Given a derivation in HA, it can be converted into normal form

Proof. The proof is an extension of the normalization procedure of Sec t ion 13.2 b elow. We note first that the conclusions of all ar ithmetical rules except Ind are never major premisses in elimination rules, because they are atomic formulas. Therefore the only new case to consider in nor-malization is when such a major premiss has been derived by rule Ind. It is readily seen that the E -rule permutes up to the derivation of the third

major premiss of rule Ind. QED.

Normal derivations begin with instances of the arithmetical rules, followed by logical rules and Ind. If in the rule of induction the arbitrary term t in the discharged assumption A(t) is a number instead of a variable (case of ‘numerical induction’), the instance of induction can be eliminated by repeated application of a conversion. The given derivation is:

.... A(0)

1

A(y),  ....

A(s y)

A(t), 1

....

C

C ^Ind,1

For simplicity, we assume the discharges simple. If t= 0, the conversion is into:

.... A(0), 

....

C

Ele me nts of the pro of theor y of ar ithme t ic 185

The der iv ation of A(s n) from the assumption A(n ) is obtained by the sub-stitution [n/ x ] in the der ivation of the second premiss. A rep etition of the conversion w ill e ventually produce a tower of composed der ivations that beg ins w ith the der ivation of A(0) and continues to A(1), . . . , A(n ), A(s n), w ith the induc tion eliminated. Note that op en assumptions in  get mul-tiplied, possibly zero t imes, in the conversion process.

Nor mal der ivabilit y in He y t ing ar ithmetic has the same definition as in pure log ic, namely that major premisses of e limination r ules are assumptions. Cont r ar y to the other ar ithmetical r ules, r ule In d cannot b e for mu -late d so that it op er ates only on atomic for mulas. As a consequence, there is no subfor mula prop er t y, a nd some authors w r ite indeed for this reason that ‘nor malization fails in ar ithmetic’. Ne ver theless, the nor mal for m for arithmetical derivations is a useful property, as we shall now see.

12.3 The existence property

The G ¨odel–Gentzen t r anslations of 1933, g iven in S ec tion 10.3, were or ig i-nally devised to reduce the consistency of classical arithmetic to intuition-istic arithmetic. The latter does not contain any of the ‘dubitable’ steps of indirect proofs with the quantifiers, but the question still remained whether intuitionistic arithmetic could be further justified as consistent. Reflecting on his incompleteness theorem, G¨odel set out in a conference talk of 1933 a number of criteria for such a justification, including the existence property:

If a formula∃x A(x) is a theorem in intuitionistic arithmetic, it should be possible to find an instance A(n) on the basis of a proof of the theorem (cf. the lecture ‘The present situation in the foundations of mathematics’, printed in the third volume of G¨odel’s Collected Works).

The existence property of Heyting arithmetic was proved by Kleene in 1945 by what is known as the realizability method. We show as an application of the normal form of theorem 12.7 that the existence property can be proved by straightforward transformations of formal derivations of an existential

theorem. The prop er t y fol lows w ith almost nothing to prove: Either the last r u le in a n or mal der ivation of ∃ x A( x ) is ∃ I or the last rule can be dropped out.

We shal l set the ar ithmetical r ules except In d aside now and consider the system of log ical r ules and r ule In d in what follows: As we have noted, the presence of ar ithmetical r ules other than In d do es not affec t the analysis of nor mal der ivations b ecause the conclusions of these ar ithmetical r ules are ne ver major premisses in e limination r ules.

Theorem 12.8. Existence prop er t y for HA. Give n a der ivat ion of ∃ x A( x ) in HA w ith no ope n assumpt ions and no free var iables in∃ x A( x ), a derivation of an instance A(n ) can b e found.

Pro o f . We may assume the der ivation to be nor mal. E -r ules would leave their major premisses as op en assumptions, so the last r ule is not an elim-ination. If it is ∃ I , the premiss is some A(t ). If t is a number, a derivation of an instance is found by deleting the last r ule, and if t is a ter m w ith a var iable y , denoted t ( y ), then the instance A(t (n )) is similarly der iv able for any v alue of n .

By the above, we may assume that the last step in the der ivation is In d , w ith an induc t ion for mula B and the der ivation:

.... free var iables, the resulting numer ical induc tion can b e removed as above.

Therefore we c an assume u to b e a t e r m t (v ) w ith a var iable v . Consider the subder ivation of the third minor premiss ∃ x A( x ) from the assumption B (u ). Becausev remains free in B (u ), it cannot b e an eigenvar iable of a r ule instance in that subder iv ation. Therefore v can b e substituted by 0 in the der ivation of∃ x A( x ) from B (u ), w ith no assumptions left, and the instance of In d that ends the der ivation deleted. QED.

The condition that there b e no fre e var iables in ∃ x A( x ) either force s t he induc t ion n ot to do any wor k, or else ∃ x A( x ) has been concluded by rule ∃ I .

We proved the disjunc tion prop er t y of intuitionistic log ic in S ec tion 3.4, and the existence prop er t y of intuitionistic pre dicate log ic in S ec tion 9.2. In

Elements of the proof theory of arithmetic 187

HA, the existence property leads to a definition of disjunction in terms of existence: