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10. Fusion of Several Complete Inductions
10. FUSION OF SEVERAL COMPLETE INDUCTIONS
In the following we intend to show that every mathematical proof in which the rule of complete induction is applied several times, can be so rearranged by means of certain simple fusions of inferences and concepts that only a single application of complete induction occurs in it. For this purpose it is not necessary to specify a particular kind of formalization of mathematical proofs. It shall merely be presupposed that the concept of a ‘mathematical proof‘ encompasses all forms of inference of ‘predicate logic’ and it must, of course, include the rule of complete induction. No further number-theoretical results need to be presupposed except for the admission of the primitive predicate ‘=’ and its associated basic formulae (axiom formulae) n = n, as well as 7(m = n), for arbitrary numerals n and rn, where n and rn are distinct. Further mathematical concepts and their associated axioms may be admitted as desired. Note that the theorem to be proved is significantprimarily for ‘elementary number theory’. If elementary number theory is extended to ‘analysis’, then, as Dedekind has shown, complete induction becomes reducible to other forms of inference and the assertion of the theorem therefore loses its significance. The proof runs as follows: Let there be given a derivation (i.e., a formalized proof) with several occurrences of formalized complete induction. Each application of complete induction is logically equivalent to an application of the ‘induction axiom’ to a certain special proposition about the natural numbers, i.e., it can be expressed formally in such a way that a formula of the following form is asserted to hold:
{W &k vg [W = 8&+ 011 = v9 SY(t))) where gv stands for an arbitrary formula with one argument place for a numeral designating a natural number, and the formula is thus the formal counterpart of a proposition about natural numbers; the subscript v serves
310
FUSION OF SEVERAL COMPLETE INDUCTIONS
to distinguish the different complete inductions occurring in the derivation, i.e., it runs through the numbers 1, 2, . . ., p, where p stands for the total number of complete inductions occurring in the derivation. (F and kl, designate arbitrary bound variables. The formulae Svcan of course also contain free variables.) We shall now deduce all of these p induction axiom formulae by applying a single formalized complete induction that fuses all of these formulae. This is done as follows: We construct the formula
[b = 1
=3
sl(a)] & [b
=
2
&.
&(a)]
. . & [b
=p
= %,,(a)],
briefly referred to as @(a). (Here a and b designate two free variables not yet occurring in the derivation.) By a single formal application of complete induction we obtain the formula:
(Here it makes no difference whether complete induction is to be admitted in the form formalized in this axiom formula or whether any other version, possibly that of an inference figure, is chosen; by means of the latter we would then simply derive the above formula.) From this formula we can now derive all of the p induction axiom formulae cited above along purely logical lines, i.e., without requiring another application of complete induction. Once this has been done, we have reached our goal. In order to prove this derivability, it should suffice to outline the main steps of the formal argument: In order to derive the induction axiom formula for %lfrom that of Q,for example, we would have to begin by substituting 1 for b in the latter formula (this is formally possible by means of successive V-introductions and V-eliminations in the case where no provision has been made for direct substitution as a permissible form of inference in the formalism). @(I), for example, yields
[l = 1
3 &(1)]
& [l = 2
3 &(1)]
&
. .. & [l = p
3
&(l)],
a formula which can be proved to be equivalent to S1(l)by means of propositional logic alone, with the additional appeal to the truth of 1 = 1 and the falsity of 1 = 2,. ., 1 = p. The same holds for @(a)and Sl(a), for an arbitrary a. The entire induction axiom formula, therefore, may be rearranged in such a way, by applications of purely logical formalized forms of inference, that eventually S1replaces every occurrence of Q,as was to be shown.
.
FUSION OF
SEVERAL COMPLETE INDUCTIONS
31 1
It might seem as though all that has been achieved is that one and the same formalized complete induction now occurs in p different places in the derivation, with the result that there are once again p individual, although identical, occurrences of complete induction. This observation, however, does not touch the heart of the matter, for we could fuse these duplicate induction inferences in a trivial way not only in thought, but also in form, so that only a single formalized induction actually occurs. In order to accomplish this we would have to prefix the induction axiom formula for Q with universal quantifiers quantifying all the free variables of Q.We might denote the resulting formula by 3, and the endformula of the derivation by 6.This would yield a purely logical derivation with several formulae of the form 9 as initial formulae (as well as possibly other mathematical axiom formulae as initial formulae; this does not affect our argument); the derivation can then be transformed in the familiar way into a (purely logical) derivation without such initial formulae and with 9 =iQ for its endformula, (‘deduction theorem’ or, in the calculus of sequents, a trivial observation) and this, with the inclusion of a single derivationfor 3, once again yields a derivation for the original endformula Q. Our result shows that the number of complete inductions occurring in a number-theoretical proof is no measure of the ‘complexity’ of the proof in the context of metamathematics; although it does have some bearing on this point, it is not the number of inductions but their ‘degree’, i.e., the complexity of the induction proposition, that counts.
In the following we intend to show that every mathematical proof in which the rule of complete induction is applied several times, can be so rearranged by means of certain simple fusions of inferences and concepts that only a single application of complete induction occurs in it. For this purpose it is not necessary to specify a particular kind of formalization of mathematical proofs. It shall merely be presupposed that the concept of a ‘mathematical proof‘ encompasses all forms of inference of ‘predicate logic’ and it must, of course, include the rule of complete induction. No further number-theoretical results need to be presupposed except for the admission of the primitive predicate ‘=’ and its associated basic formulae (axiom formulae) n = n, as well as 7(m = n), for arbitrary numerals n and rn, where n and rn are distinct. Further mathematical concepts and their associated axioms may be admitted as desired. Note that the theorem to be proved is significantprimarily for ‘elementary number theory’. If elementary number theory is extended to ‘analysis’, then, as Dedekind has shown, complete induction becomes reducible to other forms of inference and the assertion of the theorem therefore loses its significance. The proof runs as follows: Let there be given a derivation (i.e., a formalized proof) with several occurrences of formalized complete induction. Each application of complete induction is logically equivalent to an application of the ‘induction axiom’ to a certain special proposition about the natural numbers, i.e., it can be expressed formally in such a way that a formula of the following form is asserted to hold:
{W &k vg [W = 8&+ 011 = v9 SY(t))) where gv stands for an arbitrary formula with one argument place for a numeral designating a natural number, and the formula is thus the formal counterpart of a proposition about natural numbers; the subscript v serves
310
FUSION OF SEVERAL COMPLETE INDUCTIONS
to distinguish the different complete inductions occurring in the derivation, i.e., it runs through the numbers 1, 2, . . ., p, where p stands for the total number of complete inductions occurring in the derivation. (F and kl, designate arbitrary bound variables. The formulae Svcan of course also contain free variables.) We shall now deduce all of these p induction axiom formulae by applying a single formalized complete induction that fuses all of these formulae. This is done as follows: We construct the formula
[b = 1
=3
sl(a)] & [b
=
2
&.
&(a)]
. . & [b
=p
= %,,(a)],
briefly referred to as @(a). (Here a and b designate two free variables not yet occurring in the derivation.) By a single formal application of complete induction we obtain the formula:
(Here it makes no difference whether complete induction is to be admitted in the form formalized in this axiom formula or whether any other version, possibly that of an inference figure, is chosen; by means of the latter we would then simply derive the above formula.) From this formula we can now derive all of the p induction axiom formulae cited above along purely logical lines, i.e., without requiring another application of complete induction. Once this has been done, we have reached our goal. In order to prove this derivability, it should suffice to outline the main steps of the formal argument: In order to derive the induction axiom formula for %lfrom that of Q,for example, we would have to begin by substituting 1 for b in the latter formula (this is formally possible by means of successive V-introductions and V-eliminations in the case where no provision has been made for direct substitution as a permissible form of inference in the formalism). @(I), for example, yields
[l = 1
3 &(1)]
& [l = 2
3 &(1)]
&
. .. & [l = p
3
&(l)],
a formula which can be proved to be equivalent to S1(l)by means of propositional logic alone, with the additional appeal to the truth of 1 = 1 and the falsity of 1 = 2,. ., 1 = p. The same holds for @(a)and Sl(a), for an arbitrary a. The entire induction axiom formula, therefore, may be rearranged in such a way, by applications of purely logical formalized forms of inference, that eventually S1replaces every occurrence of Q,as was to be shown.
.
FUSION OF
SEVERAL COMPLETE INDUCTIONS
31 1
It might seem as though all that has been achieved is that one and the same formalized complete induction now occurs in p different places in the derivation, with the result that there are once again p individual, although identical, occurrences of complete induction. This observation, however, does not touch the heart of the matter, for we could fuse these duplicate induction inferences in a trivial way not only in thought, but also in form, so that only a single formalized induction actually occurs. In order to accomplish this we would have to prefix the induction axiom formula for Q with universal quantifiers quantifying all the free variables of Q.We might denote the resulting formula by 3, and the endformula of the derivation by 6.This would yield a purely logical derivation with several formulae of the form 9 as initial formulae (as well as possibly other mathematical axiom formulae as initial formulae; this does not affect our argument); the derivation can then be transformed in the familiar way into a (purely logical) derivation without such initial formulae and with 9 =iQ for its endformula, (‘deduction theorem’ or, in the calculus of sequents, a trivial observation) and this, with the inclusion of a single derivationfor 3, once again yields a derivation for the original endformula Q. Our result shows that the number of complete inductions occurring in a number-theoretical proof is no measure of the ‘complexity’ of the proof in the context of metamathematics; although it does have some bearing on this point, it is not the number of inductions but their ‘degree’, i.e., the complexity of the induction proposition, that counts.