course-of-values induction on 1(a) which is proved by course-of-values induction on b. 3. Accessibility of the ordinal diagrams of order n In this section, let n be a fixed positive integer. Let i be an integer satisfying 0 i n. We now define F(i; n; a) and A(i; n; a) recursively as follows: F(n; n; a) means O(n; a). A(i; n; a) is the abbreviation of (_??_ƒô(f(i; n; ƒô)_??_y(f(i; n; y)?<(i; y, ƒô)?[y])?[ƒô])?[a]). F(i n; a) is the abbreviation of F(i+l; n; a)?_??_ƒô( ¼(i+1; ƒô, a) A(i+1; n; x)), if i is less than n. In the following, we shall prove that several sequences are provable in RNN. Since the inferences other than _??_ left on an f-variable have no restriction, we have only to remark the form of _??_ left on an f-variable used in the proof-figures. Therefore we say briefly, 'A sequence _??_ follows straight from the sequences _??_1, c c,_??_n', if _??_ is provable from _??_1, c c,_??_n without using _??_ left on an f-variable. In the conclusion of the following propositions (1-16), if we prove that a sequence _??_ is pro vable in RNN, then _??_ has no free variable. If we say, 'A sequence _??_ follows straight from the proposition that _??_0 is provable in RNN', then we mean _??_ follows straight from a sequence _??_1, which is obtained from _??_0 by substituting several adequate terms for the free variables in _??_

9 9. In this section, we shall prove that the sequence is provable in RNN. It should be noticed that every inference _??_ left on an f-variable used in the proof-figure to this sequence has the chief-formula of the form A(i; n; t), which is clearly a regular formula In the following, let us use the following abbreviations: is less than n, (Thus F(i; a) and A(i; a) are abbreviations for F(i; n; a)and A(i; n; a), respectively.) 3.1. PROPOSITION 1. The following sequence is provable in RNN; PROOF. We have straight

10 10 From this follows straight By _??_ left on an f-variable, from this sequence follows 3.2. PROPOSITION 2. The following sequence is provable in RNN; PROOF. We have straight By _??_ left on an f-variable From this sequence we have straight the follwing sequences in succession; i. e PROPOSITION 3. The following sepuence is provable in RNN --> A(i; 2). PROOF The following sequence is provable in RNN; --> F(i; 2). Proof. From follows straight From this sequence follows straight On the other hand, from 1.12 follows straight - F(n; 2). From above two sequences, using cuts repeatedly, we have the sequence in question for each i, q. e. d

11 11 We now prove From 2. 6 follows straight From this sequence and 3. 2, follows Then by the help of , we have straight the sequence in question q. e. d PROPOSITION 4. The following sequence is provable in RNN;, PROOF. The following sequences follow straight in succession; On the other hand, follows straight From and follows straight Then by _??_ left on an f-variable, From this follows straight the sequence in question, q. e. d PROPOSITION 5. The following sequences are provable in RNN, PROOF. We set the first sequence as _??_1(i) and the second as _??_2(i) First we shall prove _??_1(n). From follows straight a=a(1), = (a, b), O(a) O(b). The following sequences follow straight in succession;

12 12 (by 1.13, 1.43 and 1.44). On the other hand, the following sequences follow straight in succession; From , by course-of-values induction, follows We now prove _??_2(i), provided that_??_1(i) is proved. From 3.4, follows straight Then, by 3.2 follows From this, by using _??_(i) and 3. 1, _??_(i) follows straight We now prove _??_1(i), provided that we have obtained _??_1(i+1) (hence _??_2 (i +1) by 3.5.2). The following sequences follow straight in succession; From the last sequence and _??_1(i+1), follows straight _??_1(i) for inn, q.e.d PROPOSITION 6. The following sequence is provable in RNN; The following sequence is provable in RNN; Proof. For the case i=n, it follows from We now prove

13 13. for i<n, provided that we have proved The following sequences follow straight in succession; The following sequence is provable in RNN; Proof. The sequence in question is 2.7 for i=n. Now we prove the case i<n, provided that the sequence is proved for i+1. From 2.11 follows straight Then, by our assumption we can conclude the proof The following sequence is provable in RNN; Proof. From 2.13, 3.4 and follws straight On the other hand, from 1.55 follows straight From , and follows straight the sequence in question, q.e.d The following sequence is provable in RNN; Proof. For the case i=n, it follows straight from Now we prove it for i<n, provided that F(i+1; l1a E13b) F(i+1; b) holds. From 1.35 and our assumption, it follows straight, q. e. d The following sequence is provable in RNN; Proof. From 3.5 and follows straight

14 14 From this sequence and follows straight the sequence in question, q. e. d. We now prove 3.6. The following sequences follow straight; The following sequences follow straight in succession; By _??_ left on an f-variable, From and 3.1 follows straight On the other hand, from 2.12 and follows straight From , and 2.20 by induction follows straight the sequence in question, q.e.d PROPOSITION 7. The following sequence is provable in RNN; -164-

15 15 PROOF. The following sequences follow straight From 3.7.1,3.7.2 and 2.2 follows straight hence, On the other hand, follows straight From and follows straight From this by course-of-values induction, From this and 3.2 we have straight the sequence in question, q.e.d PROPOSITION 8. The following sequence is provable in RNN; PROOF. The following sequences follow straight; From 3.8.1,3.8.2 and 2.4 follows straight From 1.29 and 1.30 follows straight On the other hand, from 3.1 and the definition of F(i; a,) follows straight

16 16 Using and repeatedly, we have the sequence Then from 3.8.5,3.8.6 and 3.7 follows straight By _??_ left on an f-variable, 3.9. PROPOSITION 9. The following sequence is provable in RNN, where i is an integer satisfying 0 <i<n; PROOF. We have straight each of the following sequences; From ,1.93 and 1.94 follows ssraight From 3.9.1,3.9.6 and 2.4 follows straight On the other hand, from 3.3 and follows straight From 3.9.7, and 3.7 follows straight the sequence in question, q.e.d PROPOSITION 10. The following sequence is provable in RNN; PROOF. From 3.9, using cuts repeatedly, we have straight

17 17 From this and 3.8 follows straight the sequence in question, q.e.d PROPOSITION 11. The following sequence is provable in RNN; PROOF. In the same way as in 3.3.1, follows straight Then in the same way as in the proof of 3.9, we have straight the sequence. Here 3.3, 3.10 and are useful for us PROPOSITION 12. The following sequence is provable in RNN; PROOF. In the same way as in the proof of 3.11, using in place of , follows straight q.e.d. From this sequence and 3.11 follows straight the sequence in question, PROPOSITION 13. The following sequence is provable in RNN; PROOF. From 3.12, by induction follows straight From this sequence and 3.3 follows straight the sequence in question, q.e.d PROPOSITION 14. The following sequence is provable in RNN; PROOF. From 3.4 and 3.13 follows straight From this sequence and 2.22 follows straight the sequence in question, q.e.d PROPOSITION 15. The following sequence is provable in RNN; PROOF The following sequence is provable in RNN;

18 18 Proof. First we prove the casei=n. From 1.17, and 2.22 follows straight then follows Similary, from 1.33 and 2.2 follows straight From , , 1.7, 1.11, and 1.27 follows straight By course-of-values induction, from this sequence follows Now we assume ¼ (i; a, b), F(j+1; b) F(j+1; a) for i<j. We have straight in the similar way as in Using , and our assump tion, we have the sequence in question in the same way as above, q.e.d. In the same way as we had and 3.7.4, we have straight the sequences and , respectively; From these sequences follows straight _??_ƒô (ƒô<b G'(i; ƒô, a, c)) _??_ƒô(ƒô< H'(i;ƒÔ,a,c)) H(i;b,a,c). Hence by course-of-values induction, From , , 1.74, 1.86 and 2.23 follows straight the sequence in question, q.e.d PROPOSITION 16. The following sequence is provable in RNN; PROOF. From 3.4, 3.5 and 1.88 follows straight From 2.11 follows straight -168-

19 From , , 1.55 and 1.84 follows straight From , 1.54, 1.55, 1.74, 1.84, 1.87 and 2.13 follows straight From 1.55, 1.68, 1.69, 1.72, 1.81 and 2.13 follows straight Then from and follows straight On the other hand, from 3.15 follows straight From and follows straight By course-of-values induction follows Then the following sequences follow in succession; by _??_ on an f-variable, Then by 2.3 and follows From this sequence and 2.22, 3.4 follows straight the sequence in question, q.e.d THEOREM.The following sequence is provable in RNN; PROOF.From 3.16 follows straight From this sequence and 3.14 follows straight the sequence

20 20 which is the sequece in question, q. e. d. Refereces [1] W. Ackermann: Konstruktiver Aufbau eines Abschnitts der zweite n Cantorschen Zahlenklasse; Math. Z. 53 (1951), [2] G. Gentzen: bie gegenwartige Lage i. der mathematischen Grund lagenforschung. Neue Fassung des Widerspruchsfreiheitsbeweises fur die reine Zahlentheorie; Leipzig, [3] A. Kino: A consistency-proof of a formal theory of Ackermann's ordi al umbers. Forthcoming in J. Math. Soc. Japan 10 (1958). [4] G. Takeuti: Ordinal diagrams; J. Math. Soc. Japan 9 (1957), [5] -: On the fundame.tal co.jecture of GLC V; J. Math. Soc. Japa n 10 (1958), [6] -: On a generalized logic calculus; Jap. J. Math. 23 (1953), Errata to 'On a generalized logic calculus'; Jap. J. Math. 24 (1954), (Received Feb. 17, 1958) Correctio. of my former paper 'Co.structio. of Ramified Real Numbers' in this Annals 1 (1956), The degree used in 2 is defined by the following modification of 1.7: substitute ƒö2i for ƒöi in and ƒö2i+1 for ƒöi+1 in