cn Journl the Mthemticl Society Jpn Vol 9, No 4, Octor, 1957 Ordinl digrms By Gi TAKEUTI (Received April 5, 1957) In h pper [2] on the constency-pro the theory nturl numrs, G Gentzen ssigned to every pro-figure n ordinl numr In modifying h method, we my do th s follows : (A) $()\frc{s_{1}s_{2}}{(b)\frc{s_{3}s_{4}}{(c)\frc{s}{s}6\underline{6}}}$ (B) $\bckslsh $1\bckslsh _{}/^{1}$ \swrrow$ 1 $c $ Fig 1 Suppose, to fix our ide, pro-figure (A) (in Fig 1) given $S_{1},$ $S_{2},$ $S_{4}$ inferences Th composed ginning sequences $(),$ $(b),$ $(c)$ To the inferences: wekening, contrction exchnge, we ssign the vlue ; to cut degree $n$, the vlue $0$ $n$ ; to n induction degree $n$, the vlue $n+1$ ; the vlue 1 to ll other inferences We denote the vlues inferences by $(),$ $(b),$ $(c)$ $,$ $b,$ $c$ respectively We replce the ginning sequences by 1, drw the figure (B) ccording to the form the pro-figure (A) If we consider $\lph\t$ $\vee$ $ (\lph,$ $\t$ ing ordinl numrs $$ $$ non-negtive integer) s opertions defining ordinl numrs (to defined properly, see low), then the figure like (B) represents itself n ordinl numr Th my clled Gentzen s numr for the pro-figure (A) Although th not the sme ordinl numr s ssigned to (A) by Gentzen himself, we cn ccomplh the constency-pro the theory nturl numrs just s in [2], in proving tht th Gentzen s numr diminhed by the reduction the pro-figure $\t$ $ $ $\vee$ The opertions descrid by Ackermnn s $$
men Ordinl digrms 387 construction in [1] We shll write for simplicity insted $(\lph, \t)$ Ackermnn s, use $(1, \lph, \t)$ $\lph+\t$ in the mening nturl sum in generl, while Ackermnn uses it only in cse $\lph\geqq\t$ $ $ $$ only for $\geqq 1$ $(, \lph+\t)$ ( $,$ $\lph 1$ respectively We put $(0, \lph)=\lph)$ Then $\lph\t$ $\vee$ $((, \lph)$ defined in [1] The purpose the present pper to construct system ordinl numrs the second Zhlenksse represented by wht we shll cll ordinl digrms Presumbly our system contins the system constructed by Ackermnn [1], but it not proved We hve in view to pply our result to constency-pro Ordinl digrms re constructed in the following wy Consider trees the following form: $e$ $g$ $$ $O^{\bckslsh _{\bullet}/^{\circ}}\bckslsh ^{o_{\bullet}}/$ $\circ\bckslsh _{\bullet}/^{o}$ $\bckslsh _{\bullet}/$ Fig 2 Such trees hve two sorts vertices, ginning vertices mrked with non-ginning vertices mrked with $0$ $\bullet$ We ssign to ech vertex positive integer clled vlue the vertex, to ech non-ginning vertex positive integer clled index the vertex, not exceeding n integer $n(>0)$ fixed once for ll, which we shll cll the order the system If we consider s opertion on digrms denote it by $(i;, \lph_{1}+\cdots+\lph_{k})(i$ the index $$ the vlue the vertex $(, i))$, then digrm like
$\omeg$ $\omeg$ $1$ 388 G TAKEUTI (C) cn descrived by $(i_{1} ; b_{1}, (i_{0} ; b_{c}, _{0}+_{1}+_{2})+_{3})+(i_{2} ; b_{2}, _{4})$ (C) $_{4}$ $(b_{1}, i_{1})$ $(b_{2}, i_{2})$ In the following lines, we shll give the forml definition ordinl digrms the ordering tween them, prove tht they re well-ordered In view pplictions to constency-pro, we should like to dd here the following remrk If we denote the system ordinl digrms order with $O(n)$, it cler tht we hve $n$ $O(1)\subset O(2)$ it will proved s ws sid bove, tht $O(n)$ wellordered $\subset\cdots$ It will lso proved tht V $0(n)$ not well-ordered $\tilde{n}$ Let some theory including the theory $N$ nturl numrs A constency-pro such theory my crried out s $\tilde{n}$ follows To ech pro-figure $P$ in $\tilde{n}$, we ssign n ordinl digrm certin order $n$, prove tht the ordinl digrm diminhed by reduction the pro-figure Th will not in contrdiction with Godel s result [3], tht the constency-pro $\tilde{n}$ not formulble in $\tilde{n},$ just s $n$ Gentzen s constency-pro $N$ not in contrdiction with $[3]$ th, even when $\tilde{n}$ rich theory, in the following sense Denote the ordinl numr firly $n$ $\omeg_{n}$ with let $Q(n)$ men the system ordinl numrs less thn $\omeg_{n}$ Then Gentzen hs ssigned to ech pro-figure $P$ in $N$ n ordinl numr $Q(n)$ for certin $n$, proved tht th ordinl
$\t$ re hs recursively hs Ordinl digrms 389 numr diminhed by reduction Although the trnsfinite induction in $Q(n)$ for given $n$, the system $\bigcup_{n}q(n)$ itself re both formulble in $N$, the trnsfinite induction in $\bigcup_{n}q(n)$ not formulble in $N$, thus Gentzen s constency-pro not in contrdiction with G\"odel s result The sme circumstnces will re when we replce $Q(n)$ by our $0(n)$ The uthor whes to express h herty thnks to Pr Iyng for h vluble dvice during the preprtion th pper \S 1 Ordinl digrm order $n$ Herefter let fixed positive integer $n$ 1 Ordinl digrm order constructed by two opertion $n$ $(i; )$ $(i=1,2, \cdots, n)$ $\#$, defined recursively s follows (If no confusion to fered, we use ordinl digrm or in plce ordinl digrm order $n$ $0$ s re denoted by $\lph,$ $\t$, $\gmm,$ $\ldots$ (possibly with suffixes) 11 If positive integer, then n $$ $$ 12 If positive integer $$ $i$ n, n integer stfying $0<i\leqq n$, then $(i;, \lph)$ n $\t$ 13 If $\lph\#\t$ s, then n $\lph,$ 2 Let $i$ s, n integer stfying $0<i\leqq n$ We $\t\subset i\lph$ $\t$ define recursively the reltion (to red : n i-section ) s follows: $\t\subset t\lph$ 21 If n integer, then never holds ( no i-section) 22 Let the form $(j;, \lph_{0})$ $\t\subset i\lph$ 221 If $j<i$, then if only if $\t\subset_{i}\lph_{0}$ $\t\subset i\lph$ $\t$ 222 If $j=i$, then if only if $\lph_{0}$ $\t\subset i\lph$ 223 If $j>i$, then never holds $\t\subset t\lph$ 23 Let the form $\lph_{1}\#\lph_{\gmm,\lrcorner}$ Then if only if either $\t\subset i\lph_{1}$ or $\t\subset t\lph_{2}$ holds $0$ 3 An $c$ clled (connected ordinl digrm), if only if the opertion used in the finl step construction not $\#$ $0$ Let n We define cornponents s follows: $c$ 31 If, then only one component which itself
$\lph_{2}$ re $\gmm_{l}$ hs such 390 G TAKEUTI 32 If $\t_{1},$ re $\t_{1},$ n $\t_{k}$ $\cdots,$ $\gmm_{1},$ $\t_{k},$ $\gmm_{1},\cdots,$ $\gmm_{l}$ $\lph_{1}\#\lph_{2}$ the form components $\lph_{1}$ $\lph_{1}\#\lph_{2}$ $\cdots,$ respectively, then components $\t_{1},$ $\t_{k}$ 4 $\cdots,$ Let $\t$ s We define recursively s follows: $\lph=\t$ 41 Let n integer Then, if only if $\lph=\t$ $\t$ the sme integer s 42 Let n the form $(i;, \lph_{0})$ Then $\lph=\t$, if only $\t$ if the form $\lph_{0}=\t_{c}$ $(i;, \t_{0})$ 43 Let non-connected with components $k$ $\lph_{1},\cdots,$ $\lph_{k}$ Then, if only if $\lph=\t$ $\t$ the sme numr components, ing these components, there exts permuttion $\cdots,$ $(l_{\tu}, \ldots, J_{k})$ $(1, \ldots, k)$ such tht $m=1,$ $\lph_{m}=\t_{lm},$ $\ldots,$ $\t=\lph$ 44 holds, if only if $\lph=\t$ 5 Let two $0$ s We define the reltions, $\lph<0\t$ $\lph<_{1}\t,$ $\cdots,$ $\lph<n\t$ recursively s follows Sometimes $\lph<0\t$ denoted $\t_{h}$ by $\lph\ll\t$ $\t$ 51 Let two integers Then $\lph<_{0}\t,$ $\lph<n\t$ $\cdots,$ ll men $\lph<\t$ in the sense integer 52 $\t$ Let the components $\lph_{1},$ $\t_{1},$ respectively $\cdots,$ $\lph_{k}$ $\ldots,$ $\lph<i\t(i=0,1, \cdots, n)$ holds, if only if one the follow- ing conditions fulfilled 521 There exts $\t_{m}(1\leqq m\leqq h)$ such tht for every $l(1\leqq l\leqq h)$ $\lph_{l}<_{i}\t_{m}$ 522 holds $k=1,$ $h>1$ $\lph_{1}=\t_{m}$ for suitble $m(1\leqq m\leqq h)$ 523 $k>1,$ $h>1$ there ext $\lph_{\iot}(1\leqq l\leqq k)$ $\t_{m}(1\leqq m\leqq l)$ $\lph_{l}=\t_{m}$ such tht $\lph<n\t$ by $\lph<\t$ $\lph_{1}\#\cdots\#\lph_{\iot-1}\#\lph_{\mthfrk{l}+1}\#\cdots\#\lph_{k}<i\t_{1}\#\cdots\#\t_{m-1}\#\t_{m+1}\#\cdots\#\t_{h}$ $\t$ $0$ $c$ $s$ 53 Let Then, $\lph<_{i}\t(i=1,2, \ldots, n)$ if only if one the following conditions fulfilled $\t$ $\t_{0}$ 531 There exts n i-section tht $\lph\leqq_{i}\t_{0}$ $\lph_{0t}<\t$ 532 for every $\lph_{0}$ i-section, $\lph<_{i-1}\t$ $\t$ 54 Let $c$ the form respectively, if only if one the following conditions $(i;, \lph_{0})$ $(j;b, \t_{0})$ $\lph\ll\t$ fulfilled 541 $<b$ 542 $=b$ $j<i$ 543 $=b,$ $i=j$ $\lph_{0}<_{i}\t_{0}$ $k$
in in 55 Let $$ positive integer $\t$ Ordinl digrms 391 $c$ $(j;b, \t_{0})$ Then $ \ll\t$, if only if $\leqq b$ And $\t\lngle\lngle if $b<$ $, if the form only Under these definitions the following propositions re esily proved PROPOSITION 1 $=$ n equivlence reltion tween $0d$ $s,$ $\lph=\lph$ $ie$ imply $\lph=\gmm$ $\lph=\t,$ $\t=\gmm$ $(i;, \lph_{2})$ $PROPOSI^{\prime}rION2$ $\lph_{1}=\lph_{2},$ $\t_{1}=\t_{2}$ imply PROPOSITION 3 $\lph_{1}=\lph_{2},$ $\t_{1}=\t_{2},$ $\lph_{1}<_{i}\t_{1}$, $\lph_{1}\#\t_{1}=\lph_{2}\#\t_{2},$ $(i;, \lph_{1})=$ imply $\lph_{2}<_{i}\t_{2}$ PROPOSITION 4 Everyone the reltions $<_{\iot}(i=0,1, \ldots, n)$ defines hner order tween s, $i$ $e$ $\lph<i\t,$ $\t<tr$ imply $\lph<t\gmm$ ; one only one reltion $\lph<i\t,$ $\lph=\t,$ $\t<i\lph$ holds for every pir $\lph,$ $\t$ s \S 2 Trnsfinite induction $\mthfrk{s}$ $\mthfrk{s}$ 1 Let system $s$ with liner order An element clled ccessible in th system (or ccessible for th order), if $\mthfrk{s}$ the subsystem consting elements, which re not greter thn $s$, well-ordered The following propositions re esily proved PROPOSITION 1 Let n If every Jess thn the sense $<_{i}$ ccessible, then $for<_{\iot}$ ccessible $for<_{i}$ $0$ PROPOSITION 2 Let n If ccessible $for<_{i}$, then every less thn the sense $<_{i}$ ccessible $for<_{i}$ $\lph_{1},$ $\lph_{k}$ PROPOSITION 3 Let $\cdots,$ $\lph_{1},$ $\lph_{k}$ s $\ldots,$ If re ccessible $\lph_{1}\#\cdots\#\lph_{k}$ $for<_{i}$, then ccessible $for<_{i}$ $i$ 2 Let n n integer stfying $0\leqq i\leqq n$ We define recursively n i-fn $i$-ccessibje s follows : 21 Every n n-fn 22 i-ccessible, if only if n i-fn ccessible for $<_{i}$ in the system i-fns 23 n i-fn, $(0\leqq i\leqq n)$ if only if n $(i+1)$-fn every $(i+1)$-section $(i+1)$-ccessible Every O-fn lso clled fn A fn sid to ccessible in the sense fn, if O-ccessible We see clerly tht propositions 1, 2, 3 remin correct, if we replce
$\gmm$ hs for hs such such 392 G TAKEUTI with i-fn ccessible for with i-ccessible $<_{i}$ We obtin esily the following propositions PROPOSITION 4 The following two conditions on n $$ re equivlent : 24 ccessible for $<$ 25 n-ccessible PROPOSITION 5 If n i-fn, then n $(i+1)$ -fn PROPOSITION 6 If every positive integer i-ccessible, then every i-fn i-ccessible PROOF Let n i-fn the mximl numr $$ integers, which composed Then clerly $\lph<_{i}(+1)$, whence the proposition 6 follows directly PROPOSITION 7 Every fn ccessible in the sense fn 3 Now we shll prove the following proposition PROPOSITION 8 If every $(i-1)$ -fn (i-l)-ccessible, then every i-fn i-ccessible $(i=1,2, \cdots, n)$ PROOF Let n rbitrry $(i-1)$-fn By the proposition 6 we hve only to prove tht i-ccessible Without loss generlity, we my ssume the following condition 31 on : $\t$ $\t$ 31 i-ccessible, if n $(i-1)$-fn $\t<_{i-1}$ $\gmm$ Now, let n rbitrry connected i-fn suppose $\gmm<i\lph$ $\gmm$ We hve only to prove tht i-ccessible We prove th by induction on the numr opertions in the construction $\gmm$ If $\gmm$ no i-section, then n $(i-1)$-fn one the following $\gmm<_{i}\lph$ conditions follows from : 32 $\gmm<i-1\lph$ $\delt$ $\gmm\leqq 33 There exts n i-section tht In cse 32, the proposition 8 follows from 31 In cse 33, the proposition 8 follows from the condition tht n $(i-1)$-fn $\gmm$ $\gmm$ Now, suppose n i-section Since every i-section less thn n $i$-fn, it follows from the hypothes $<_{i}$ $\gmm$ $\gmm$ the induction, tht every i-section i-ccessible Hence n $\gmm<_{i}\lph$ $(i-1)$-fn Therefore, from one the following conditions follows: 34 $r<i-1$ $\delt_{0}$ 35 There exts i-section tht $\gmm\leqq_{i}\delt_{0}$ In cse 34, the proposition 8 follows from 31 In cse 35, the proposition 8 follows from the condition tht n $(i-1)$-fn $$ i\delt$
other such Ordinl digrms 393 From propositions 7 8 follows: THEOREM The system ll the $s$ well-ordered $for<$ \S 3 Some properties o d s The following propositions on bove s follow esily from the $i$ $\t$ PROPOSITION 1 Let c o d s n integer stfying $0<i\leqq n$ If holds for every stfying for every $ _{0}<j\t$ $j$ $j\leqq i$ j-section $\lph_{0}$ $\lph\ll\t$, then $\lph<i\t$ PROPOSITION 2 Let n i-section Then $\t<i\lph$ $c$ $\t$ $\t$ $i,$ $k$ PROPOSITION 3 Let $c$ s integers stfying $0<i\leqq n$, $0<k\leqq i$ $\lph_{0}$ respectively If k-section the following cmditims 11 13 re fulfilled, then $\lph<i\t$ 11 Let ny integer stfying $j$ $0<j\leqq i$ $\lph_{1}$ $\t_{1}$ Then there exts j-section thn $\lph_{0}$ $\lph_{1}\leqq J\t_{1}$ 12 $\lph_{0}<k\t$ 13 $\lph\ll\t$ j-section tht PROPOSITION 4 In the nottion the introduction V $O(n)$ not well-ordered $n$ PROOF Th esily seen by the following exmple 2 2 2 $(1,2) $ $(1,3) $ $(1,4) $ $\t$ $(2\rfloor_{1})$ $>$ $(1,2) $ $>$ $(1,3) >\cdots$ $(2,1) $ $(1,2) $ $(2,1) $
394 G TAKEUTI References [1] W Ackermnn: Konstruktiver Aufbu eines Abschnitts der zweiten Cntorschen Zhlenklsse; Mth Z 53 (1951), 403-413 [2] G Gentzen: Die Widerspruchsfreiheit der reinen Zhlentheorie; Mth Ann 112 (1936), 493-565 [3] K G\"odel: \"Ur forml unentscheidbre S\"tze der Principi Mthemtic und verwter System I; Montsh f Mth Phys 38 (1931), 173-198