By Alfred Tarski (Eds.)

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3. We could try to improve that part of Theorem 9 which refers to Theory Q, by constructing some finitely axiomatizable subtheories of Q which would be still essentially undecidable. The simplest way of constructing subtheories of Q consists of course in omitting some of the axioms. In the next theorem we shall show, however, that none of the theories thus obtained is essentially undecidable (though, by Corollary 10, all of them are undecidable). THEOREM 11. No axiomatic subtheory of Q obtained by removing anyone of the seven axioms from the axiom system is essentially undecidable.

0 ~ Vy(u =Sy). We shall now derive all the sentences D1-Do from the axioms of Q. , (1) A,. -tf> for all n,p E N are shown to be valid in Q by induction on p. ) In fact, for p = 0, (1) is a particular instance of 8 4 ; when passing from p to p + 1, we apply 8 0 , Similarly, by an induction on p based upon 8 8 , 8 7 , and (1), we derive the sentences D2 : (2) L1,. " for all n,p EN. Consider now the sentences Da: (3) L1,. #;. L1" for all n,p We can clearly assume that n E < p. N with n =1= p. For n = 0 (and every p > n) 54 UNDECIDABILITY IN ARITHMETIC (3) is a particular case of e2 ; when passing from n to n apply e1 • e s and e5 imply z x 0 II X '# 0 ~ Vy[x Sy II S(z y) 0]; = + hence, by e = + +1 we = 2, (4) Similarly, for every natural number n we have byes and z + x =LI"+1 hence, by (5) '# 0 ~ II X =Sy Vy[x e1> x

By the definition of E, there is a natural m such that m2 and hence + En = n < (m + 1)2 m ~ n and En ~ 2· m, By QI and Q2' we conclude that the sentence LIm < LI" 1\ LIEn < LIm + LIm 1\ LI" = (LIm LIm) + LIEn 0 58 UNDECIDABILITY IN ARITHMETIC is valid in R; in view of (4), this clearly implies the validity of (5). 10' + v) v ... 1",+ v). 1" L1 m ' +21 with m ~ nand p ~ 2· m. En is uniquely determined as the natural number p such that n = m 2 + p and p ~ 2· m for some mEN. 1" #= L1 m' +2l with p #- En and p ~ 2·m are valid in R.