ACCESSIBLE INDEPENDENCE RESULTS FOR PEANO ARITHMETIC

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1 ACCESSIBLE INDEPENDENCE RESULTS FOR PEANO ARITHMETIC LAURIE KIRBY AND JEFF PARIS Recetly some iterestig first-order statemets idepedet of Peao Arithmetic (P) have bee foud. Here we preset perhaps the first which is, i a iformal sese, purely umber-theoretic i character (as opposed to metamathematical or combiatorial). The methods used to prove it, however, are combiatorial. We also give aother idepedece result (uashamedly combiatorial i character) proved by the same methods. The first result is a improvemet of a theorem of Goodstei [2]. Let m ad be atural umbers, > 1. We defie the base represetatio of m as follows: First write m as the sum of powers of. (For example, if m = 266, = 2, write 266 = ) Now write each expoet as the sum of powers of. (For example, 266 = ) Repeat with expoets of expoets ad so o util the represetatio stabilizes. For example, 266 stabilizes at the represetatio 2* +l l +2 l. We ow defie the umber G (m) as follows. If m = 0 set G (m) = 0. Otherwise set G (m) to be the umber produced by replacig every i the base represetatio of m by +1 ad the subtractig 1. (For example, G 2 (266) = ). Now defie the Goodstei sequece for m startig at 2 by m 0 = m, m x = G 2 {m 0 ), m 2 = G^mJ, m 3 = G^m 2 ),.... So, for example, 266 O = 266 = X = ~ 1O = l ~ = 5 s ~ Similarly we ca defie the Goodstei sequece for m startig at for ay > 1. THEOREM 1. (i) (Goodstei [2]) Vm 3/c m k = 0. More geerally for ay m, > 1 the Goodstei sequece for m startig at evetually hits zero. (ii) Vm 3k m k = 0 (formalized i the laguage of first order arithmetic) is ot provable i P. Received 1 February, The first-amed author was supported by a SRC Research Fellowship. Bull. Lodo Math. Soc, 14 (1982),

2 286 LAURIE KIRBY AND JEFF PARIS So, belyig its early form, the sequece m k evetually hits zero. However despite this fact beig expressible i first order arithmetic we caot give a proof of it i Peao Arithmetic P. As we shall see later the reaso for this is the immese time it takes for the sequece m k to reach zero. (For example, the sequece 4 k first reaches zero whe k = 3 x "-3, which is of the order of io ) Before provig Theorem 1 we state our secod result. A hydra is a fiite tree, which may be cosidered as a fiite collectio of straight lie segmets, each joiig two odes, such that every ode is coected by a uique path of segmets to a fixed ode called the root. For example: top ode root A top ode of a hydra is oe which is a ode of oly oe segmet, ad is ot the root. A head of the hydra is a top ode together with its attached segmet. A battle betwee Hercules ad a give hydra proceeds as follows: at stage ( ^ 1), Hercules chops off oe head from the hydra. The hydra the grows "ew heads" i the followig maer: From the ode that used to be attached to the head which was just chopped off, traverse oe segmet towards the root util the ext ode is reached. From this ode sprout replicas of that part of the hydra (after decapitatio) which is "above" the segmet just traversed, i.e., those odes ad segmets from which, i order to reach the root, this segmet would have to be traversed. If the head just chopped off had the root as oe of its odes, o ew head is grow. Thus the battle might for istace commece like this, assumig that at each stage Hercules decides to chop off the head marked with a arrow: after stage 2 after stage 3

3 ACCESSIBLE INDEPENDENCE RESULTS FOR PEANO ARITHMETIC 287 Hercules wis if after some fiite umber of stages, othig is left of the hydra but its root. A strategy is a fuctio which determies for Hercules which head to chop off at each stage of ay battle. It is ot hard to fid a reasoably fast wiig strategy (i.e. a strategy which esures that Hercules wis agaist ay hydra). More surprisigly, Hercules caot help wiig: THEOREM 2. (i) Every strategy is a wiig strategy. We ca code hydras as umbers ad thus talk about battles i the laguage of first order arithmetic. We caot formalise Theorem 2(i) as a statemet of this laguage, as strategies are ifiitary objects. However, we ca if we restrict ourselves to recursive strategies. I that case: THEOREM 2. (ii) The statemet "every recursive strategy is a wiig strategy" is ot provable from P. I provig the theorems we rely o work of Ketoe ad Solovay [3] o ordials below e 0, which i tur develops earlier work by the preset authors, Harrigto, Waier, ad others. Getze [1] showed that usig trasfiite iductio o ordials below e 0 oe ca prove the cosistecy of P, ad the Ketoe-Solovay machiery we use here ca be viewed as illumiatig i more detail the relatioship betwee e 0 ad P. (Note: Goodstei proved the followig: if h : N -> N is a o-decreasig fuctio, defie a fi-goodstei sequece b Q,b x,... by lettig b i + l be the result of replacig every h(i) i the base h(i) represetatio of b { by h{i +1), ad subtractig 1. The the statemet "for every o-decreasig h, every Ji-Goodstei sequece evetually reaches 0" is equivalet to trasfiite iductio below e 0.) To prove Theorem 1 we first defie the base represetatio more formally, at the same time defiig the ordial o (m), i Cator Normal Form, which results from replacig every i the base represetatio of m by a>. Suppose m,e N (the set of atural umbers), > 1 ad m = k a k + kl a k _ l + l 0 For x e iv or x = co, set k f m '"(x) = a,-x /l(x). i = 0 (This defiitio is by iductio o m, startig with / 0> "(x) = 0.) The for m > 0, G (m) = / m '"(+l)-l ad o (m) = f m ' {(o). Set G B (0) = o (0) = 0. Fially for e N we defie a operatio <<x>() o ordials a < e 0 by iductio o a: ad for <5 > 0,

4 288 LAURIE KIRBY AND JEFF PARIS LEMMA 3. (i) For m ^ 0, > 1, if a = o + l {m) the o + 1 (m 1) = <<X>(H). (ii) For > 1, <o (m)>() = o +1 {G {m)). Proof, (i) Cosider the base +1 represetatio of m: let with 0 < a,- < w, ad (sice we may suppose m ^ 0) let j be miimal such that cij =/= 0. The result is clear if j = 0 so we may assume that; > 0 ad that the result holds for all 0 < m' < m. The whilst Usig the iductive hypothesis it is easy to see that these are equal. (ii) P Let m = b i p '" () where 0 ^ b { < ad bj-, j= 0. If j = 0 the it is clear that <o (m)>() = o + 1 (G (m)) so assume j > 0. The ad By(i) ad o. + i(( + l) /AlI( " +l) - 1 -l) = <G)<''-" ( w >(), which gives the required result. Thus for each Goodstei sequece b o,b 1,b 2,--- there is a correspodig sequece o {b 0 ), O+iibJ, o + 2 {b 2 ),...

5 ACCESSIBLE INDEPENDENCE RESULTS FOR PEANO ARITHMETIC 289 of ordials. (E.g. i the example give before we would have If we write <a>( l5 2,..., k ) for C-X < ( a )( i)x 2)---) > ( fc) sequece as we ca write this o (b 0 ) = a, <a>(), <a>(,+l), <a>(, +1, + 2),.... It is ot hard to see that for ay a < e 0 ad e N, <a>() < a if a > 0. Now we already have Theorem 1 (i) (followig Goodstei). For suppose m, were such that the Goodstei sequece for m startig at was always positive. The the correspodig sequece of ordials would be a ifiite, strictly decreasig sequece which is a impossibility (by trasfiite iductio below e 0 ). I order to prove (ii) we itroduce the Ketoe-Solovay machiery (see [3], [4] for details). First we defie aother operatio {<x}() for a < e 0 ad en by iductio o a: ad for limit <5, () = 0, Note that (a}() < a for a > 0. Now defie the otio of a-large fiite sets for 0 < a < e 0 by iductio o a: If X a N is fiite, eumerate the elemets of X i ascedig order as X o, X y,..., X m _,. X is 1-large if ad oly if \X\ ^ 2; X is a-large if ad oly if X {X 0 ] is {a.}(x^aarge. Now write {a}( 1,..., k ) for {... {{a}( 1 )}( 2 )... }( k ). The by iductio o a we ca show that X is a-large if ad oly if {cc}(x 1,X 2,..., X m _ { ) = 0. For details see Lemma 11 of [4]. Defie co 0 = w, (o + l = of". The above cocepts ca be put i the laguage of first order arithmetic (with ordials < 0 replaced by suitable otatios for them) ad thus make sese i a ostadard model of P (see [4]). THEOREM 4 (Ketoe-Solovay; see [3], [4]). (i) The fuctio Y(a, b) = the greatest c such that [a, b~\ is co c -large is a idicator for models of P. (ii) The statemet Va Vc 3b ([a, b~] is a> c -large) is idepedet of P ad is equivalet i P to Co (P+ T t ) where T x is the set of the true I^ seteces. (iii) The fuctios g (x) = least y ^ x such that [x, y~\ is co -large are provably (i P) total recursive fuctios, ad for ay provably total recursive fuctio f there exists e N such that f(x) < g {x)for all sufficietly large x e N.

6 290 LAURIE KIRBY AND JEFF PARIS Write p -* a if ad oly if for some j x,..., j k <, a = {P}Ui,-,Jk)'> P => a if ad oly if the same holds with j Y =... = j k =. The followig lemma is a stadard applicatio of these cocepts. LEMMA 5. (i) ///?=> a ad > 0 f/ie a/ => a/. (ii) //0<i<7<rte{j3}O-)=f W(i). (iii) P => txif ad oly if P-* a.. (iv) Suppose P = oj Pi co P, y = a> co ym, ad T/ie i/y -» 5 t/ie j9 + y -> P + d. I particular P + y -» /?. Havig itroduced this machiery we shall apply it to lik the operatios {a}() ad <a>() ad hece obtai our result. LEMMA 6. Suppose P -> a ad 0 < ^ t < 2 <... < k. The Proof. The proof is by iductio o p. Assume the result holds below p. By Lemma 5 (iii), so (js}(«i) -* {a}^). Hece by iductive hypothesis PROPOSITION 7. For a// a < e 0 ad j e N, <a>(j) -* {a}(j). Proof by iductio o a. If a is 0 or a successor, the result is trivial. If <x = co y+i (p + \) the from the defiitios {a}(;) = w y+1 /? + a//, ad 7) = {«}(./) + < '>(;) Applyig Lemma 5 (iv), <a>(;) -* {«}(;) If a = co s (P+1), 5 limit, the by iductive hypothesis 7 WO")-

7 ACCESSIBLE INDEPENDENCE RESULTS FOR PEANO ARITHMETIC 291 By Lemma 5(i), (o <5>U) -> (D m \ Thus <a>(;) = co^ + co <5>U) j + (a) <s>u) y{j) -> co 5 p + co <3>U) by Lemma 5 (iv) PROPOSITION 8. Let b o,b l,b 2,... be the Goodstei sequece for m startig at ad let k be miimal such that b k = 0. The [ 1, + k~\ is o (m)-large. Proof. Cosider the correspodig sequece of ordials o {m) = o (b 0 ) = a By Lemma 6 ad Propositio 7, o +k {b k ) = o +k {0) = 0 = <a>(, ^ ^ < > (,, ) = 0. Hece [ 1, + /c] is a-large. Now to prove (ii) of Theorem 1 suppose we had P h Vm 3k m k = 0. (*) By Theorem 4 ad the methods of idicator theory (see [5]) we ca fid M = P ad ostadard ce M such that M\= -i 3y ([1, y~\ is co c -large). (Briefly, this is doe by takig a coutable ostadard model J of P ad ostadard c,aej such that Y(c, a) is ostadard but less tha c 1, where y is as i Theorem 4 (i). Now the idicator Y havig ostadard value o (c, a) meas precisely that there is a iitial segmet of J which is a model of P ad lies "betwee" c ad a, that is, cotais but ot a. We ca let M be such a iitial segmet.) I M, take d = with c iterated expoetiatios, so o 2 (d) = a> c. By (*) take e M such that d e = 0. Sice the proof of Propositio 8 ca be carried out i P (see the discussio precedig Lemma 4 i [4]), we have i M a cotradictio. [1, 2 + e] is w c -large,

8 292 LAURIE KIRBY AND JEFF PARIS We sketch the proof of Theorem 2, which is similar to that of Theorem 1. First assig to each ode i a give hydra a ordial below e 0 as follows: To each top ode assig 0. To each other ode assig oj ai (o a ", where <Xj ^. assiged to the odes immediately "above", {o 0 = 1). Thus our origial example would have the assigmets o a are the ordials 0 O The ordial of a hydra is the ordial assiged to its root. For ay strategy a, we ca defie a operatio [a] CT () which maps the ordial of the hydra after stage 1 to the ordial of the hydra after stage, where a is the strategy beig used. To show Theorem 2 (i) it will suffice for the reader to check that for ay strategy a, ay 0 < a < 0 ad s N, [a]» < a. For (ii) we shall produce a recursive strategy T for which [«] t () = {<*}(«+ 1). The a proof like that of Theorem 1 will show that i P we caot prove that T is a wiig strategy. A algorithm for T is as follows: startig from the root, we travel "up" the tree i such a way that, havig reached a ode, we travel to the ode immediately above it which has miimal assiged ordial amog all the odes immediately above it. (If more tha oe of them has miimal ordial we choose, say, the leftmost.) Evetually we reach a top ode ad the head it is attached to is the oe to chop off. Thus i the previous diagram the head determied by x is starred, ad the battle determied by x begis thus: after stage 1 after stage 2

9 ACCESSIBLE INDEPENDENCE RESULTS FOR PEANO ARITHMETIC 293 (Note: A proof that T is a wiig strategy is equivalet to a proof of "e 0 -iductio with respect to the predecessor fuctio p(cc, ) = (a}()" which amouts i tur to a proof that the fuctio Xx. g (x) of Theorem 4 (iii) is total recursive, ad this of course is impossible i P.) REMARK. Let /I fc deote Peao's axioms with iductio restricted to S k ' formulae. The usig results i [4] we ca refie Theorem 1 to give, for k e N, k ^ 1, THEOREM 1'. (i) For each fixed pen, /I k \- Vm, > 1 {if m < "" P [where occurs k times) the the Goodstei sequece for m startig at evetually hits zero). (ii) I'L k \-f Vm, > 1 (if m < """ (where occurs k + l times) the the Goodstei sequece for m startig at evetually hits zero). Similarly if we restrict ourselves i Theorem 2 to hydras of height k + l (i.e. o ode is more tha k+l segmets away from the root) the we caot prove that "every recursive strategy is a wiig strategy" usig just /X fc. I particular the fuctio givig the legths of battles for hydras of height 2 with strategy x is ot provably total i /Z x ad hece is ot primitive recursive. Refereces 1. G. GENTZEN, 'Die Widerspruchsfreiheit der reie Zahletheorie', Math. A., 112 (1936), R. L. GOODSTEIN, 'O the restricted ordial theorem', J. Symbolic Logic, 9 (1944), J. KETONEN ad R. SOLOVAY, 'Rapidly growig Ramsey fuctios', A. of Math., 113 (1981), J. PARIS, 'A hierarchy of cuts i models of arithmetic', Model theory of algebra ad arithmetic, Proceedigs, Karpacz, Polad Lecture Notes i Mathematics 834 (Spriger, Berli) pp J. PARIS, 'Some idepedece results for Peao arithmetic', J. Symbolic Logic, 43 (1978), Departmet of Mathematics, The Uiversity of Machester, Machester M13 9PL.