Appendix 143 142 Appendix DEFINITION 4 A qualllifier Q is a function that assigns to each universe A I. II is a runtioll from (()a into (CI~I. a quantifier on A. QA' and such that if A. A' are universes of the same 2. Ii is onto ((i'j'11. and cardinality, then Q and QA' have the samc cardinality countcrparl. A 3. h is one-to-one. (I.a) First I prove that" is a function. Let Oa be an a-operator of type Chapter 2, Section 5 (II' ...• 1,,). Let (s" ... , s,,) be a sequence such that for I ::; i ~ k, Sj E A if Ii = 0, and Sj ~ An if 'i = n =f. O. I have to show that [(i(J:l)' ... , ;(s,,»] DEFINITION 5 Let A be a set. A 2-place quantifier 011 A is a function exists and is unique. Existence is obvious. To prove uniqueness, let J, I' be q: P(A) x P(A) -+ {T, F} two indexings of A bya. Let i(51), ... , ;(s,J and ;'(s.), ... , i'(s,,) be the such that ifm : A -+ A is an automorphism of A. then for every IJ. C s; A, index images 0[.\", ... , SIc under 1and I' respectively. 1'-1 0 J is a permuta­ q(m(B), m(C» = q(B, C), tion of a and (;'(.\',)•... , i'(s,,» is the image of (i(Sl)' ... , ;(s,,» under /,-1 /. I-Iellce. (i(sd• ... , ;(s,,» and (i'(sd, , ..• i'(s,,» are similar and where m(B) and m(C) are the images of Band C under m. 0 l(;(s,).... , i(s,J)] = l(i'(sd, ... , i'(s,,»]. That is, [(;(S1)"'" ;(.'1,,»] is DEFINITION 6 Let a be a cardinal number. A 4-parlithm (~r a is a quadruple unique. ({1, y, 0, e) of cardinals such that fl + }' + b + I: = a. ( I.b) Next I prove that h is into (61'11. Let 0(1. be an a-operator of type (1\, ... , I,). Let Bl x ... X B" be a Cartesian product such that for I ~ DEFINITION 7 Let (fl, y, (5, e)a be the class of4-parlitiolls of ex. A cardinality i ~ /(. B = A if Ii = O. and lJj = p(An) if 'i = n #- O. Let C\ll be a func­ jimction on 4-partitions of ex is a function j tion from /J, x ... X B" into {T, F} such that for every (SI' ... , SIc) E 1: (/1, y. b, e)a -+ {T, F}. 1)0111(('.11), C,,,(s\ •...• sd = O(1.[(;(s.), ... , i(s,J)]. where for some indexing lor A by a. i(.\j). I sj 5 k, is the index image of Sj under I. By definition THEOREM 2 (Lindstr61ll 1966.) Let A he a set. Lct:1 bc thc sct ofc:ndinal­ It, ('11 /i(o(J;)' (By (I.a) above. C'.II is well del1ned.) 'We have to show ity functions on 4-partitiolls of a = IA I. Let :j, bc the set of 2-placc tillallti­ or that C'11 is indeed a logical term restricted to '11. In particular, we have to fiers on A. Then there exists a one-to-one function II from ,'1 onto ;?) defined as follows: show that C'.II satisfies the restriction of condition (E) of chapter 3. section 6 to'll. That is, if (SI' ... , s,,) and (s~, ...• s~) are in Dom(C9,) and For any ( E Y, h(t) = the 2-place quantifier q on A such that for any (A, s" .... .'I,,) ~ (A. s~ • ... , s~), then C'1I(Sl"'" SIc) = C'lI(S~, ... , s~). B. C ~ A, q(B, C) = (IBn CI, IB - CI, IC BI, fA (Bu C)I). Take any indexing 1 of A by ex. For I 5}::; k, let i(sj) be the index image Given a 2-place quantifier on A. Q2. I will call the cardinality function t of Sj under I. By delinitioll, satisfying the above equation the cardinality counterpart (~l Q2 and SYIll­ l:>olize it as IQ. C ~I (.~ ,. .. , Sk) = () 0: [ ( i (.'11 ), ... , i (Sk ) ) ]. C\l\(s;, ... , s~) = oo:[ (;(s;), ... , ;(s~ »1· DEFINITION 8 A 2-place quantifier Q2 is a function that assigns to each It sullkes to show that l(;(s,)•... , ;(sd)l = l(;(s'.), ...• i(s~»l· Let f universe A a 2-place quantifier on A, Q!, and such that if A. A' are he an isomorphism of (A. ,\'\ •...• ,\'k) onto (A,,\'~, ...• s~). Thus.r is a universes of the same cardinality. then Q~ and Q~, have the same cardinal­ permutation or A. Define a permutation m of a as follows: for all ity counterpart. /1 E a, Chapter 4, Section 2 m(/I) )' ilfI(op) = lly. ('Ieady. (;(J~) •...• i(s~» is the image of (;(SI)' ... , i(Sk» under the per­ Proofoftheorem 1 The proof is straightforward because we have already mutation induced hy m. llence. (;(SI)' ... , ;(Sk» and (i(s;), ... , i(5~» are introduced all the concepts connecting the ordinal structures over which similar. Therefore, l(i(sd..... ;(Sk»] = l(i(s;), ...• i(s~»l· a-operators are defined with structures within ~( over which logical terms (2) The next step is to prove that h is onto (efl~l. Take any C~ll E rei'll· restricted to'll are defined. I will prove The claim is that there is an 0a E (()a such that 11(0(1) C'1t· Let the type of 145 Appendix 144 Appendix C'11 be </1"'" t,). Consider the totality of a arguments R(IX) <r1 (IX), I: a -+ (/1, ... , rlc(a» ofthe same mark. Let 19l(a)] be the set ofall equivalence classes and Ict .'F he the set of sllch functions. For any'/; I' E ff, the functions f of R(a)'s under the relation of similarity. Define and/' arc similor ifl' there is an automorphism 111 of a such that for every (}a : [9l(a)] -+ {T, F} () E a, as follows: Let I be some indexing of A by IX. For any [<'1 (a), ... , rk(a»] E ) = I' (111«) Dom(oa), 0a[ (a), ... , rlc(a»] T iff for some structure <A, sI' ... , '~k) <r l = DEFINITION 12 Let a be a cardinal number. Consider the functions f such that s;, 1 $; i .:s; k, is of mark t , (I) and (2) below hold: i defined above. Let [:~] be the set of equivalence classes [f] under the (I) rj(a) i(sj), where I .:s;j .:s; k, and i(sj) is the index image of Sj under I relation of similarity defined above. Then a hinary cardinality function on (2) C 91 (S" ••• , Sic) = T ex is a function I have to show that 0a is a well-defined a-operator. Let <rl (a) • ... , r/«a» k'J : [,'17] -+ {T, F}. and <r~ (a), ... , r~(a» be two similar a-arguments of mark <111 , ••• , I1I ). 1 k TIIEOREM 2 (Higginbotham and May 1981.) Let A be a set. Let f be the Let the corresponding structures within VI be <.'II' ... , .'I,) and <J~, {jk) .... set of binary cardinality functions ka on a = 1A I. Let!l be the set of I-place respectively. It suffices to show that Cyl(SI •... , = Cy,(s;, ... , s.). S,r> quantifiers on binary relations over A satisfying the invariance condition But this follows from the fact that C'11 is a logical term (restricted to 'II) (h. I ) (p. XX). Then there exists a one-to-one function It from J{ onto !l and the fact that the structures (A, .'II' ... , s/<) and (A, s;, ...• ."k) are defined thus: isomorphic. (That the structures <A, SI' ... , s/<) and <A, s~, ... , sk) are isomorphic is obvious: By definition of similarity. there is a permutation For every ka E .~', h(ka) the quantifier qA E !l such that for any R ~ ;12, l/A(R) ka([./~])' where.l~: a -+ (P, Y)a is defined, relative m of a such that <a. r l (a), ... , r/«a» ;; <a, r; (a), ...• r~(a». where for to somc one-to-onc and onto indexing i of A by a, as foll'ows: for every I .:s;j $; k, rl(a) is the image of rj(a) under m. Letfbe the corresponding () E a, t~({» (1:, (> ifl' 1-: 1 {a: <1I~, a) E R}I and ( = I{a: <a~, a) ¢ R}I. permutation of A. That is, under the given im.lexingf(ap ) = {i}, iffm(//) 2 y. Now <A, .'II' ••• , Sic) ;; <a, r l (a), "', rlc(a», and <A, s~, ... , Jk) ;; I say that.l~ represenls the cardi1lalilies ofR ill A (relative to i). <a, r; (a), ... , r~(a». Therefore, <A, SI' ••. , Sic) ~ <A, .'I;, ... , sk).) LEMMA I Let A be a set. Let m be a I-automorphism of A2, and let In t be (3) Finally, I prove that h is one-to-one. Let 0 la :f= 02a • Then for some an automorphism of A such that for all a, hE A, mea, h) = (ml (a), hi) for R(a), ola[R(a)J ::/; 02a[R(a)J. Say olaIR(a)] = T, 02a[R(a)J = F. Suppose some h' E A. Let R, R' be two binary relations on A such that R' = meR). "(ola) = CI \/" II(02a) C2<]1' We have to show that C).~I ::/; e2 yl . Let I be an indexing of A by a and let S be the structure within VI determined by Then for cvery a E A, R(a) through I. Then by definition of fl, CI <lI(S) = T, C2 yl (s) = F. llence, (I) l{hEA:<lI,h)ER}I= E A: <1111 (a), hi) E R'}I, el ::/; y,.