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Chapter 5 Σ-Definability of algebraic structures
Chapter 5 2-Definability of Algebraic Structures Yu. L. Ershov
The notion of the N-definability of an algebraic structure in an admissible set was introduced in the author's article [4] as a generalization of the notion of constructivization. Below, we use notions and facts from the theory of admissible sets or, generally speaking, KpU-models, following [5]. All the model-theoretic notions we use can be found in [2, 6]. Let A be a KpU-model and u" B -+ M be an A-numbering, i.e., a mapping of a N-subset B of A onto M. D e f i n i t i o n 1 An n - a r y predicate P C_ M n on M is called (i) a 2~-predicate if {(b,,...,
b , ) ] b G B", (ub,, . . . , ub,) e P} -
u-l(P)
is a E-predicate; (ii) a A~-predicate if P and M " \ P are F~-predicates. Let 9)t = (M, P0~, . . . , Pff) be an algebraic structure with signature (p~o . . . , p ~ . ) (for the sake of simplicity we consider here only the case of relational signatures). D e f i n i t i o n 2 An A-numbering u 9 B -+ M of the universe of ffJ[ is called an A-constructivization of the structure 9I[ if the equality predicate and all the predicates P0~, . . . , P ~ are A~-predicates. A structure ff)t for which there exists at least one A-constructivization is called A-constructivizable. A pair ( ~ , u ) , where u 9 B --+ M is an A-constructivization of the structure 9J[, is called an A-constructive structure. H A N D B O O K OF R E C U R S I V E M A T H E M A T I C S Edited by Yu. L. Ershov, S. S. Goncharov, A. N e r o d e , a nd J. B. R e m m e l O 1998 Elsevier Science B.V. All rights reserved.
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R e m a r k 1 In the case where A is IHIF(0), the notion of an A-constructivizable structure coincides with that of a constructivizable structure, while (if IHIF(0) and w are identified as in Section 2.1 of [5]) the notions of an A-constructivization and of an A-constructive structure differ from the well-known ones only in the fact that an arbitrary nonempty recursively enumerable set (and not only w) can be taken as the domain of the constructivization. It is possible to give an equivalent definition of an A-constructivizable structure which is based on the notion of "definability" well-known in model theory. Let 9)t = ( M , P0~, . . . , P ~ ) be an algebraic structure with signature ( P o ~ . . . , PuSh). ffYr is called definable in A if there exist formulas
Definition 3
~o(Xo) , ~i(Xo,
xi),
~o(Xo, ...,
X~o-i) , ...
, ~(Xo,
...,
x~o-i)
(with parameters in A), such that _/lIo - - ~oA[Xo] r 0 and 7/-- ~)[Xo, Xl]NMg is a congruence on the algebraic structure
(Mo,
P o),
where p~o
=
r
N Mo' ,
i~
and the structure 9)t is isomorphic to the quotient structure 9Yr (In this case we say that the system of formulas ~o, 1J~l , l ~ O , ' ' ' , O n defines 9)t in A.) We recall that 71 is a congruence on 9)I0 if 7/ is an equivalence on M0 and, for any i <~ n and d o , . . . , am,-1, bo, ... , bin,-1 E ~]o such that (aj, bj) E for j < mi, we have
(a)
p o.
Definition 4 A structure 9)l is called E-definable in A if there exists a system of E-formulas ~o,
~1, r
...,
~,
~,
* (I)o, ... ,
(I)*
such that ~o, ~1, ~o, ... , (I)~ defines 9)l in A, and ~A[Xo, x~] N Mo2 *A (I) i [ X 0 ,
where M o -
...
,
Xmi_l] N M o'
~oA[Xo].
-
Mo2 \ ~ [ X o , Xl],
Mo'-.
and
xm,-1],
i~n,
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R e m a r k 2 Let Mo C A, and let P C M~ be an n - a r y predicate. If we refer to P as a A-predicate on Mo in the case that there exist E-formulas (I)(xo, . . . , x~_,) and (I)*(Xo, . . . , Xn-l) such that p
_
q~A[g] n
M~
and
M~ \ P -
r
n Mg,
then it is possible to give another equivalent definition of E-definability. (The E-formulas ~T, (I)~), . . . , (I)~ in the above definition "confirm" the equivalence.) D e f i n i t i o n 5 A structure 9Jl is called E-definable in A if there exists a sequence of E-formulas qJo, qJl, (I)o, ... , (I)n which defines 9)t in A, and for which the predicates ~lA[xo, xl], . . . , O/A[~], . . . , i ~< n, are A-predicates on M o - OoA[Xo]. 1 An algebraic structure ~}'A is A-constructivizable if and only if it is E-definable in A.
Proposition
P r o o f . Let ~ be E-definable in A, and qJo, ~1, (I)o, ... , (I)~ be a sequence of N-formulas which defines 9~J[ in A;
Mo-~ ~o~[Xo], -~ ~ [ x o , x,] n Mg, p~o ~ e t [ x o , . . . , xm,_l] n Mo',
i <~ n,
PY").
Let T : ~YAo/q --+ ~J[ be an isomorphism. We define an A-numbering u : Mo --+ M by setting u(mo) ~ ~([mo]), mo E Mo, where [mo] is the class of elements of Mo that are r/-equivalent to the element too. A routine verification shows that the A-numbering constructed in such a way is an A-constructivization of the structure gJt. Conversely, let ~)t be A-constructivizable, and let u " B --+ M be some A-constructivization of the structure 9Jt. Since B is a E-subset of A, there exists a E-formula qo(xo) such that B - qoA[Xo]. Since u " B --+ M is an A-constructivization of 9Jt, the relations
Yu. L. Ershov
238
flu ~
{ (bo, bl) I bo, bl e B, ubo - vbl},
Qo ~
{(bo,...,
Q~
---" {(bo, . . . ,
are A , - p r e d i c a t e s .
bmo-1) I bo, . . . , bmo-1 C B , (t.'bo, . . . , ubmo-1) E PogX},
bin,-1)I bo, . . . , bmo-1 E B, (t/bo, . . . ,
t,'bmn_l) C Pn9Jr}
Consequently, there exist E - f o r m u l a s I'I/I(Xo, X l ) , kI/1 (Xo, Xl), r
...
, X~o-1),
%(xo, . . . , Xmo-~),
r
. . . , xm~_l),
(~;(Xo, . . . ,
Xmn-1)
such t h a t
Qo
Qn
--
II)IA[Xo , Xl] ,
-
(I)oA[Xo,...,/too-l],
U 2 ~ 7-]
--
III1A [Xo, Xl],
B m~ \ Qo
--
r
9
o
9
o
--
r n[xO, --- , Xmn-1],
B m" \ Qn
-
. . . , Xmo_l] '
(I)nA[xo, -.. , Xmn-1]"
It is not hard to verify t h a t the s y s t e m of E - f o r m u l a s ~ o , ~ 1 , ~ o , . . . , (I)~ defines g)t in A. T h e existence of E - f o r m u l a s q ~ , q)~, . . . , (I)~ shows t h a t 93t is E-definable in A. El T h e following claim is an interesting consequence of this (uninteresting) proposition. Let al - (0, E2).
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P r o p o s i t i o n 2 If an algebraic structure 9Yt of signature a is A-constructivizable, then there exists a KPU-model I~ of signature al U a with set of urelements M , such that (~ r M ) I (r = 9~, and ]B is E-definable in A and "has the same ordinals" as A does. If A is an admissible set, then ]~ is also an admissible set. For the proof of this proposition we refer the reader to Section 3.4 of [5]. C o r o l l a r y 3 If A is an admissible, set and the algebraic structure 93I is A-constructivizable, then the ordinal o(A) is 9'Jr-admissible. The definition of an 9~-admissible ordinal can be found in [1]. It is wellknown that if 9~t is a constructivizable algebraic structure, then the 3-theory Th(gJt)3 of this structure is recursively enumerable. It is easy to see that if 9Jr is constructivizable, then 1HIF(gY0 is also constructivizable. Hence the 3-theory Th(IH[F(gY0) 3 (and moreover, the E-theory Th(lHIF(9~t))r.)of the admissible set ]HIF(gY0 is recursively enumerable. If 9Y~ is A-constructivizable, and A is a sufficiently "large" admissible set, it is possible to indicate, using infinite formulas, a larger fragment of the theory of the structure 97t which is effectively "enumerable" in A. We outline a description of the syntax and semantics of this E*-logic of an admissible set ~ of signature (0, C2;cb, b E B , P o ~ p~n). The variables VB = {xb I b C B} are coded as follows: Lxba --~ (0, b), b C B. The constants are coded as follows: Lcb- ~ (1, b), b E B (we assume that co = 0). We first introduce the notion of a A~-formula. Simultaneously, we define the code of a A~-formula. (1) By elementary formulas we mean formulas of the forms to = t l, to E t l, and P i ( t l , . . . , tin,) for i < n; they are A;-formulas and have the following codes: Lt0 = tlJ --- (2, 0, L/0J, LtlJ), Lt0Etlz ...,
t,.,),
= (2, 1 , L t 0 J , - t l j ) , =
(2,/+
2, , t , , ,
...,
,t=,,),
i<~ n.
(2) If (I) is a A~)-formula, then --,0 is a A~)-formula and its code is (3, LO_J).
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Yu. L. Ershov
(3) If 9 is a family of A~)-formulas such that LO_J "-- {L(I)J I o c o} c 8, then A(O) and V(4~) are A~)-formulas, and their codes are defined as follows:
~A(e)~ "--(4, ~e~), ,v(e)~ ~-(5, ~e~). (4) If (I) is a A~)-formula, x is a variable, and t is a term, then Vx E t (I) and =Ix E t (I) are A ; - f o r m u l a s , and their codes are defined as follows: LVx E t (I)j "- (6, LX_J, LtJ, L(I)J),
L~tX E t (I)_l ~
(7, LZA, Lt..J, L(I)_I).
Something is a A~)-formula if and only if it is contained in the set generated by rules (1)-(4). It is easy to establish that the set A o of all codes of A~)-formulas is a A - s u b s e t of B. We now introduce the notion of a E*-formula. (1) Every A~)-formula is a E*-formula, and its [;*-code coincides with its A;-code. (2) If 4~ is a family of s then h(4~) and V(4~) are s as follows:
such that L~_J ~ {L(I)J I ,:I:,c ~} c B, and their N*-codes are defined
~A(e)~ ~ - ( 4 , . e ~ ) , ~v(~)~--(5,
~).
(3) If (I) is a E*-formula, x is a variable, and t is a term, then Vx E t (I) and =Ix (5 t (I) are P~*-formulas and their codes are defined as follows" ~Vx E t (I)~ - -
(6, ~x~, LtJ, ~'I)-),
L3X E t (I)j ~
(7, LXJ, LtJ, L(ID/).
(4) If (I) is a E*-formula, and X is a set of variables such that LXJ {LXJ IX C X} E B, then (3, X , (I)) is a E*-formula, and its code is defined as follows"
,(3, x , ~), -- (8, , x ~ , ~ ) .
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241
The set E* of all codes of E*-formulas is a A-subset of B. For any E*-formula (I), it is natural to define the set FV((I)) of free variables of (I). For any such formula, LFV((I))j C B. To any element a C B, we can assign a mapping % : V --+ B from the set of all variables {Xb [ b E B } into B by the following rule: given xb, if there exists a unique element c C B such that (LXb_J, C) = ((0, b), c) C a, then %(Xb) ~ c, otherwise 7a(Xb) ~ f). By induction on E*-formulas, we define the relation B ~ 4)[%] of the truth of a formula (I) in B under the interpretation %. This is defined as usual. We must only specify the case in which (I) = (3, X , ~). B ~ (I)[')'a] is true if and only if there is a function f : X --+ B such that f C B and B ~ ~[%,] for a' --~ (a \ X x B) U f . The following theorem on the E-definability of the truth of E*-formulas is not hard to prove. T h e o r e m 4 The predicate Trs. ~ {(-(I)-,a) I (I) is a E*-formula, It~ ~ (I)[%1} is a E-predicate in B. C o r o l l a r y 5 If 9)t, A, and I~ = l~(gJl) are the same as in Proposition 2, then the E*-theory of g(9)l) is a "E-subset" of A. In the sequel, our attention will be focused mainly on the E-definability of structures in admissible sets of the form IHIN(gJt) for models 93t of sufficiently simple theories T. It is useful to obtain beforehand information on definability in such structures. The following assertion turns out to be useful. P r o p o s i t i o n 6 If g) ~ HF(gY0, where g) is an elementary submodel of HF(9)I), then g) has the form IHIF(9)t') for a suitable model g)t' 4 9)t. We give a sketch of the proof of tile fact that predicates of the system 9)t which are first-order definable in HF(gN) can be described by (infinite) formulas of the language L~, ,~. Let n E co. We set
2)n ~
IHIN(n(= { i l i C n})),
.% =
HIF(O)
U .~,, = ~ ( ~ ) .
nEo.,,
c_ .9~ c_ . . . ,
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Yu. L. Ershov
For any n E w, x C Dn, ~ E M" we define an element x ( ~ ) C H F ( M ) in the following way. Let A~" n --+ M be defined as follows" A~(i) - m i , i < n, where ~ - (m0, ... , ran-l). The mapping A~- can be uniquely extended to a mapping A~ " .~n - H F ( n ) --+ H F ( M ) , so that A~({ao,...,ak}
--
{/~(ao),...,
/~(ak)}
for any set (not u r e l e m e n t ) { a o , . . . , ak} e IHIF(n). Let
Given any element x E D~, it is possible to define effectively a term ...
,xn)
of the signature (0, { } ' , U:) such that, for any m,)
-
ml,
...
, /7/n e
M,
,(m).
Let T(5, ~) be a formula of the language of the signature al U cr', where a' is the signature of the structure 9)t. We assume that the variables in the list correspond to urelements of the structure IHIF(9~) (i.e., to elements of the structure 93t), and variables in the list y correspond to arbitrary elements of IHIF(ff)t). The formula ~ . ( 5 , y ) of the language n~, ,~ of the signature o" U (~, e 2 , { }1, U 2) is constructed from the formula ~o as follows: (1) if r does not contain quantifiers, then ~. --" r (2) if ~ - ~o q Vl, then ~p. - - (90o). q (~1)., for q @ {A, V, -+}, (3) if ~o - -'~0, then ~. ~ -~(~o0)., (4) if ~ - Q x~o, then V. - - Q x ( ~ o ) . , (5 / if V - 3 y~o, then r
for Q c {v, 3},
Y --" V,,e~ ( V,,e~. 3 x , . . . ::Ix,., ((Vo)*)t,.(~)),
(6) if ~o - Vyvo, then ~% ~ Ane~ ( Axe~. v/x "-" Vxn ((qo0).) t,,(e))" y If qa(~) does not contain variables from the list ~, then neither does T.(~). For any terms to and tl of the signature a ' U (r { }~, U2), with variables from the list ~, it is possible to write effectively quantifier-free (finite) formulas
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E-Definability of Algebraic Structures
9 to, * t~ (7) and (fro,it(-2) of the s i g n a t u r e ~r' such t h a t to - t, - ~ ( ~ ) ~*to, t] and to r tl - - ~ - ( ~ ) d~to,t ~. For any pair of t e r m s to a n d tl of the s i g n a t u r e ( 0 , { }1, U2), w i t h variables f r o m the list ~, we define f o r m u l a s (bto,t, and ~to,t~ of the e m p t y s i g n a t u r e such t h a t
FV(to)u VV(t ) and t(oH~'(~),{ },u)[3'] E t~ H~(~)'{ },u)[3'] r
19y~[ ~ (I)to,tl[3'],
t(o~'(~),{ },u)[3'] c_ t~ M~'(~)' { },u)[3'] r
or
for any i n t e r p r e t a t i o n 3' " F V ( t o - tl) --+ [971~[. Let 7- d e n o t e the s e n t e n c e 3 u(u - u), and let Uo a n d Ul r e p r e s e n t variables f r o m the list 7. We define for to - Uo: if
Ul,
then
(I)uo ,u, ~
if tl - O,
then
(I)uo,O ---" --,7", ~uo ,0 ---" T,
if t l -
then
(I)uo,t, ---" ~t~,~,o,
tl
-
{t~},
if t~ - (t~ U t~'), t h e n
-7,
[I/uo ,Ul ~
(uo - Ul),
~uo,t, ---" 7-,
(I)~o ,t, ~ (I)~o ,t~ V (I)~o ,t~', qJuo ,t, ~ r;
for to - r if tx
-
then
(I)~,,ttl
if
-- r
then
(I)~,~ ---" - r ,
if ta - {t~},
then
(I)~,t, ~ ~t~ ,~ A ~ r
tl
Ul,
if tl - (t~ U t~), t h e n
._..a ....IT ,
l~I)~,~l
~,~
~_..-..IT ,
~ r, qJ~,t, ~ r,
(I)o,t, --" (I)o,t~ V (I)o,t7 , ~ r
---" 7-;
for to - {t~)}: if tl - Ul,
then
(I)to ,~,~ ~ --,7", ~to ,~,~ ~ -~7,
if tl - 0,
then
(I)to,O ~ --,7-, ~to,O ~ --,7",
if tx - {t]},
then
(~to ,t~ ~
if t~ - (t] u t~'), t h e n
~to,t~ /~ ~t'~ ,to, ~to ,t~ -- (~t~ ,t~,
(~to ,t, "- (bto ,t~ V (~to ,tT, q~to ,t, " - (~t~ ,t,;
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Yu. L. Ershov
for to - (t; 0 tg)" if tl - 721,
then d~to,u1 ~ "nT, I~to,u , ~ "nT,
if tl - r
then Oto,O ~ - - r ,
if tl - {tl},
then d~to,t 1 ~ l~to,t~,
klIto,r
kX/t~,r A II/t~,r ,
l~to,t 1 ~-- IJ~t~,t 1 A ~t~l,tl,
if t, - (t~ U t~), then Oto ,t, ~ Oto ,t~ V (IIto ,t~', /Ilto , t 1
.._.x l~tp0 ,tl A l~ttoPt 1 9
W e set ~*to ,t, --7~--IX/to,t, A It/t, ,to. Using t h e f o r m u l a s ~to ,t, a n d (~to,t,, we transform the formula ~ . ( : ) into a formula ~*(:) of the language L~, ,~ of the signature a ~ such that
Then ~HF(~)[g] _ qp.gn[g]. Using this transformation and Karp's theorem [7, Theorem 9.10], we arrive at the following proposition. P r o p o s i t i o n 7 Let 93to and ~}:J[1 be w-saturated models. If 93lo- ~ 1 , IH[F(gJ/o) - I[-IIF(~J[1); if 93to ~ ~ 1 , then IHIF(gJto) 4 IHIF(gJ/a).
then
D e f i n i t i o n 6 A model 9)1o is called saturated enough if there exists an w s a t u r a t e d model ~f~l such that 93t0 4 ~J~l and IHIF(9~0) 4 IHIF(~J~ 1). C o r o l l a r y 8 Let 9Jto and 9Jh be saturated enough models. If 9)lo - 93tl, then InIF(gYto) - I[tIF(9~I). If 9Jto ~ 93tl, then InlF(gJto) ~ HF(g)t1). P r o o f . The assertion follows from the definition, Proposition 7, and the existence of w-saturated models, rn R e m a r k 3 In any saturated enough model 97t, any (not necessarily complete) arithmetic type of formulas with a finite number of quantifier alterations is realized over a finite M0 C_ [9")t[. The author does not know whether this condition is equivalent to being saturated enough. R e m a r k 4 Any complete theory T (of a countable language) with infinite models has a countable saturated enough model. This fact follows from the Lgwenheim-Skolem theorem, Proposition 6, and the existence of w-saturated models of the theory T. Any complete categorical theory T has sufficiently many w-saturated models.
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245
P r o p o s i t i o n 9 If a theory T is complete and w-categorical, then any model
of T is w-saturated. If the theory T is complete and Wl-categorical, then any uncountable model of T is saturated (and, in addition, w-saturated). P r o o f . The first assertion follows from the Ryll-Nardzewski theorem on the characterization of w-categorical theories (see [2, Theorem 2.3.13]), or more exactly, from the fact that there is a finite number of types over any finite set. The second assertion is well-known. (see, for example, [2, Corollary 7.1.15]). E! R e m a r k 5 Not all countable models of Wl-categorical theories are saturated enough. For example, the following claim is true.
Let T be the theory of algebraically closed fields of characteristic O. If Fo, F1 ~ T, then tlI F(Fo) - tlIIF(F1) if and only if either Fo and F1 both have infinite transcendence degree over Q, or the transcendence degrees of Fo and F1 over Q are finite and equal. If ho, hl E IHIF(ff)t), then the types t~(~)(ho) and t~(~t)(hl) coincide if and only if there exist n E w, x E ~,~, ~ o , ~ : E M '~ such that ho - x(-~o), h: - X ( ~ l ) , and Theorem
10 Let 93t be a saturated enough model.
Proof.
Necessity. Let ho, hx E IHIF(9)t) and t~(~)(ho) = t~(~)(hl). Let sp ho (the support of ho) have n elements: sp ho - {rn~ ... , mn_lO}. It is easy to see that the support of h l also has n elements, and there exists x E ~)~ such that ho - x(m---o),~-o ~ (too, ... , ran_ o 1), and h: - ~ f ( ~ l ) for a suitable ~ 1 ~ ( m l , ... , ran_:). 1 Furthermore, for any formula 99(g) (5 t~(~-~ there exists a permutation a E Sn of the set n such that q0(g) E tgyt(a(ml)) where ( r ( m 1) ~ (m~(0) 1 ~ . . . ~ m a1( n - 1 ) ) and hi - - X(O'(~I)) 9 But then there exists a E S~ such that h~ = x ( a ( N 1 ) ) a n d t ~ ( m --~ - t ~ ( a ( N ' ) ) . Indeed, if such a does not exist, then for any a E Sn such that hi = x ( a ( ~ l ) ) there exists qp~(g) belonging to t ~ ( ~ -~ but not to t ~ ( a ( N 1 ) ) . Let ~P(x)--" A ~ s z ~p~(x), where
246
Yu. L. Ershov
Since ~po(5) E t ~ ( m --~ for all a E Sn~, it follows that ~(5) E t~(m---~ As already mentioned, there exists ao E S~ such that ~(g) E t~t(ao(m~)), but ~,o(E) ~ t~(ao(Nl)). We arrive at a contradiction. Hence there exists a E Sg such that =
and h, = x ( a ( N ' ) ) . SuiOiciency. Let z E . ~ , ~---0,~-1 E M ~, and t ~ ( ~ -~ = t ~ ( ~ ) . Using Propositions 5.1.7 (ii) and 5.1.8 of [2], we find an elementary extension 0Jr' of the model 0Jr which is a special model.
Since 9)t' is w-saturated, it follows that IHIF(gJt) ~ IHIF(gYr Then t~,(m --~
= t~(m ---~ = t ~ ( ~ 1) = t ~ , ( ~ 1)
and, since 9Yt~ is special, there exists an automorphism ~ of 9)t~ such that ~(m---o) = ~1. We extend p to an automorphism ~ of the model ~ = IHIF (9~'). Then =
Consequently, t~)(x(m---~ = t g ( x ( ~ ) ) ,
and
The theorem is proved. By special admissible sets we mean admissible sets of the form IHIF(gYt), where OR is an algebraic structure which is a model of a simple theory T. The preceding shows that many questions about admissible sets of the form II-IIF(FJt) can be reduced to the corresponding ones about the structure 9Yr On the other hand, such admissible sets present examples of structures for which there is a natural notion of computability without conditions on the cardinality of structures. Thus, if IR denotes the field of real numbers, then the notion of E-definability in IHIF(R) is a reasonable "computability over IR". However, the notion of "simplicity of a theory" requires refinement. One of the required features of a simple theory is its regularity in the sense of the following definition: D e f i n i t i o n 7 A theory T of the (finite) signature a ~ is called regular if it is decidable and model complete.
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R e m a r k 6 In a model complete theory T any formula is equivalent to an 3-formula (see [2, Proposition 3.17]). If r is also decidable, then such an 3-formula can be found effectively. The theories Th(R) and Th(Qp) of the fields of real and p-adic numbers, respectively, provide examples of regular theories. However, it is hard to consider such theories as simple ones because they have many nonconstructivizable countable models. D e f i n i t i o n 8 A theory T is called simple if it is a regular w-categorical theory whose set of complete formulas is decidable. The condition of w-categoricity means the uniqueness (up to an isomorphism) of a countable model of the theory. The model completeness and the decidability of the set of complete formulas guarantee the autoequivalence of all constructivizations of this countable model (see [31), i.e., the "uniqueness" of the computability theory for countable models of such theories. We now turn to the question of the possibility of (E-)definition of classical objects, the fields C and N of complex and real numbers, in special admissible sets. We first consider the case of the simple theory To of an infinite structure with empty signature, i.e., models of the theory To are presented by infinite sets without any additional structure. We assume that some uncountable algebraic structure 9Yt = ( M , ... ) is defined in IHIF(S), where S is an infinite set. Let do, ... , am C S be all the urelements appearing in the definition of 9J[ as parameters and let ~ ~ (a0, . . . , am). Let ~o(X0) and ~l(X0, xx) be formulas from the definition of 9)t in ]I-]IF(S) such that Mo --" ~t~(s) ~o [Xo], r/ ~, [Xo, x,] M Mo2 is an equivalence on Mo, and there exists a oneto-one correspondence (induced by an isomorphism between ~ and 9Jto/r/) between M and Mo/y. With any n E w and x E -~n, we associate the sets
T.
e s",
M,, ---" { x(~, b) I ~ E T,,} / ((~ r {x(~, and the relation
b) lb e
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248
Then M,~ C_ M0, 5,~ is an equivalence on T,~, and IM,~[ - [T,~/5,~ I. We note that since S is saturated enough, Theorem 10 implies that the set T,~ and the relation 5,~ are closed with respect to types over ~, i.e., if b C Tx, g E S" and t(s,~)(-b) - t(s,~)(-b'), then b' E T,~, and if (~o,~) E 5,~ and t(s,~)(b~ ~) - t(s,~)(~,-d ~) for some ~0,31 e S", then (3o,~) C 5x. It should be noted that the following obvious relation is valid: Mo -
U
U
M~ -
n ~,~
M,,.
U ~ e H ~(~o)
Since M is uncountable, we can find n C w and x C -~n such that M,~ is uncountable. Then there exist pairwise unequal b~
( b l , . . . ,b~) e S \
{ao,...,am}
so that 1 4- m 4- k = n, x(K, b~ x(~, b 1) E i 0 , and (x(~, b~ x(K, bl)) q~ r/, w h e r e -0 b ~ (hi~ . . . , b~), 51 _.~ (bl, . . . , b~). Without loss of generality, we can assume that there exists a unique i, 1 ~< i ~< k, such that b~ r b1. Indeed, let us consider the following sequence of collections of k elements. b~
b~ . . . ,
bl, b~
b~, ... , bl, bi~
~1
b~, . . . ,
b~
...,
bOk;
b~r
Each collection differs from the neighboring one by at most one element, so if for each neighboring pair b, b' the pair (x(~, b), x(~, b')) belongs to q, then the pair (x(~, ~0), x(K, b~)) must also belong to 7/since 7/is an equivalence. We note that x(~, b) E M0 for any collection b from this sequence. This follows from the fact that the types of the elements x(~, ~o) and ~(~, ~1) coincide.
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249
Hence we can assume that there exist b~ , . . . , bk-~ , bk , b'k or_ S \ {ao, . . . ,am} such that
bk-1, bk ) , ~'(-d, bl , . . . , bk-1, b~k) } ~ 7].
Now for any b 7(: b' ~ {g, b l , . . . , b k - l } , the fact that
t(~,~(b,,..., b~_,, b~, b,,..., b~_,, b~) - t(~,~(b,,..., b~_,, b, b,,..., b~_,, t,') implies (as mentioned above) that
(~(~, b,,..., b~_,, b), ~(~, b,,..., b~_,, b')) ~ ,7It should be noted that any permutation a of the set S uniquely defines (lifts up to) an automorphism a* of the admissible set IHIF(S) where
~'({~,,... , ~ } ) ~ {~'(~,),... ,~'(~)},
{~,,... , ~ } e ~ ( s ) . .
s.
If a r {a} - id{~}, then a* induces an automorphism on the structures 93t0 and 9Jl:0/rl. The following proposition is a consequence of the above considerations. P r o p o s i t i o n 11 I f an uncountable algebraic structure flit is definable in IHIF(S), then the a u t o m o r p h i s m group Aut(gJl) o f the structure 9)t contains a subgroup G that is i s o m o r p h i c to the s y m m e t r i c group Sym(S) of all perm u t a t i o n s of the set S .
P r o o f . Indeed, if x, ~, bl, ... , bk-1 have the same meaning as above, then any nontrivial permutation a0 of the set So ~
S \ { ~ , b l , . . . ,bk-1}
extended to S by letting ao(~) - ~, ao(b,) = b,, . . . , induces a nontrivial automorphism on 9)lo/7/. Thus Aut(glto/r/) "" Aut(gJl:) contains a subgroup isomorphic to Sym(So) "" Sym(S).
ao(bk-1) -
bk-1,
250
Yu. L. Ershov
C o r o l l a r y 12 The fields C and R are not definable in I-]IF(S) for any set S. P r o o f . Indeed, the field ]R has no automorphisms, and the field C has no automorphisms of finite order greater than 2. rn R e m a r k 7 It is easy to see that if the field IR is definable (E-definable) in the Keu-model A, then C is also definable (E-definable) in A. Hence Proposition 11 is a serious obstacle to the definability of uncountable structures in admissible sets of the form HF(S), where S is an infinite set. We now consider the simple theory T1 the theory of dense linear orders without endpoints. Let L be a dense linear order without endpoints. We describe a general model-theoretic construction in terms of which we state necessary conditions for the definability in H F ( L ) of an uncountable structure (see Proposition 16 below), and a necessary and sufficient condition for the E-definability in H F ( L ) of uncountable models of a given theory (see Theorem 17 below). The category *w is defined as follows. Its objects are the sets of the form [hi ~ {0, 1 , . . . , n - 1}, n E w ([0] ~ 0), and its morphisms are orderpreserving embeddings. It should be noted that there is a unique morphism from [0] into ["l for any n E w. D e f i n i t i o n 9 By a *w-spectrum, we mean any functor S from the category *w into the category Mod~ of algebraic structures (of some fixed signature a) whose morphisms are all possible embeddings. To define a *w-spectrum $, it is necessary to give an infinite sequence !YJlo, 9X1, ... of algebraic structures of signature a, and associate with each order-preserving embedding # : [n] -4 [m] an embedding p. : 9Jr, --4 9Xm so that, if # o : In] --4 Ira] and p, : [m] -9 [k], n ~< m E k E w, are morphisms of the category *w, then (#,#o). = p,.#o., and if # : In] -4 [n] is the unique morphism from In] into In] (= id [,~]), then #. = id ~ . : 9X, --4 9X,, n E w. If the *w-spectrum .5' = {gJt,,/~. I n E w, # E Mor*w} has been defined, then for any linearly ordered set L, it is possible to define the algebraic structure 9XL (if)ts) as a direct limit li_.mg~o of the spectrum L0
{~k0,
~Lo,L,
I L0 C_L,
C L, L1 is finite},
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where 9Ytko --" 9)tn if Lo C_ L is finite and ILol- n, and the embedding ~Lo ,L, 9 ~ l Lo -+ 9Jt~ 1 is defined for finite Lo C_ L1 (C_ L) as follows: if Ll-{/o
and
(in which case 0 ~< io < ix < . . - < as #(j) ~ ij for j < n, then ~L0,L,
--~
Lo-{/io
<...
i~_x ~< m), and # ' [ n ] l
/2-" ~JC~L0 -- ~rf~n --~ ~ m
-~ [m] is defined /
-- ~ L , "
If L C_ L' are linearly ordered sets, then the structure with a substructure of 9Y~L, in a natural way.
~)~g c a n
be identified
R e m a r k 8 Any isomorphism between linearly ordered sets L and L' induces an isomorphism between ~J~g and PYRE,. P r o p o s i t i o n 13 If L C_ L ~ are dense linear orders without endpoints, then ~ f t L --~ ~ f t g , "
P r o o f . For countable L' the assertion follows from [6, Proposition 24.2]. The case of an arbitrary L' is proved by the application of the Tarski-Vaught theorem on elementary embeddings. D C o r o l l a r y 14 If L and L' are dense linear orders without endpoints, then ~[~ L -~ ~ L ' .
Let #o and ~1 be morphisms from [11 into [21 such that po(0) ~-- 0 and #1(0) ~ 1. The condition
r
(*)
is sufficient for 19JtSLI >/ ILl to hold for any linearly ordered set L. Indeed, let ~ C M1 be such that p0.(() 7~ #a.((). For any linearly ordered set L, and 1 E L, we define the corresponding embedding tit 9 ifJilt } = 9Ytl --+ 931L, and let ~, ~ p,(~). Then 1 r l' implies ~t r ~t, and [LI = [{~t I1 E L}[ ~< IML[. The following model-theoretic claim is stated without proof. P r o p o s i t i o n 15 Let S be a *w-spectrum such that #0.(~) r #,.(~) for some E M1, and let L be a dense linear order without endpoints. Then
is a set of indiscernible elements of the structure 9Jtc (with order induced
bu L).
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P r o p o s i t i o n 16 Let 93t be an uncountable algebraic structure defined in NF(L) for some dense linear order L without endpoints. Then there exists a *w-spectrum S such that tto. 5r #x., all the models 9Jto, 9 ' A 1 , . . . of this spectrum are countable and isomorphic, 9J[o ~ 9)tl ~ ... ~ 9)l, and for any embeddin9 tt :[n] --+ [rn], the embedding p. :92~n --+ 92~m is elementary. P r o o f . The preceding analysis of definable sets shows that there exist k E w, ~r C IHIF(k), and sequences g,b C L k that have the same type over (L,~), where ~ are constants appearing in the definition of 9)t in I[-IIF(L) (we can assume that {~} C_ {g}N{b}), such that ~r x ( b ) E Mo and (x(g), x(b)) ~ r/x. Then g -7(:b, and we can show (in a similar way to the proof of Proposition 11) that there exist such g and b so that ai 7~ bi for a single number i < k (aj - by for j < k, j :fi i). For definiteness, we will assume that ai < hi. We suppose that a0 < al < " " < ak-1, and 0 < i < k - 1. Then ai < bi < ai+l. m
_
m
g
By the Lgwenheim-Skolem theorem, we can find a countable elementary submodel 1HIF(L~ 4 II-IIF(L) such that {~} C_ {g} U {b~} C_ L ~ Since L ~ is a dense linear order, there are elements d2, d 3 , . . . , dn, ... such that bi < d2 < d3 < . . . < dn < . . . < ai+l. We set do "--- ai and dl "-hi. Then a i - 1 < do < d~ < d2 < . . .
< d~ < . . .
< ai+l.
We set 8k "-- [dk, dk+l ), k C a,,, and Ln
~
(-oo, do) U ~ o U - " U ~n-1 U [ai+l,OO) -- (-oo, dn) U [1:::/i+1,oo)
(define L0 - (-cx~, do) U [ai+l,
OO)).
From the above definition, it is clear that L~ is a countable dense linear order without endpoints which is isomorphic to L ~ over {~}. Let 92t~ be an algebraic structure definable in IHIIF(Ln) by the same formulas (and parameters ~) as 9R in IHIF(L). For any n E w we fix an isomorphism between linearly ordered sets c2n : ~,~ ~ [dn, dn+~) --4 8o -~ [do, d~) (such an isomorphism exists because 8n and 60 are countable dense linear orders with least elements but without largest elements). For n = 0 we set q;0 ~ ids0. For n, k E w let ~,~,k : 8~ -+ 8k be an isomorphism between linearly ordered sets which is defined as follows: ~ , k - ~k-1 ~ (we note that ~n,n = ids, and ~p,,t = ~Pk,l~,,k for n, k, l E w). Let tt : [n] --+ [m] be a monotone embedding of [n] = {0, ... , n - 1} into [m] = {0, . . . , m - 1} (i.e., p is a morphism of the category *w).
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253
Corresponding to this embedding, we assign the monotone embedding g " Ln -4 Lm uniquely defined by the following relations: I L0 -
idL0,
-fi I Sk -- ~k,,(k),
fork
(we note that L~ - Lo U (~o U --. U 8n-1). It is easy to verify that # ~ ~ is a functor from *ca into the category of linearly ordered sets with monotone embeddings (i.e., i-d[,q - idL. and #A - ~ for # ' [ h i -4 [m], A'[/] -4 [n]). For a monotone embedding # 9 [hi -4 Ira], the embedding ~ " L~ -4 Lm (identical on {~}!) induces elementary embeddings/z'" ]HIIF(Ln) -4 ]HIIF(Lm), and #. 9 9Jt~ -4 9Jtm. It is easy to check that # ~-~ #. i's a functor from 9ca into the category of algebraic structures that are isomorphic to 9910 with elementary embeddings as morphisms. It remains to check that #o, ~- /21-. By definition (do - ai and dl bi), we see that x(~) e L1, tt~(x(~)) - x(~) and tt~(x(~)) - x(b) (since ~l(d0) - dl - bi and ~l(dj) - aj for j < k, j ~- i). Since (x(~), x(b)) ~ r/, for IH[F(L), this fact remains valid for IHI~(L2) ~ lHI~(L). Thus, #0,([x(~)]) [,7"t'(a)lrl r [,7'r -- t21.([~t,(~)]). [[] u
Proposition 16 gives a necessary condition for definability in HIF(L), where L is a dense linear order without endpoints. It turns out that the effectivization of this condition is already a necessary and sufficient condition for E-definability in H ~ ( L ) . D e f i n i t i o n 10 A system of numberings u,~ 9 ca -4 M~, n C ca, is called a computable sequence of constructivizations . . . ,
. . . ,
n e
if the following conditions hold (we assume that the signature a of the structures 9~to, 9)tl, ... is finite and without function symbols): (1) E ~ {(n, mo, ml) l n, mo, ml e w, u n ( m o ) = un(ml)} is a A-predicate on ca, (2) Np ~ { n - ( n 0 , n a , . . . ,nk) i ~ e wk+~, ( v , 0 ( n , ) , . . . ,Uno(nk))EP ~'~ } is a A-predicate on ca for any (k-ary) predicate symbol P C a; (3) for any constant symbol c E a there exists a E-function fc : ca -4 ca such that c~" = vnfc(n).
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254
Every morphism # : [n] --+ [rn] of the category *co is uniquely defined by the n u m b e r m and the subset g([n]) c_ [m]. This remark allows one to define a one-to-one correspondence #* : A -+ Nor*co between the subset A ~ {n In E co, r(n) < 2 t(n)} C_ co and the set Nor*w, provided that n E A is assumed to code the m o r p h i s m p : [k] -~ [l] such that l = l(n) and r(n) is the n u m b e r of the subset #([k]) C_ [l] = [l(n)] in some standard listing of finite subsets of w. (Here 1 and r are such that n = (l(n), r(n)).) It is evident that A is a A - s u b s e t of w. Let S - {9Jtn,#. In C w,p E Nor*co} be a *co-spectrum. D e f i n i t i o n 11 By a constructivization of S we mean any computable sequence of constructivizations
(~o,-o), ( ~ , , - , ) ,
..., (~,,-,),
...,
, e ~,
together with a E-function f : A x w --+ w such that, for any n, m, k C co and tt : [n] -+ [re] e Nor*w, if n* E A is such that p*(n*) = #, then # . r , . ( k ) -- r,~f(n*, k). A *w-spectrum S is called constructivizable if there exists a constructivization for it. Theorem
17 Let L be a dense linear order without endpoints. A theory T
h ~ ~,~ uncountabt~ modal r~-d~fi.~bl~ in H~(L) if and o.lu if t h ~ ~x~t~ a constructivizable *w-spectrum S satisfying condition (,), and such that 99ts ~ T. P r o o f . Let 9)t be an uncountable model of the theory T which is E-definable in IHIF(L) for some dense linear order L without endpoints. Acting as in the proof of Proposition 16, we find x , g , b, L ~ , d2 , d3 , . . . , and so on. Since L ~ is a countable dense linear order without endpoints, it is isomorphic to the set Q of rational numbers with the natural order. From this, the existence of a constructivization u 9 w -~ H F ( L ~ of the admissible set w
R~(L ~
(_~ MF(Q, ~<))
follows. Since M0 is E-definable, the set u-~(Mo) - {n I n e w, u(n) E M0} is a E - s u b s e t of w. The predicate
{(n, m) l n,m e ~, ~(n),~(m) e M0, (~(n), ~(m)) e ~}
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N-Definability of Algebraic Structures
is a A - p r e d i c a t e o n / / - l ( M 0 ) . The predicate
nk>lrti e
//-l(pffJto) __, { ( n l , . . . ,
03, (//(T/l),
. . - ,
//(T/k)) e p ~ o }
is a A - p r e d i c a t e on//-l(M0) for any ( k - a r y ) predicate symbol P C or. The elements d2 < .-. < d~ < . . . , for n E 03, of the set L ~ (for which bi < dn < ai+l for all n C a;) can be chosen so t h a t there exists a G-function d : a; --+ a; such t h a t ud(n) = dn for all n E a;. If the substructures ffJln, n C 03, of the s t r u c t u r e 9Jr are defined as in Proposition 16, then it is obvious that the corresponding *03-spectrum has a constructivization. (We only need to choose uniformly effective isomorphisms W~ : 8n ~ 80, n E 03.) Conversely, let S be a *03-spectrum such t h a t 9Yts ~ r for (any) dense linear order L without endpoints. Let . . . ,
and let f 9 A x w ~ ,J be a constructivization of this *03-spectrum. We sketch the proof of the ~-definability of the model 9)15s in H F ( L ) . Let Lo {/0 < "'" < ln-i} C_ L be a finite subset of the set L,
Lo (for n
--"
-- 0 we set Lo --" 0). Let
ilo
Ik e
'Lo --~ {((io,~o), (Lo,]r P~0
]~O,kl e 03, //n(]%)- //n(~:l)},
{(('E0, kl), . . . , (Lo, km))I kl, . . . , ~m E 0d, (//n(]r
for each ( m - a r y ) predicate symbol P E a.
. . . , //n(km)) E P ~ ' }
For c C a we choose k E 03 so
that c ~n - / / n ( k ) and set c ~ o ~ (Lo, k). T h e n rlLo is a congruence on the structure 9~io - - ( M ~ o , . . . , p ~ o , . . . , c ~ o , . . . ) and the quotient structure ffJtio --" 9 ~ i o /7/50 is isomorphic to 9Jtn. Lo C_ L1 C_ L be finite subsets of the set L, Lo =
{/o < " "
< ln-1},
L, = {t; < . . .
<
!
Let
Yu. L. Ershov
256
and 0 ~< j0 < "'" < jn-1 < m be such that li - l}, for i < n. Now let # ~ PLo,L, 9 [n] --+ [m] be the morphism of *w such that #(i) - ji for i < n. Then the embedding #, 9 9)~n -+ 9Jim defines the embedding #---, " 9J~L0 --+ 9YtL1. On the set M ~ --" (.J {M~o ]Lo C_ L is a finite subset of L}, we define the equivalence relation 77 C (M~ ~ as follows. Then for (Lo, k}, (L,, l) E M ~ we set ((To, k), (L,, l)) e 7] if
(PLo,L~). ([(Lo, K)]nLo) -
(,L,,L~). ([(Ll,l>lvL,)
for L2 ~ Lo U L1. It is easy to verify that 7] N(M~o)2 _ r/g~ for finite Lo C_ L. Taking into account that Lo C_ L1 C_ L2 C_ L implies/Ago,L2 [2L,,L2/Ago,L, (for a finite L2) and using the definition of a *w-spectrum, we obtain the following equivalence" -
-
((Lo, k), (L1,1)) E r/ ca there exists a finite subset L2 C_ L, such that Lo U L1 C_ L2, and (pgo,g2).([(Lo, k)],Lo) -- (#L,,g2).([(Ll,1}],7,., ). Owing to this equivalence, it is not hard to check that r/is an equivalence relation on M ~ Furthermore, 7/is a congruence on the algebraic structure 9~
~
(M~
]LoC_Lisfinite},...,c
O,...).
The quotient structure 9Yt~ isomorphic to 9)~s. It is not hard to verify that the set M ~ is a A-subset of ]I-]I~(L), and the relation r / a n d predicates [.J {p~Lo [ Lo C L is finite}, P E a, are A predicates on H I~'(L). To this end, it is necessary to use the properties of constructivization of the *w-spectrum S, and the remark that a A-subset (E-subset) and a A-predicate (E-predicate)in ~ ~ (w, 0, s, + , - , <) are the same in any admissible set. Theorem 17 is proved. [::] Regarding this theorem, the following conjecture arises.
If a theory T has an uncountable model that is ~-definable in ]['][F(9~) for an algebraic structure 9Jr with a simple theory, then the theory T also has an uncountable model that is ~-definable in I-]IF(L) for some dense linear order L.
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257
We indicate an example of the application of Theorem 17. Let Q(a0, a l , ... ) denote the purely transcendental extension of the field Q of rational numbers regarded as a linearly ordered field such that for n E w, all elements of the field Qn ~ Q(a0, ... , an-l) (we set Q0 ---" Q) are infinitesimal with respect to the element an. Such an ordering is unique, and there exists a single-valued constructivization -~
-+ (Q(a0, . . . , a~, . . . I~ ~ ~ ) , -< )
such that n -+ u - l ( a n ) , n E w, is a E-function on f~. Let IR* be the real closure of the ordered field (Q(a~ In E w), ~ ). Then there exists a constructivization u* 9 w ~ IR* that "extends" u. We note that the embedding Q(an In E w) --+ IR* is a morphism from (Q(a~ [ n E w),u) into (R*, v*). If R ~ , n E w, is the algebraic closure of Q,, in R*, then n --+ (u*)-l(R;), n E co, is the computable numbering of a 2-subset of w, which allows us to construct a computable sequence of constructivizations
(R0..0)....,
(R;,,~),
...
such that the embeddings N~ C_ R* are morphisms from (IR~ , u , ) i n t o (R*, u*). Let tt 9 [hi --+ Ira] be a morphism of *w. The embedding # " Q , --+ Qm such that I.tt(ai) - att(i ) can be uniquely determined from #. It is an orderpreserving embedding. Consequently, it induces the uniquely defined embedding ~ ' I R ~ --+ lR~. Therefore, the structure
(R"n , ~ I n E w ,
#EMor*w
}
is a *w-spectrum which is constructivizable. Indeed, the sequence of constructivizations
(R0. ~ 0 ) , . . . ,
(R~, , , ) , . . . ,
~ E ~,
was given above. We can assume that all the constructivizations un are singlevalued. Then the functions j r such that - f i ( u , ( 1 ) ) - umj',(1) for any I E ~o are uniquely determined from p E Mor*w. These functions are E-functions and the corresponding function f " A x w --+ w such that f ( n * , k ) - fu.(n.)(k), n* E A , k E w, is a E-function. Hence the *w-spectrum {IR:, g l n E w, p E Mor*w} is construcfivizable and, by Theorem 17, the structure 1RL~ *s being the realclosed field of cardinality ILl, is E~-definable in IHIF(L).
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C o r o l l a r y 18 If L is a dense linear order without endpoints of the power of the continuum, then the field C of complex numbers is E-definable in IHIF(L). P r o o f . Indeed, in HF(L), some real-closed field R of the power of the continuum is E-definable. Therefore, its finite extension R(i), the algebraic closure of N, is also definable in HF(L), and R(i) is isomorphic to C. [] Neither Proposition 16 nor Theorem 17 can help in solving the question of the definability (2-definability) of a concrete uncountable model. By Corollary 18, the field C of complex numbers is E-definable in IHIF(L) for any dense linear order L of the power of the continuum. But what about the definability of the field R of real numbers? P r o p o s i t i o n 19 The field R is not definable in IHIF(L) for any dense linear order L. P r o o f . If ]R is definable in H F ( L ) for some dense linear order L, then it is possible to choose two countable suborders L0 -~ L1 (-~ L) such that L0 contains all the parameters from the definition of I~ in ]HIF(L), and if IR0 and RI are subfields of N defined in ]I-]IF(L0) and HF(L1) respectively (in the same way as ]R in HF(L)), then R0 < IRI < R. Since IR0 and I~ 1 are different countable subfields of IR, they are not isomorphic (since It( is Archimedean). On the other hand, there exists an isomorphism between L0 and L1 (preserving parameters). Consequently, it must be that IR0 ~- N1. The contradiction thus obtained proves the proposition. [] R e m a r k 9 Using a similar argument, we can prove that the field Qp of p--adic numbers is not definable in ][-IIF(L) for any dense linear order L. We establish a general fact which implies that R is not ~-definable in IHIF (L) for any linear order L. D e f i n i t i o n 12 An algebraic structure 9Jr of finite (recursive) signature is called locally constructivizable if for any finite family of elements a0, . . . , an E M, the =t-theory Th3(gYt, ~) of the structure (ff)l, a0, . . . , an) is recursively enumerable. R e m a r k 10 A structure 9Jr is locally constructivizable if and only if, for any a 0 , . . . , an E M, there exist a constructivizable structure 91 and b0, . . . , bn E N such that Th3(fflt, ~) - Th3(91, b).
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259
The decidability of the theory Th(9)l) and its categoricity with respect to some cardinality is a sufficient condition for the local constructivizability of 9)I:. In particular, all models of simple theories are locally constructivizable. P r o p o s i t i o n 20 If fib is locally constructivizable and 91 is E-definable in ]HIF(~), then 91 is locally constructivizable. Proof. The assertion is obtained from the above remark and the following general fact:
If ff)t is constructivizable, and a0, . . . , a~-i C M, then the N-theory of the structure ( H F ( ~ ) , >r is recursively enumerable,
rn
R e m a r k 11 Any two infinite linearly ordered sets have the same 3-theory, whence any linearly ordered set is locally constructivizable. C o r o l l a r y 21 The field R is not E-definable in HF(L) for any linear order L. In conclusion, we give an example of a complete decidable theory only the simple model of which is locally constructivizable. Let E ~ {0, 1}* be the set of all finite sequences formed by 0 and 1. The signature a ~ {P] I e C E} consists only of unary predicates. The theory T is defined by the choice of an infini~;e recursive binary tree D(C_ E) with no infinite recursive branches (an example of such a tree can be found, e.g., in [8]) and by the following system of axioms:
Vx PA(~), Vx
(P,o(x) v
P,,
(x) -~
p,(x)),
~ e E,
VX ((Peo(X)--+ rePel(X)) /~ (Pel(X)--~ mPeo(X))), 3x (P~(x) A -'P~o(X) A-,P~, (x)), Vx-,P~(x),
c e E,
r E D,
e E E \ D,
w v~ (p,(x) A -~p,o(~) A -~p,, (x) A p,(~) ^ -~P,o(~) A -,p,, (~) -~ ~ - ~),
tEE.
260
Yu. L. Ershov
References [1] J. Barwise, (ed.), Handbook of Mathematical Logic, Stud. Logic Found. Math., 90 (1977). [2] C.C. Chang and H. J. Keisler, Model Theory, 3rd. edn., Stud. Logic Found. Math., 73 (1990); [lst. edn. 1973, 2nd. edn. 1977]. [3] Yu. L. Ershov, Decision Problems and Constructivizable Models (Russian), (Mathematical Logic and Foundations of Mathematics, Nauka, Moskva, 1980). [4] Yu. L. Ershov, E-definability in admissible sets (Russian), Dokl. Akad. Nauk sssR, 285 (1985) 259-262; [translated in: Soviet Math.- Dokl., 32 (1985) 767-770]. [5] Yu. L. Ershov, Definability and Computability (Russian), (Sibirsk. Shk. Alg. i Log., NII MIOONGU, Novosibirsk, 1996); [translated in: Siberian School of Algebra and Logic, (Consultants Bureau, Plenum, New York, 1996)]. [6] Yu. L. Ershov and E. A. Palyutin, Mathematical Logic (Russian), 2nd. edn., (Nauka, Moskva, 1987). [English translation of 1st. edn., translated by V. Shokurov, (MIR Publishers, Moscow, 1984)]. [7] H. J. Keisler, Fundamentals of Model Theory, in: J. Barwise, (ed.), Handbook of Mathematical Logic, Stud. Logic Found. Math., 90 (1977) 47-103. [8] M. G. Peretyat'kin, Strongly constructive models and enumerations of the Boolean algebra of recursive sets (Russian), Algebra i Logika, 10 (1971) 535-557; [translated in: Algebra and Logic, 10 (1971) 332-345].
Yu. L. Ershov Sobolev Institute of Mathematics pr. Akademika Koptuga, 4 Novosibirsk, 630090 Russia
The notion of the N-definability of an algebraic structure in an admissible set was introduced in the author's article [4] as a generalization of the notion of constructivization. Below, we use notions and facts from the theory of admissible sets or, generally speaking, KpU-models, following [5]. All the model-theoretic notions we use can be found in [2, 6]. Let A be a KpU-model and u" B -+ M be an A-numbering, i.e., a mapping of a N-subset B of A onto M. D e f i n i t i o n 1 An n - a r y predicate P C_ M n on M is called (i) a 2~-predicate if {(b,,...,
b , ) ] b G B", (ub,, . . . , ub,) e P} -
u-l(P)
is a E-predicate; (ii) a A~-predicate if P and M " \ P are F~-predicates. Let 9)t = (M, P0~, . . . , Pff) be an algebraic structure with signature (p~o . . . , p ~ . ) (for the sake of simplicity we consider here only the case of relational signatures). D e f i n i t i o n 2 An A-numbering u 9 B -+ M of the universe of ffJ[ is called an A-constructivization of the structure 9I[ if the equality predicate and all the predicates P0~, . . . , P ~ are A~-predicates. A structure ff)t for which there exists at least one A-constructivization is called A-constructivizable. A pair ( ~ , u ) , where u 9 B --+ M is an A-constructivization of the structure 9J[, is called an A-constructive structure. H A N D B O O K OF R E C U R S I V E M A T H E M A T I C S Edited by Yu. L. Ershov, S. S. Goncharov, A. N e r o d e , a nd J. B. R e m m e l O 1998 Elsevier Science B.V. All rights reserved.
235
Yu. L. Ershov
236
R e m a r k 1 In the case where A is IHIF(0), the notion of an A-constructivizable structure coincides with that of a constructivizable structure, while (if IHIF(0) and w are identified as in Section 2.1 of [5]) the notions of an A-constructivization and of an A-constructive structure differ from the well-known ones only in the fact that an arbitrary nonempty recursively enumerable set (and not only w) can be taken as the domain of the constructivization. It is possible to give an equivalent definition of an A-constructivizable structure which is based on the notion of "definability" well-known in model theory. Let 9)t = ( M , P0~, . . . , P ~ ) be an algebraic structure with signature ( P o ~ . . . , PuSh). ffYr is called definable in A if there exist formulas
Definition 3
~o(Xo) , ~i(Xo,
xi),
~o(Xo, ...,
X~o-i) , ...
, ~(Xo,
...,
x~o-i)
(with parameters in A), such that _/lIo - - ~oA[Xo] r 0 and 7/-- ~)[Xo, Xl]NMg is a congruence on the algebraic structure
(Mo,
P o),
where p~o
=
r
N Mo' ,
i~
and the structure 9)t is isomorphic to the quotient structure 9Yr (In this case we say that the system of formulas ~o, 1J~l , l ~ O , ' ' ' , O n defines 9)t in A.) We recall that 71 is a congruence on 9)I0 if 7/ is an equivalence on M0 and, for any i <~ n and d o , . . . , am,-1, bo, ... , bin,-1 E ~]o such that (aj, bj) E for j < mi, we have
(a)
p o.
Definition 4 A structure 9)l is called E-definable in A if there exists a system of E-formulas ~o,
~1, r
...,
~,
~,
* (I)o, ... ,
(I)*
such that ~o, ~1, ~o, ... , (I)~ defines 9)l in A, and ~A[Xo, x~] N Mo2 *A (I) i [ X 0 ,
where M o -
...
,
Xmi_l] N M o'
~oA[Xo].
-
Mo2 \ ~ [ X o , Xl],
Mo'-.
and
xm,-1],
i~n,
Chapter 5
E-Definability of Algebraic Structures
237
R e m a r k 2 Let Mo C A, and let P C M~ be an n - a r y predicate. If we refer to P as a A-predicate on Mo in the case that there exist E-formulas (I)(xo, . . . , x~_,) and (I)*(Xo, . . . , Xn-l) such that p
_
q~A[g] n
M~
and
M~ \ P -
r
n Mg,
then it is possible to give another equivalent definition of E-definability. (The E-formulas ~T, (I)~), . . . , (I)~ in the above definition "confirm" the equivalence.) D e f i n i t i o n 5 A structure 9Jl is called E-definable in A if there exists a sequence of E-formulas qJo, qJl, (I)o, ... , (I)n which defines 9)t in A, and for which the predicates ~lA[xo, xl], . . . , O/A[~], . . . , i ~< n, are A-predicates on M o - OoA[Xo]. 1 An algebraic structure ~}'A is A-constructivizable if and only if it is E-definable in A.
Proposition
P r o o f . Let ~ be E-definable in A, and qJo, ~1, (I)o, ... , (I)~ be a sequence of N-formulas which defines 9~J[ in A;
Mo-~ ~o~[Xo], -~ ~ [ x o , x,] n Mg, p~o ~ e t [ x o , . . . , xm,_l] n Mo',
i <~ n,
PY").
Let T : ~YAo/q --+ ~J[ be an isomorphism. We define an A-numbering u : Mo --+ M by setting u(mo) ~ ~([mo]), mo E Mo, where [mo] is the class of elements of Mo that are r/-equivalent to the element too. A routine verification shows that the A-numbering constructed in such a way is an A-constructivization of the structure gJt. Conversely, let ~)t be A-constructivizable, and let u " B --+ M be some A-constructivization of the structure 9Jt. Since B is a E-subset of A, there exists a E-formula qo(xo) such that B - qoA[Xo]. Since u " B --+ M is an A-constructivization of 9Jt, the relations
Yu. L. Ershov
238
flu ~
{ (bo, bl) I bo, bl e B, ubo - vbl},
Qo ~
{(bo,...,
Q~
---" {(bo, . . . ,
are A , - p r e d i c a t e s .
bmo-1) I bo, . . . , bmo-1 C B , (t.'bo, . . . , ubmo-1) E PogX},
bin,-1)I bo, . . . , bmo-1 E B, (t/bo, . . . ,
t,'bmn_l) C Pn9Jr}
Consequently, there exist E - f o r m u l a s I'I/I(Xo, X l ) , kI/1 (Xo, Xl), r
...
, X~o-1),
%(xo, . . . , Xmo-~),
r
. . . , xm~_l),
(~;(Xo, . . . ,
Xmn-1)
such t h a t
Qo
Qn
--
II)IA[Xo , Xl] ,
-
(I)oA[Xo,...,/too-l],
U 2 ~ 7-]
--
III1A [Xo, Xl],
B m~ \ Qo
--
r
9
o
9
o
--
r n[xO, --- , Xmn-1],
B m" \ Qn
-
. . . , Xmo_l] '
(I)nA[xo, -.. , Xmn-1]"
It is not hard to verify t h a t the s y s t e m of E - f o r m u l a s ~ o , ~ 1 , ~ o , . . . , (I)~ defines g)t in A. T h e existence of E - f o r m u l a s q ~ , q)~, . . . , (I)~ shows t h a t 93t is E-definable in A. El T h e following claim is an interesting consequence of this (uninteresting) proposition. Let al - (0, E2).
Chapter 5
E-Definabilitg of Algebraic Structures
239
P r o p o s i t i o n 2 If an algebraic structure 9Yt of signature a is A-constructivizable, then there exists a KPU-model I~ of signature al U a with set of urelements M , such that (~ r M ) I (r = 9~, and ]B is E-definable in A and "has the same ordinals" as A does. If A is an admissible set, then ]~ is also an admissible set. For the proof of this proposition we refer the reader to Section 3.4 of [5]. C o r o l l a r y 3 If A is an admissible, set and the algebraic structure 93I is A-constructivizable, then the ordinal o(A) is 9'Jr-admissible. The definition of an 9~-admissible ordinal can be found in [1]. It is wellknown that if 9~t is a constructivizable algebraic structure, then the 3-theory Th(gJt)3 of this structure is recursively enumerable. It is easy to see that if 9Jr is constructivizable, then 1HIF(gY0 is also constructivizable. Hence the 3-theory Th(IH[F(gY0) 3 (and moreover, the E-theory Th(lHIF(9~t))r.)of the admissible set ]HIF(gY0 is recursively enumerable. If 9Y~ is A-constructivizable, and A is a sufficiently "large" admissible set, it is possible to indicate, using infinite formulas, a larger fragment of the theory of the structure 97t which is effectively "enumerable" in A. We outline a description of the syntax and semantics of this E*-logic of an admissible set ~ of signature (0, C2;cb, b E B , P o ~ p~n). The variables VB = {xb I b C B} are coded as follows: Lxba --~ (0, b), b C B. The constants are coded as follows: Lcb- ~ (1, b), b E B (we assume that co = 0). We first introduce the notion of a A~-formula. Simultaneously, we define the code of a A~-formula. (1) By elementary formulas we mean formulas of the forms to = t l, to E t l, and P i ( t l , . . . , tin,) for i < n; they are A;-formulas and have the following codes: Lt0 = tlJ --- (2, 0, L/0J, LtlJ), Lt0Etlz ...,
t,.,),
= (2, 1 , L t 0 J , - t l j ) , =
(2,/+
2, , t , , ,
...,
,t=,,),
i<~ n.
(2) If (I) is a A~)-formula, then --,0 is a A~)-formula and its code is (3, LO_J).
240
Yu. L. Ershov
(3) If 9 is a family of A~)-formulas such that LO_J "-- {L(I)J I o c o} c 8, then A(O) and V(4~) are A~)-formulas, and their codes are defined as follows:
~A(e)~ "--(4, ~e~), ,v(e)~ ~-(5, ~e~). (4) If (I) is a A~)-formula, x is a variable, and t is a term, then Vx E t (I) and =Ix E t (I) are A ; - f o r m u l a s , and their codes are defined as follows: LVx E t (I)j "- (6, LX_J, LtJ, L(I)J),
L~tX E t (I)_l ~
(7, LZA, Lt..J, L(I)_I).
Something is a A~)-formula if and only if it is contained in the set generated by rules (1)-(4). It is easy to establish that the set A o of all codes of A~)-formulas is a A - s u b s e t of B. We now introduce the notion of a E*-formula. (1) Every A~)-formula is a E*-formula, and its [;*-code coincides with its A;-code. (2) If 4~ is a family of s then h(4~) and V(4~) are s as follows:
such that L~_J ~ {L(I)J I ,:I:,c ~} c B, and their N*-codes are defined
~A(e)~ ~ - ( 4 , . e ~ ) , ~v(~)~--(5,
~).
(3) If (I) is a E*-formula, x is a variable, and t is a term, then Vx E t (I) and =Ix (5 t (I) are P~*-formulas and their codes are defined as follows" ~Vx E t (I)~ - -
(6, ~x~, LtJ, ~'I)-),
L3X E t (I)j ~
(7, LXJ, LtJ, L(ID/).
(4) If (I) is a E*-formula, and X is a set of variables such that LXJ {LXJ IX C X} E B, then (3, X , (I)) is a E*-formula, and its code is defined as follows"
,(3, x , ~), -- (8, , x ~ , ~ ) .
Chapter 5
E-Definabilitg of Al9ebraic Structures
241
The set E* of all codes of E*-formulas is a A-subset of B. For any E*-formula (I), it is natural to define the set FV((I)) of free variables of (I). For any such formula, LFV((I))j C B. To any element a C B, we can assign a mapping % : V --+ B from the set of all variables {Xb [ b E B } into B by the following rule: given xb, if there exists a unique element c C B such that (LXb_J, C) = ((0, b), c) C a, then %(Xb) ~ c, otherwise 7a(Xb) ~ f). By induction on E*-formulas, we define the relation B ~ 4)[%] of the truth of a formula (I) in B under the interpretation %. This is defined as usual. We must only specify the case in which (I) = (3, X , ~). B ~ (I)[')'a] is true if and only if there is a function f : X --+ B such that f C B and B ~ ~[%,] for a' --~ (a \ X x B) U f . The following theorem on the E-definability of the truth of E*-formulas is not hard to prove. T h e o r e m 4 The predicate Trs. ~ {(-(I)-,a) I (I) is a E*-formula, It~ ~ (I)[%1} is a E-predicate in B. C o r o l l a r y 5 If 9)t, A, and I~ = l~(gJl) are the same as in Proposition 2, then the E*-theory of g(9)l) is a "E-subset" of A. In the sequel, our attention will be focused mainly on the E-definability of structures in admissible sets of the form IHIN(gJt) for models 93t of sufficiently simple theories T. It is useful to obtain beforehand information on definability in such structures. The following assertion turns out to be useful. P r o p o s i t i o n 6 If g) ~ HF(gY0, where g) is an elementary submodel of HF(9)I), then g) has the form IHIF(9)t') for a suitable model g)t' 4 9)t. We give a sketch of the proof of tile fact that predicates of the system 9)t which are first-order definable in HF(gN) can be described by (infinite) formulas of the language L~, ,~. Let n E co. We set
2)n ~
IHIN(n(= { i l i C n})),
.% =
HIF(O)
U .~,, = ~ ( ~ ) .
nEo.,,
c_ .9~ c_ . . . ,
242
Yu. L. Ershov
For any n E w, x C Dn, ~ E M" we define an element x ( ~ ) C H F ( M ) in the following way. Let A~" n --+ M be defined as follows" A~(i) - m i , i < n, where ~ - (m0, ... , ran-l). The mapping A~- can be uniquely extended to a mapping A~ " .~n - H F ( n ) --+ H F ( M ) , so that A~({ao,...,ak}
--
{/~(ao),...,
/~(ak)}
for any set (not u r e l e m e n t ) { a o , . . . , ak} e IHIF(n). Let
Given any element x E D~, it is possible to define effectively a term ...
,xn)
of the signature (0, { } ' , U:) such that, for any m,)
-
ml,
...
, /7/n e
M,
,(m).
Let T(5, ~) be a formula of the language of the signature al U cr', where a' is the signature of the structure 9)t. We assume that the variables in the list correspond to urelements of the structure IHIF(9~) (i.e., to elements of the structure 93t), and variables in the list y correspond to arbitrary elements of IHIF(ff)t). The formula ~ . ( 5 , y ) of the language n~, ,~ of the signature o" U (~, e 2 , { }1, U 2) is constructed from the formula ~o as follows: (1) if r does not contain quantifiers, then ~. --" r (2) if ~ - ~o q Vl, then ~p. - - (90o). q (~1)., for q @ {A, V, -+}, (3) if ~o - -'~0, then ~. ~ -~(~o0)., (4) if ~ - Q x~o, then V. - - Q x ( ~ o ) . , (5 / if V - 3 y~o, then r
for Q c {v, 3},
Y --" V,,e~ ( V,,e~. 3 x , . . . ::Ix,., ((Vo)*)t,.(~)),
(6) if ~o - Vyvo, then ~% ~ Ane~ ( Axe~. v/x "-" Vxn ((qo0).) t,,(e))" y If qa(~) does not contain variables from the list ~, then neither does T.(~). For any terms to and tl of the signature a ' U (r { }~, U2), with variables from the list ~, it is possible to write effectively quantifier-free (finite) formulas
Chapter 5
243
E-Definability of Algebraic Structures
9 to, * t~ (7) and (fro,it(-2) of the s i g n a t u r e ~r' such t h a t to - t, - ~ ( ~ ) ~*to, t] and to r tl - - ~ - ( ~ ) d~to,t ~. For any pair of t e r m s to a n d tl of the s i g n a t u r e ( 0 , { }1, U2), w i t h variables f r o m the list ~, we define f o r m u l a s (bto,t, and ~to,t~ of the e m p t y s i g n a t u r e such t h a t
FV(to)u VV(t ) and t(oH~'(~),{ },u)[3'] E t~ H~(~)'{ },u)[3'] r
19y~[ ~ (I)to,tl[3'],
t(o~'(~),{ },u)[3'] c_ t~ M~'(~)' { },u)[3'] r
or
for any i n t e r p r e t a t i o n 3' " F V ( t o - tl) --+ [971~[. Let 7- d e n o t e the s e n t e n c e 3 u(u - u), and let Uo a n d Ul r e p r e s e n t variables f r o m the list 7. We define for to - Uo: if
Ul,
then
(I)uo ,u, ~
if tl - O,
then
(I)uo,O ---" --,7", ~uo ,0 ---" T,
if t l -
then
(I)uo,t, ---" ~t~,~,o,
tl
-
{t~},
if t~ - (t~ U t~'), t h e n
-7,
[I/uo ,Ul ~
(uo - Ul),
~uo,t, ---" 7-,
(I)~o ,t, ~ (I)~o ,t~ V (I)~o ,t~', qJuo ,t, ~ r;
for to - r if tx
-
then
(I)~,,ttl
if
-- r
then
(I)~,~ ---" - r ,
if ta - {t~},
then
(I)~,t, ~ ~t~ ,~ A ~ r
tl
Ul,
if tl - (t~ U t~), t h e n
._..a ....IT ,
l~I)~,~l
~,~
~_..-..IT ,
~ r, qJ~,t, ~ r,
(I)o,t, --" (I)o,t~ V (I)o,t7 , ~ r
---" 7-;
for to - {t~)}: if tl - Ul,
then
(I)to ,~,~ ~ --,7", ~to ,~,~ ~ -~7,
if tl - 0,
then
(I)to,O ~ --,7-, ~to,O ~ --,7",
if tx - {t]},
then
(~to ,t~ ~
if t~ - (t] u t~'), t h e n
~to,t~ /~ ~t'~ ,to, ~to ,t~ -- (~t~ ,t~,
(~to ,t, "- (bto ,t~ V (~to ,tT, q~to ,t, " - (~t~ ,t,;
244
Yu. L. Ershov
for to - (t; 0 tg)" if tl - 721,
then d~to,u1 ~ "nT, I~to,u , ~ "nT,
if tl - r
then Oto,O ~ - - r ,
if tl - {tl},
then d~to,t 1 ~ l~to,t~,
klIto,r
kX/t~,r A II/t~,r ,
l~to,t 1 ~-- IJ~t~,t 1 A ~t~l,tl,
if t, - (t~ U t~), then Oto ,t, ~ Oto ,t~ V (IIto ,t~', /Ilto , t 1
.._.x l~tp0 ,tl A l~ttoPt 1 9
W e set ~*to ,t, --7~--IX/to,t, A It/t, ,to. Using t h e f o r m u l a s ~to ,t, a n d (~to,t,, we transform the formula ~ . ( : ) into a formula ~*(:) of the language L~, ,~ of the signature a ~ such that
Then ~HF(~)[g] _ qp.gn[g]. Using this transformation and Karp's theorem [7, Theorem 9.10], we arrive at the following proposition. P r o p o s i t i o n 7 Let 93to and ~}:J[1 be w-saturated models. If 93lo- ~ 1 , IH[F(gJ/o) - I[-IIF(~J[1); if 93to ~ ~ 1 , then IHIF(gJto) 4 IHIF(gJ/a).
then
D e f i n i t i o n 6 A model 9)1o is called saturated enough if there exists an w s a t u r a t e d model ~f~l such that 93t0 4 ~J~l and IHIF(9~0) 4 IHIF(~J~ 1). C o r o l l a r y 8 Let 9Jto and 9Jh be saturated enough models. If 9)lo - 93tl, then InIF(gYto) - I[tIF(9~I). If 9Jto ~ 93tl, then InlF(gJto) ~ HF(g)t1). P r o o f . The assertion follows from the definition, Proposition 7, and the existence of w-saturated models, rn R e m a r k 3 In any saturated enough model 97t, any (not necessarily complete) arithmetic type of formulas with a finite number of quantifier alterations is realized over a finite M0 C_ [9")t[. The author does not know whether this condition is equivalent to being saturated enough. R e m a r k 4 Any complete theory T (of a countable language) with infinite models has a countable saturated enough model. This fact follows from the Lgwenheim-Skolem theorem, Proposition 6, and the existence of w-saturated models of the theory T. Any complete categorical theory T has sufficiently many w-saturated models.
Chapter 5
E - D e f i n a b i l i t y o f Algebraic S t r u c t u r e s
245
P r o p o s i t i o n 9 If a theory T is complete and w-categorical, then any model
of T is w-saturated. If the theory T is complete and Wl-categorical, then any uncountable model of T is saturated (and, in addition, w-saturated). P r o o f . The first assertion follows from the Ryll-Nardzewski theorem on the characterization of w-categorical theories (see [2, Theorem 2.3.13]), or more exactly, from the fact that there is a finite number of types over any finite set. The second assertion is well-known. (see, for example, [2, Corollary 7.1.15]). E! R e m a r k 5 Not all countable models of Wl-categorical theories are saturated enough. For example, the following claim is true.
Let T be the theory of algebraically closed fields of characteristic O. If Fo, F1 ~ T, then tlI F(Fo) - tlIIF(F1) if and only if either Fo and F1 both have infinite transcendence degree over Q, or the transcendence degrees of Fo and F1 over Q are finite and equal. If ho, hl E IHIF(ff)t), then the types t~(~)(ho) and t~(~t)(hl) coincide if and only if there exist n E w, x E ~,~, ~ o , ~ : E M '~ such that ho - x(-~o), h: - X ( ~ l ) , and Theorem
10 Let 93t be a saturated enough model.
Proof.
Necessity. Let ho, hx E IHIF(9)t) and t~(~)(ho) = t~(~)(hl). Let sp ho (the support of ho) have n elements: sp ho - {rn~ ... , mn_lO}. It is easy to see that the support of h l also has n elements, and there exists x E ~)~ such that ho - x(m---o),~-o ~ (too, ... , ran_ o 1), and h: - ~ f ( ~ l ) for a suitable ~ 1 ~ ( m l , ... , ran_:). 1 Furthermore, for any formula 99(g) (5 t~(~-~ there exists a permutation a E Sn of the set n such that q0(g) E tgyt(a(ml)) where ( r ( m 1) ~ (m~(0) 1 ~ . . . ~ m a1( n - 1 ) ) and hi - - X(O'(~I)) 9 But then there exists a E S~ such that h~ = x ( a ( N 1 ) ) a n d t ~ ( m --~ - t ~ ( a ( N ' ) ) . Indeed, if such a does not exist, then for any a E Sn such that hi = x ( a ( ~ l ) ) there exists qp~(g) belonging to t ~ ( ~ -~ but not to t ~ ( a ( N 1 ) ) . Let ~P(x)--" A ~ s z ~p~(x), where
246
Yu. L. Ershov
Since ~po(5) E t ~ ( m --~ for all a E Sn~, it follows that ~(5) E t~(m---~ As already mentioned, there exists ao E S~ such that ~(g) E t~t(ao(m~)), but ~,o(E) ~ t~(ao(Nl)). We arrive at a contradiction. Hence there exists a E Sg such that =
and h, = x ( a ( N ' ) ) . SuiOiciency. Let z E . ~ , ~---0,~-1 E M ~, and t ~ ( ~ -~ = t ~ ( ~ ) . Using Propositions 5.1.7 (ii) and 5.1.8 of [2], we find an elementary extension 0Jr' of the model 0Jr which is a special model.
Since 9)t' is w-saturated, it follows that IHIF(gJt) ~ IHIF(gYr Then t~,(m --~
= t~(m ---~ = t ~ ( ~ 1) = t ~ , ( ~ 1)
and, since 9Yt~ is special, there exists an automorphism ~ of 9)t~ such that ~(m---o) = ~1. We extend p to an automorphism ~ of the model ~ = IHIF (9~'). Then =
Consequently, t~)(x(m---~ = t g ( x ( ~ ) ) ,
and
The theorem is proved. By special admissible sets we mean admissible sets of the form IHIF(gYt), where OR is an algebraic structure which is a model of a simple theory T. The preceding shows that many questions about admissible sets of the form II-IIF(FJt) can be reduced to the corresponding ones about the structure 9Yr On the other hand, such admissible sets present examples of structures for which there is a natural notion of computability without conditions on the cardinality of structures. Thus, if IR denotes the field of real numbers, then the notion of E-definability in IHIF(R) is a reasonable "computability over IR". However, the notion of "simplicity of a theory" requires refinement. One of the required features of a simple theory is its regularity in the sense of the following definition: D e f i n i t i o n 7 A theory T of the (finite) signature a ~ is called regular if it is decidable and model complete.
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247
R e m a r k 6 In a model complete theory T any formula is equivalent to an 3-formula (see [2, Proposition 3.17]). If r is also decidable, then such an 3-formula can be found effectively. The theories Th(R) and Th(Qp) of the fields of real and p-adic numbers, respectively, provide examples of regular theories. However, it is hard to consider such theories as simple ones because they have many nonconstructivizable countable models. D e f i n i t i o n 8 A theory T is called simple if it is a regular w-categorical theory whose set of complete formulas is decidable. The condition of w-categoricity means the uniqueness (up to an isomorphism) of a countable model of the theory. The model completeness and the decidability of the set of complete formulas guarantee the autoequivalence of all constructivizations of this countable model (see [31), i.e., the "uniqueness" of the computability theory for countable models of such theories. We now turn to the question of the possibility of (E-)definition of classical objects, the fields C and N of complex and real numbers, in special admissible sets. We first consider the case of the simple theory To of an infinite structure with empty signature, i.e., models of the theory To are presented by infinite sets without any additional structure. We assume that some uncountable algebraic structure 9Yt = ( M , ... ) is defined in IHIF(S), where S is an infinite set. Let do, ... , am C S be all the urelements appearing in the definition of 9J[ as parameters and let ~ ~ (a0, . . . , am). Let ~o(X0) and ~l(X0, xx) be formulas from the definition of 9)t in ]I-]IF(S) such that Mo --" ~t~(s) ~o [Xo], r/ ~, [Xo, x,] M Mo2 is an equivalence on Mo, and there exists a oneto-one correspondence (induced by an isomorphism between ~ and 9Jto/r/) between M and Mo/y. With any n E w and x E -~n, we associate the sets
T.
e s",
M,, ---" { x(~, b) I ~ E T,,} / ((~ r {x(~, and the relation
b) lb e
Yu. L. Ershov
248
Then M,~ C_ M0, 5,~ is an equivalence on T,~, and IM,~[ - [T,~/5,~ I. We note that since S is saturated enough, Theorem 10 implies that the set T,~ and the relation 5,~ are closed with respect to types over ~, i.e., if b C Tx, g E S" and t(s,~)(-b) - t(s,~)(-b'), then b' E T,~, and if (~o,~) E 5,~ and t(s,~)(b~ ~) - t(s,~)(~,-d ~) for some ~0,31 e S", then (3o,~) C 5x. It should be noted that the following obvious relation is valid: Mo -
U
U
M~ -
n ~,~
M,,.
U ~ e H ~(~o)
Since M is uncountable, we can find n C w and x C -~n such that M,~ is uncountable. Then there exist pairwise unequal b~
( b l , . . . ,b~) e S \
{ao,...,am}
so that 1 4- m 4- k = n, x(K, b~ x(~, b 1) E i 0 , and (x(~, b~ x(K, bl)) q~ r/, w h e r e -0 b ~ (hi~ . . . , b~), 51 _.~ (bl, . . . , b~). Without loss of generality, we can assume that there exists a unique i, 1 ~< i ~< k, such that b~ r b1. Indeed, let us consider the following sequence of collections of k elements. b~
b~ . . . ,
bl, b~
b~, ... , bl, bi~
~1
b~, . . . ,
b~
...,
bOk;
b~r
Each collection differs from the neighboring one by at most one element, so if for each neighboring pair b, b' the pair (x(~, b), x(~, b')) belongs to q, then the pair (x(~, ~0), x(K, b~)) must also belong to 7/since 7/is an equivalence. We note that x(~, b) E M0 for any collection b from this sequence. This follows from the fact that the types of the elements x(~, ~o) and ~(~, ~1) coincide.
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249
Hence we can assume that there exist b~ , . . . , bk-~ , bk , b'k or_ S \ {ao, . . . ,am} such that
bk-1, bk ) , ~'(-d, bl , . . . , bk-1, b~k) } ~ 7].
Now for any b 7(: b' ~ {g, b l , . . . , b k - l } , the fact that
t(~,~(b,,..., b~_,, b~, b,,..., b~_,, b~) - t(~,~(b,,..., b~_,, b, b,,..., b~_,, t,') implies (as mentioned above) that
(~(~, b,,..., b~_,, b), ~(~, b,,..., b~_,, b')) ~ ,7It should be noted that any permutation a of the set S uniquely defines (lifts up to) an automorphism a* of the admissible set IHIF(S) where
~'({~,,... , ~ } ) ~ {~'(~,),... ,~'(~)},
{~,,... , ~ } e ~ ( s ) . .
s.
If a r {a} - id{~}, then a* induces an automorphism on the structures 93t0 and 9Jl:0/rl. The following proposition is a consequence of the above considerations. P r o p o s i t i o n 11 I f an uncountable algebraic structure flit is definable in IHIF(S), then the a u t o m o r p h i s m group Aut(gJl) o f the structure 9)t contains a subgroup G that is i s o m o r p h i c to the s y m m e t r i c group Sym(S) of all perm u t a t i o n s of the set S .
P r o o f . Indeed, if x, ~, bl, ... , bk-1 have the same meaning as above, then any nontrivial permutation a0 of the set So ~
S \ { ~ , b l , . . . ,bk-1}
extended to S by letting ao(~) - ~, ao(b,) = b,, . . . , induces a nontrivial automorphism on 9)lo/7/. Thus Aut(glto/r/) "" Aut(gJl:) contains a subgroup isomorphic to Sym(So) "" Sym(S).
ao(bk-1) -
bk-1,
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Yu. L. Ershov
C o r o l l a r y 12 The fields C and R are not definable in I-]IF(S) for any set S. P r o o f . Indeed, the field ]R has no automorphisms, and the field C has no automorphisms of finite order greater than 2. rn R e m a r k 7 It is easy to see that if the field IR is definable (E-definable) in the Keu-model A, then C is also definable (E-definable) in A. Hence Proposition 11 is a serious obstacle to the definability of uncountable structures in admissible sets of the form HF(S), where S is an infinite set. We now consider the simple theory T1 the theory of dense linear orders without endpoints. Let L be a dense linear order without endpoints. We describe a general model-theoretic construction in terms of which we state necessary conditions for the definability in H F ( L ) of an uncountable structure (see Proposition 16 below), and a necessary and sufficient condition for the E-definability in H F ( L ) of uncountable models of a given theory (see Theorem 17 below). The category *w is defined as follows. Its objects are the sets of the form [hi ~ {0, 1 , . . . , n - 1}, n E w ([0] ~ 0), and its morphisms are orderpreserving embeddings. It should be noted that there is a unique morphism from [0] into ["l for any n E w. D e f i n i t i o n 9 By a *w-spectrum, we mean any functor S from the category *w into the category Mod~ of algebraic structures (of some fixed signature a) whose morphisms are all possible embeddings. To define a *w-spectrum $, it is necessary to give an infinite sequence !YJlo, 9X1, ... of algebraic structures of signature a, and associate with each order-preserving embedding # : [n] -4 [m] an embedding p. : 9Jr, --4 9Xm so that, if # o : In] --4 Ira] and p, : [m] -9 [k], n ~< m E k E w, are morphisms of the category *w, then (#,#o). = p,.#o., and if # : In] -4 [n] is the unique morphism from In] into In] (= id [,~]), then #. = id ~ . : 9X, --4 9X,, n E w. If the *w-spectrum .5' = {gJt,,/~. I n E w, # E Mor*w} has been defined, then for any linearly ordered set L, it is possible to define the algebraic structure 9XL (if)ts) as a direct limit li_.mg~o of the spectrum L0
{~k0,
~Lo,L,
I L0 C_L,
C L, L1 is finite},
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where 9Ytko --" 9)tn if Lo C_ L is finite and ILol- n, and the embedding ~Lo ,L, 9 ~ l Lo -+ 9Jt~ 1 is defined for finite Lo C_ L1 (C_ L) as follows: if Ll-{/o
and
(in which case 0 ~< io < ix < . . - < as #(j) ~ ij for j < n, then ~L0,L,
--~
Lo-{/io
<...
i~_x ~< m), and # ' [ n ] l
/2-" ~JC~L0 -- ~rf~n --~ ~ m
-~ [m] is defined /
-- ~ L , "
If L C_ L' are linearly ordered sets, then the structure with a substructure of 9Y~L, in a natural way.
~)~g c a n
be identified
R e m a r k 8 Any isomorphism between linearly ordered sets L and L' induces an isomorphism between ~J~g and PYRE,. P r o p o s i t i o n 13 If L C_ L ~ are dense linear orders without endpoints, then ~ f t L --~ ~ f t g , "
P r o o f . For countable L' the assertion follows from [6, Proposition 24.2]. The case of an arbitrary L' is proved by the application of the Tarski-Vaught theorem on elementary embeddings. D C o r o l l a r y 14 If L and L' are dense linear orders without endpoints, then ~[~ L -~ ~ L ' .
Let #o and ~1 be morphisms from [11 into [21 such that po(0) ~-- 0 and #1(0) ~ 1. The condition
r
(*)
is sufficient for 19JtSLI >/ ILl to hold for any linearly ordered set L. Indeed, let ~ C M1 be such that p0.(() 7~ #a.((). For any linearly ordered set L, and 1 E L, we define the corresponding embedding tit 9 ifJilt } = 9Ytl --+ 931L, and let ~, ~ p,(~). Then 1 r l' implies ~t r ~t, and [LI = [{~t I1 E L}[ ~< IML[. The following model-theoretic claim is stated without proof. P r o p o s i t i o n 15 Let S be a *w-spectrum such that #0.(~) r #,.(~) for some E M1, and let L be a dense linear order without endpoints. Then
is a set of indiscernible elements of the structure 9Jtc (with order induced
bu L).
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252
P r o p o s i t i o n 16 Let 93t be an uncountable algebraic structure defined in NF(L) for some dense linear order L without endpoints. Then there exists a *w-spectrum S such that tto. 5r #x., all the models 9Jto, 9 ' A 1 , . . . of this spectrum are countable and isomorphic, 9J[o ~ 9)tl ~ ... ~ 9)l, and for any embeddin9 tt :[n] --+ [rn], the embedding p. :92~n --+ 92~m is elementary. P r o o f . The preceding analysis of definable sets shows that there exist k E w, ~r C IHIF(k), and sequences g,b C L k that have the same type over (L,~), where ~ are constants appearing in the definition of 9)t in I[-IIF(L) (we can assume that {~} C_ {g}N{b}), such that ~r x ( b ) E Mo and (x(g), x(b)) ~ r/x. Then g -7(:b, and we can show (in a similar way to the proof of Proposition 11) that there exist such g and b so that ai 7~ bi for a single number i < k (aj - by for j < k, j :fi i). For definiteness, we will assume that ai < hi. We suppose that a0 < al < " " < ak-1, and 0 < i < k - 1. Then ai < bi < ai+l. m
_
m
g
By the Lgwenheim-Skolem theorem, we can find a countable elementary submodel 1HIF(L~ 4 II-IIF(L) such that {~} C_ {g} U {b~} C_ L ~ Since L ~ is a dense linear order, there are elements d2, d 3 , . . . , dn, ... such that bi < d2 < d3 < . . . < dn < . . . < ai+l. We set do "--- ai and dl "-hi. Then a i - 1 < do < d~ < d2 < . . .
< d~ < . . .
< ai+l.
We set 8k "-- [dk, dk+l ), k C a,,, and Ln
~
(-oo, do) U ~ o U - " U ~n-1 U [ai+l,OO) -- (-oo, dn) U [1:::/i+1,oo)
(define L0 - (-cx~, do) U [ai+l,
OO)).
From the above definition, it is clear that L~ is a countable dense linear order without endpoints which is isomorphic to L ~ over {~}. Let 92t~ be an algebraic structure definable in IHIIF(Ln) by the same formulas (and parameters ~) as 9R in IHIF(L). For any n E w we fix an isomorphism between linearly ordered sets c2n : ~,~ ~ [dn, dn+~) --4 8o -~ [do, d~) (such an isomorphism exists because 8n and 60 are countable dense linear orders with least elements but without largest elements). For n = 0 we set q;0 ~ ids0. For n, k E w let ~,~,k : 8~ -+ 8k be an isomorphism between linearly ordered sets which is defined as follows: ~ , k - ~k-1 ~ (we note that ~n,n = ids, and ~p,,t = ~Pk,l~,,k for n, k, l E w). Let tt : [n] --+ [m] be a monotone embedding of [n] = {0, ... , n - 1} into [m] = {0, . . . , m - 1} (i.e., p is a morphism of the category *w).
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253
Corresponding to this embedding, we assign the monotone embedding g " Ln -4 Lm uniquely defined by the following relations: I L0 -
idL0,
-fi I Sk -- ~k,,(k),
fork
(we note that L~ - Lo U (~o U --. U 8n-1). It is easy to verify that # ~ ~ is a functor from *ca into the category of linearly ordered sets with monotone embeddings (i.e., i-d[,q - idL. and #A - ~ for # ' [ h i -4 [m], A'[/] -4 [n]). For a monotone embedding # 9 [hi -4 Ira], the embedding ~ " L~ -4 Lm (identical on {~}!) induces elementary embeddings/z'" ]HIIF(Ln) -4 ]HIIF(Lm), and #. 9 9Jt~ -4 9Jtm. It is easy to check that # ~-~ #. i's a functor from 9ca into the category of algebraic structures that are isomorphic to 9910 with elementary embeddings as morphisms. It remains to check that #o, ~- /21-. By definition (do - ai and dl bi), we see that x(~) e L1, tt~(x(~)) - x(~) and tt~(x(~)) - x(b) (since ~l(d0) - dl - bi and ~l(dj) - aj for j < k, j ~- i). Since (x(~), x(b)) ~ r/, for IH[F(L), this fact remains valid for IHI~(L2) ~ lHI~(L). Thus, #0,([x(~)]) [,7"t'(a)lrl r [,7'r -- t21.([~t,(~)]). [[] u
Proposition 16 gives a necessary condition for definability in HIF(L), where L is a dense linear order without endpoints. It turns out that the effectivization of this condition is already a necessary and sufficient condition for E-definability in H ~ ( L ) . D e f i n i t i o n 10 A system of numberings u,~ 9 ca -4 M~, n C ca, is called a computable sequence of constructivizations . . . ,
. . . ,
n e
if the following conditions hold (we assume that the signature a of the structures 9~to, 9)tl, ... is finite and without function symbols): (1) E ~ {(n, mo, ml) l n, mo, ml e w, u n ( m o ) = un(ml)} is a A-predicate on ca, (2) Np ~ { n - ( n 0 , n a , . . . ,nk) i ~ e wk+~, ( v , 0 ( n , ) , . . . ,Uno(nk))EP ~'~ } is a A-predicate on ca for any (k-ary) predicate symbol P C a; (3) for any constant symbol c E a there exists a E-function fc : ca -4 ca such that c~" = vnfc(n).
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254
Every morphism # : [n] --+ [rn] of the category *co is uniquely defined by the n u m b e r m and the subset g([n]) c_ [m]. This remark allows one to define a one-to-one correspondence #* : A -+ Nor*co between the subset A ~ {n In E co, r(n) < 2 t(n)} C_ co and the set Nor*w, provided that n E A is assumed to code the m o r p h i s m p : [k] -~ [l] such that l = l(n) and r(n) is the n u m b e r of the subset #([k]) C_ [l] = [l(n)] in some standard listing of finite subsets of w. (Here 1 and r are such that n = (l(n), r(n)).) It is evident that A is a A - s u b s e t of w. Let S - {9Jtn,#. In C w,p E Nor*co} be a *co-spectrum. D e f i n i t i o n 11 By a constructivization of S we mean any computable sequence of constructivizations
(~o,-o), ( ~ , , - , ) ,
..., (~,,-,),
...,
, e ~,
together with a E-function f : A x w --+ w such that, for any n, m, k C co and tt : [n] -+ [re] e Nor*w, if n* E A is such that p*(n*) = #, then # . r , . ( k ) -- r,~f(n*, k). A *w-spectrum S is called constructivizable if there exists a constructivization for it. Theorem
17 Let L be a dense linear order without endpoints. A theory T
h ~ ~,~ uncountabt~ modal r~-d~fi.~bl~ in H~(L) if and o.lu if t h ~ ~x~t~ a constructivizable *w-spectrum S satisfying condition (,), and such that 99ts ~ T. P r o o f . Let 9)t be an uncountable model of the theory T which is E-definable in IHIF(L) for some dense linear order L without endpoints. Acting as in the proof of Proposition 16, we find x , g , b, L ~ , d2 , d3 , . . . , and so on. Since L ~ is a countable dense linear order without endpoints, it is isomorphic to the set Q of rational numbers with the natural order. From this, the existence of a constructivization u 9 w -~ H F ( L ~ of the admissible set w
R~(L ~
(_~ MF(Q, ~<))
follows. Since M0 is E-definable, the set u-~(Mo) - {n I n e w, u(n) E M0} is a E - s u b s e t of w. The predicate
{(n, m) l n,m e ~, ~(n),~(m) e M0, (~(n), ~(m)) e ~}
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255
N-Definability of Algebraic Structures
is a A - p r e d i c a t e o n / / - l ( M 0 ) . The predicate
nk>lrti e
//-l(pffJto) __, { ( n l , . . . ,
03, (//(T/l),
. . - ,
//(T/k)) e p ~ o }
is a A - p r e d i c a t e on//-l(M0) for any ( k - a r y ) predicate symbol P C or. The elements d2 < .-. < d~ < . . . , for n E 03, of the set L ~ (for which bi < dn < ai+l for all n C a;) can be chosen so t h a t there exists a G-function d : a; --+ a; such t h a t ud(n) = dn for all n E a;. If the substructures ffJln, n C 03, of the s t r u c t u r e 9Jr are defined as in Proposition 16, then it is obvious that the corresponding *03-spectrum has a constructivization. (We only need to choose uniformly effective isomorphisms W~ : 8n ~ 80, n E 03.) Conversely, let S be a *03-spectrum such t h a t 9Yts ~ r for (any) dense linear order L without endpoints. Let . . . ,
and let f 9 A x w ~ ,J be a constructivization of this *03-spectrum. We sketch the proof of the ~-definability of the model 9)15s in H F ( L ) . Let Lo {/0 < "'" < ln-i} C_ L be a finite subset of the set L,
Lo (for n
--"
-- 0 we set Lo --" 0). Let
ilo
Ik e
'Lo --~ {((io,~o), (Lo,]r P~0
]~O,kl e 03, //n(]%)- //n(~:l)},
{(('E0, kl), . . . , (Lo, km))I kl, . . . , ~m E 0d, (//n(]r
for each ( m - a r y ) predicate symbol P E a.
. . . , //n(km)) E P ~ ' }
For c C a we choose k E 03 so
that c ~n - / / n ( k ) and set c ~ o ~ (Lo, k). T h e n rlLo is a congruence on the structure 9~io - - ( M ~ o , . . . , p ~ o , . . . , c ~ o , . . . ) and the quotient structure ffJtio --" 9 ~ i o /7/50 is isomorphic to 9Jtn. Lo C_ L1 C_ L be finite subsets of the set L, Lo =
{/o < " "
< ln-1},
L, = {t; < . . .
<
!
Let
Yu. L. Ershov
256
and 0 ~< j0 < "'" < jn-1 < m be such that li - l}, for i < n. Now let # ~ PLo,L, 9 [n] --+ [m] be the morphism of *w such that #(i) - ji for i < n. Then the embedding #, 9 9)~n -+ 9Jim defines the embedding #---, " 9J~L0 --+ 9YtL1. On the set M ~ --" (.J {M~o ]Lo C_ L is a finite subset of L}, we define the equivalence relation 77 C (M~ ~ as follows. Then for (Lo, k}, (L,, l) E M ~ we set ((To, k), (L,, l)) e 7] if
(PLo,L~). ([(Lo, K)]nLo) -
(,L,,L~). ([(Ll,l>lvL,)
for L2 ~ Lo U L1. It is easy to verify that 7] N(M~o)2 _ r/g~ for finite Lo C_ L. Taking into account that Lo C_ L1 C_ L2 C_ L implies/Ago,L2 [2L,,L2/Ago,L, (for a finite L2) and using the definition of a *w-spectrum, we obtain the following equivalence" -
-
((Lo, k), (L1,1)) E r/ ca there exists a finite subset L2 C_ L, such that Lo U L1 C_ L2, and (pgo,g2).([(Lo, k)],Lo) -- (#L,,g2).([(Ll,1}],7,., ). Owing to this equivalence, it is not hard to check that r/is an equivalence relation on M ~ Furthermore, 7/is a congruence on the algebraic structure 9~
~
(M~
]LoC_Lisfinite},...,c
O,...).
The quotient structure 9Yt~ isomorphic to 9)~s. It is not hard to verify that the set M ~ is a A-subset of ]I-]I~(L), and the relation r / a n d predicates [.J {p~Lo [ Lo C L is finite}, P E a, are A predicates on H I~'(L). To this end, it is necessary to use the properties of constructivization of the *w-spectrum S, and the remark that a A-subset (E-subset) and a A-predicate (E-predicate)in ~ ~ (w, 0, s, + , - , <) are the same in any admissible set. Theorem 17 is proved. [::] Regarding this theorem, the following conjecture arises.
If a theory T has an uncountable model that is ~-definable in ]['][F(9~) for an algebraic structure 9Jr with a simple theory, then the theory T also has an uncountable model that is ~-definable in I-]IF(L) for some dense linear order L.
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E-Definability of Algebraic Structures
257
We indicate an example of the application of Theorem 17. Let Q(a0, a l , ... ) denote the purely transcendental extension of the field Q of rational numbers regarded as a linearly ordered field such that for n E w, all elements of the field Qn ~ Q(a0, ... , an-l) (we set Q0 ---" Q) are infinitesimal with respect to the element an. Such an ordering is unique, and there exists a single-valued constructivization -~
-+ (Q(a0, . . . , a~, . . . I~ ~ ~ ) , -< )
such that n -+ u - l ( a n ) , n E w, is a E-function on f~. Let IR* be the real closure of the ordered field (Q(a~ In E w), ~ ). Then there exists a constructivization u* 9 w ~ IR* that "extends" u. We note that the embedding Q(an In E w) --+ IR* is a morphism from (Q(a~ [ n E w),u) into (R*, v*). If R ~ , n E w, is the algebraic closure of Q,, in R*, then n --+ (u*)-l(R;), n E co, is the computable numbering of a 2-subset of w, which allows us to construct a computable sequence of constructivizations
(R0..0)....,
(R;,,~),
...
such that the embeddings N~ C_ R* are morphisms from (IR~ , u , ) i n t o (R*, u*). Let tt 9 [hi --+ Ira] be a morphism of *w. The embedding # " Q , --+ Qm such that I.tt(ai) - att(i ) can be uniquely determined from #. It is an orderpreserving embedding. Consequently, it induces the uniquely defined embedding ~ ' I R ~ --+ lR~. Therefore, the structure
(R"n , ~ I n E w ,
#EMor*w
}
is a *w-spectrum which is constructivizable. Indeed, the sequence of constructivizations
(R0. ~ 0 ) , . . . ,
(R~, , , ) , . . . ,
~ E ~,
was given above. We can assume that all the constructivizations un are singlevalued. Then the functions j r such that - f i ( u , ( 1 ) ) - umj',(1) for any I E ~o are uniquely determined from p E Mor*w. These functions are E-functions and the corresponding function f " A x w --+ w such that f ( n * , k ) - fu.(n.)(k), n* E A , k E w, is a E-function. Hence the *w-spectrum {IR:, g l n E w, p E Mor*w} is construcfivizable and, by Theorem 17, the structure 1RL~ *s being the realclosed field of cardinality ILl, is E~-definable in IHIF(L).
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C o r o l l a r y 18 If L is a dense linear order without endpoints of the power of the continuum, then the field C of complex numbers is E-definable in IHIF(L). P r o o f . Indeed, in HF(L), some real-closed field R of the power of the continuum is E-definable. Therefore, its finite extension R(i), the algebraic closure of N, is also definable in HF(L), and R(i) is isomorphic to C. [] Neither Proposition 16 nor Theorem 17 can help in solving the question of the definability (2-definability) of a concrete uncountable model. By Corollary 18, the field C of complex numbers is E-definable in IHIF(L) for any dense linear order L of the power of the continuum. But what about the definability of the field R of real numbers? P r o p o s i t i o n 19 The field R is not definable in IHIF(L) for any dense linear order L. P r o o f . If ]R is definable in H F ( L ) for some dense linear order L, then it is possible to choose two countable suborders L0 -~ L1 (-~ L) such that L0 contains all the parameters from the definition of I~ in ]HIF(L), and if IR0 and RI are subfields of N defined in ]I-]IF(L0) and HF(L1) respectively (in the same way as ]R in HF(L)), then R0 < IRI < R. Since IR0 and I~ 1 are different countable subfields of IR, they are not isomorphic (since It( is Archimedean). On the other hand, there exists an isomorphism between L0 and L1 (preserving parameters). Consequently, it must be that IR0 ~- N1. The contradiction thus obtained proves the proposition. [] R e m a r k 9 Using a similar argument, we can prove that the field Qp of p--adic numbers is not definable in ][-IIF(L) for any dense linear order L. We establish a general fact which implies that R is not ~-definable in IHIF (L) for any linear order L. D e f i n i t i o n 12 An algebraic structure 9Jr of finite (recursive) signature is called locally constructivizable if for any finite family of elements a0, . . . , an E M, the =t-theory Th3(gYt, ~) of the structure (ff)l, a0, . . . , an) is recursively enumerable. R e m a r k 10 A structure 9Jr is locally constructivizable if and only if, for any a 0 , . . . , an E M, there exist a constructivizable structure 91 and b0, . . . , bn E N such that Th3(fflt, ~) - Th3(91, b).
Chapter 5
E-Definabilityof Algebraic Structures
259
The decidability of the theory Th(9)l) and its categoricity with respect to some cardinality is a sufficient condition for the local constructivizability of 9)I:. In particular, all models of simple theories are locally constructivizable. P r o p o s i t i o n 20 If fib is locally constructivizable and 91 is E-definable in ]HIF(~), then 91 is locally constructivizable. Proof. The assertion is obtained from the above remark and the following general fact:
If ff)t is constructivizable, and a0, . . . , a~-i C M, then the N-theory of the structure ( H F ( ~ ) , >r is recursively enumerable,
rn
R e m a r k 11 Any two infinite linearly ordered sets have the same 3-theory, whence any linearly ordered set is locally constructivizable. C o r o l l a r y 21 The field R is not E-definable in HF(L) for any linear order L. In conclusion, we give an example of a complete decidable theory only the simple model of which is locally constructivizable. Let E ~ {0, 1}* be the set of all finite sequences formed by 0 and 1. The signature a ~ {P] I e C E} consists only of unary predicates. The theory T is defined by the choice of an infini~;e recursive binary tree D(C_ E) with no infinite recursive branches (an example of such a tree can be found, e.g., in [8]) and by the following system of axioms:
Vx PA(~), Vx
(P,o(x) v
P,,
(x) -~
p,(x)),
~ e E,
VX ((Peo(X)--+ rePel(X)) /~ (Pel(X)--~ mPeo(X))), 3x (P~(x) A -'P~o(X) A-,P~, (x)), Vx-,P~(x),
c e E,
r E D,
e E E \ D,
w v~ (p,(x) A -~p,o(~) A -~p,, (x) A p,(~) ^ -~P,o(~) A -,p,, (~) -~ ~ - ~),
tEE.
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References [1] J. Barwise, (ed.), Handbook of Mathematical Logic, Stud. Logic Found. Math., 90 (1977). [2] C.C. Chang and H. J. Keisler, Model Theory, 3rd. edn., Stud. Logic Found. Math., 73 (1990); [lst. edn. 1973, 2nd. edn. 1977]. [3] Yu. L. Ershov, Decision Problems and Constructivizable Models (Russian), (Mathematical Logic and Foundations of Mathematics, Nauka, Moskva, 1980). [4] Yu. L. Ershov, E-definability in admissible sets (Russian), Dokl. Akad. Nauk sssR, 285 (1985) 259-262; [translated in: Soviet Math.- Dokl., 32 (1985) 767-770]. [5] Yu. L. Ershov, Definability and Computability (Russian), (Sibirsk. Shk. Alg. i Log., NII MIOONGU, Novosibirsk, 1996); [translated in: Siberian School of Algebra and Logic, (Consultants Bureau, Plenum, New York, 1996)]. [6] Yu. L. Ershov and E. A. Palyutin, Mathematical Logic (Russian), 2nd. edn., (Nauka, Moskva, 1987). [English translation of 1st. edn., translated by V. Shokurov, (MIR Publishers, Moscow, 1984)]. [7] H. J. Keisler, Fundamentals of Model Theory, in: J. Barwise, (ed.), Handbook of Mathematical Logic, Stud. Logic Found. Math., 90 (1977) 47-103. [8] M. G. Peretyat'kin, Strongly constructive models and enumerations of the Boolean algebra of recursive sets (Russian), Algebra i Logika, 10 (1971) 535-557; [translated in: Algebra and Logic, 10 (1971) 332-345].
Yu. L. Ershov Sobolev Institute of Mathematics pr. Akademika Koptuga, 4 Novosibirsk, 630090 Russia