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25 Short, uncountable models of set theory
25 Short, uncountable models of set theory
A model % of Zermelo set theory is said to be an w-model iff the element a of % such that % t (a = w ) has the property that ( a E ,E ) has order type ( w , <).
COROLLARY B (KEISLER and MORLEY[1968]). Let % be a countable o-model of Zermelo set theory plus ‘wi is regular’. Then there is an omodel 115 > 8 such that if 115 t (b = ol),lbFl = w l .
We shall use the following lemma to obtain extensions having exactly the same ordinals as the original model in many situations. LEMMA C. Let % be a model of ZF, % < 23, and let C = {b E B: For some c1 E A, 23 I. c1 is an ordinal ~b E R(c1)). Then if 6 is the model with universe C,
138
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PROOF.We show that for every formula cp(uo . . . u,) and all cl, . . ., c,
E c,
.
If 23 k (3uo)p[cl . . c,] then for some co E C,B k cp[co . . . c,]. (1) It will then follow that 0.i 23. Since also B i 23 and 'Z c Q, we can conclude from (1) that the required formula B i 0.i2' 3 holds. For some a E A , 23 k (a is an ordinal), and
B C c1 E R(Ix)A.. . A c, E R(a). By the reflection principle in 8 there exists fl E A such that 8 k (fl is an ordinal) A a 5 /?,and both the formulas cp, (3uo)cp are absolute for B and the R(P) of B;that is, 8 k u o , . . ., u, E R(B) + (cp(uo . . . u,)
c-)
cpR'B)(~o. . . on)),
(2)
and
B t. u l , . . ., u, E R(B) -+
((3uo)cp(u . . . u,,)c-)(3uo E R(/3))cpR'B'(uo. . . u,)). (3)
Since 'Z i 23 the universal closures of (2) and (3) also hold in 8.Since a
IP,
23 k c1 E R(fl)A . . . A c, E R(P).
(4)
Thus (2) and (3) hold for 23 with u l , . . ., u, replaced by cl,. . ., c,. By assumption, 23 k (3uo)cp[c . c,]. Hence by (3) there exists co E B such that 8 C co E R(P)A rpR(')[c0 . . . c,].
..
Using (2),
23 C c0 E R ( / ~ ) A ~ [.C* cn]. ~.
Therefore co E C and (1) holds for 'p. -I COROLLARY D. Let B be a countable model of ZF plus 'every union of a well-orderable set of well-orderable sets is well-orderable'. Then there is a model 23 > B such that (i) B has exactly the same ordinals as B. (ii) For every b E B such that 23 C (b is not well-orderable), we have lbFl
=
01.
Moreover, if
B is well-founded so is B.
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REMARK. The conclusion (i) implies that every well-orderable u E A is fixed, and also if b E B and 8 k ( b is well-orderable), then lbFl 5 w. Conclusion (ii) implies that every non well-orderable a E A is enlarged. Examples of well-founded models M satisfying the hypotheses of the corollary and in which not every set is well-orderable (i.e. choice fails) have been given, e.g., by MATHIAS [1970]. PROOFOF COROLLARY D. Let q ( u ) be the formula ‘ u is not well-orderable’ and $ ( u ) be ‘ u is well-orderable’. It then follows from the hypotheses that q(M) is regular over $(a). Hence by Theorem 36 there exists 8’> CU such that every ordinal of M is fixed and for all non-well-orderable b E B’, IbPl = ol.By Lemma c the submodel B of all b E 23’ such that 93’.It follows that for some CI E A , B’ k b E R(a), is such that 9l < B i 8 has exactly the same ordinals as a, i.e. (i) holds. Since bF = bF,for all b E B, (ii) also holds. The ‘moreover’ clause follows at once from (i). -I Our next aim is to generalize the last corollary to arbitrary models of ZF. LEMMA E. There is a formula W(u) of ZF such that in every model % of ZF, W(M) is the least class in such that: (i) Every singleton of % belongs to W(M). (ii) W(M) is closed under unions of well-orderable subsets, that is, if uE c W(M) and % C (a is well-orderable), then % k W ( u a). PROOF.For each ordinal a, define W, to be the least subset of R(a+ 1) which contains all singletons in R(a+ 1) and is closed under unions of well-orderable subsets. This definition can be expressed in ZF. If a < j, then W, n R ( a + 1) is obviously closed under unions of well-orderable subsets and contains all singletons in R(a + l), whence W, c W,. Define W ( v )c, (3a)(a is an ordinal A v E W,). Then W(u)clearly satisfies (i) and (ii). Let M be a model of ZF and suppose a class q(M) satisfies (i) and (ii). Since each R(a+ 1) is closed under unions of subsets, each set q(%) n R(a+ 1) is closed under unions of well-orderable subsets. Therefore each W, c cp(%), whence W(%) c q(%). This shows that W(M) is the least class satisfying (i) and (ii). i
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LEMMA F. There is a formula W"(u)of ZF with variables c(, u, such that in ZF: (i) Wo((u)t)u is either empty or a singleton. (ii) If a is a limit ordinal, W"(u)c-f (3p cc)Ws(v),intuitively W" = U B < a Ws(iii) For any ordinal a, W"+'(u)c-f u = x for some well-orderable x such that (Vy E x)W"(y), intuitively W a + l = x: x c W" and x is well-orderable}. (iv) W ( u )c,For some ordinal a, W"(u).
-=
u
{u
The proof of (i)-(iii) is by restricting to R(P+ 1) as in Lemma E, and then (iv) follows at once. G . For any model W of ZF, the class A - W(W) is regular over the LEMMA class of ordinals of W.
PROOF. If x
=
us<=y s and each ys
E
W ( a ) ,then x E W(94).i
COROLLARY H. Let W be a countable model of ZF. Then there exists an elementary extension 23 > W such that 23 has exactly the same ordinals as '21 and for all b E B - W(23),lbFl = wl.
PROOF. By Theorem 36 there exists 23' > W such that each ordinal of &L is fixed and for all b E B'- W ( B ' ) ,IbF.l = w1 (using Lemma G ) . Let 8 be the submodel of 23' such that B
=
(b E B': For some ordinal a of
2,23' b b E R(a)}.
By Lemma c, W < 23 < 23'. It is easy to see that 23 has exactly the same ordinals as W and that for all b E B, bF = bF'. The result follows. i COROLLARY I. Let W be a well-founded model of ZF and let 23 > W have exactly the same ordinals as W. Then every U E W(W) is fixed. Thus if 9 k ZFC or even W k (Vu)W(u), then we must have 23 = W.
PROOF. By induction on the ordinals a of W, show that every a E W"(W) is fixed. i A set d is said to be Dedekind iff neither Id] < w nor w 5 (dl. That is, d is neither finite nor infinite. In ZFC it can be proved that there are no
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125
Dedekind sets. Halpern and Levy, and later Mathias, have given examples of countable €-models of ZF in which every set can be linearly ordered but the set of real numbers has a Dedekind subset. These models satisfy the hypotheses of the next corollary. COROLLARY J. Let % be a countable model of ZF in which the following holds: If 0 < Ix,I < o for all n < o,then the sequence (xn: n < o) has a choice function. Then there is a model 23 > % which has exactly the same ordinals as 9.l and such that for all b E B, if 23 C b is Dedekind then lbFl = ol.Thus Dedekind sets are ‘much larger’ than the class of all ordinals when % is viewed from the outside. PROOF.In view of our previous arguments it suffices to show that the class cp(%) of all Dedekind sets in 2 is regular over the class I)(%) of all ordinals of a. Let x E cp(%), y E I)(%), and
fl C (fis a function and x
=
u
{ f ( z ) :z E y } ) .
Suppose that for all zEy, % k If(z)l < o.Since 2 C ( y is an ordinal), we may assume without loss of generality that
% C ( z E z‘
Ey
+ f ( z ) , f ( z ’ )are disjoint and non-empty).
For we may replace f ( z ’ ) by f ( z ’ ) - u z s z , f ( z )and ‘delete’ z’s with f ( z ) empty. If k y < o,then % C 1x1 < o,contradicting % C x is Dedekind. Thus 2 C y 2 w . Using our choice assumption,
8 C (3g)(g is a choice function for ( f ( n ) :n < 0)). Let % C u = { g ( n ) :n < w}. Since the f ( n ) ’ s are disjoint, % k IuI = to A u c x . But this contradicts 2 I= (x is Dedekind). We cannot have Yl k If(z)l 2 w. Hence for some zEy, % C ( f ( z ) is Dedekind), i.e. ‘!I k cp(f(z)).-I Corollaries H and I can be explained in terms of transitive E-models. If \u = ( A , E) is a transitive €-model of ZF, then by ord(%) we mean the
251
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143
least ordinal not in A , which coincides with the set of all ordinals u E A . Using Corollaries H and I we can draw the following conclusions: Let T be a complete extension of ZF. Then T has a transitive ernode123 of power u1such that ord(23) is countable if and only if T has a countable transitive €-model and T k (3u) W(u). T has a transitive €-model 23 of power o1with ord(23) = a, (a a given countable ordinal), if and only if T has a countable transitive model 91 with ord(9l) = u and T b (30) 1W(u). Theorem 36 above appeared in KEISLER [1970], Corollaries A and B appeared earlier in KEISLER and MORLEY [1968]. The models mentioned in connection with Corollary J are constructed in HALPERN and LEVY [1970] and MATHIAS [1970]. The first examples of transitive €-models 23 of ZF such that ord(23) is countable but 93 has power w1 were constructed by Cohen and were not published. His proof used the forcing construction. His method was refined by Easton, Solovay, and Sacks to obtain transitive €-models 23 of ZF such that ord(23)is countable and 23 has power 2" (also unpublished). They evidently did not settle exactly which complete extensions of ZF have such models, as in Corollaries H and r above. The precise connection between the models of this kind constructed via forcing and via the Omitting Types Theorem has not been worked out. The question of whether Corollary H holds with 2" in place of o1is open.
A model % of Zermelo set theory is said to be an w-model iff the element a of % such that % t (a = w ) has the property that ( a E ,E ) has order type ( w , <).
COROLLARY B (KEISLER and MORLEY[1968]). Let % be a countable o-model of Zermelo set theory plus ‘wi is regular’. Then there is an omodel 115 > 8 such that if 115 t (b = ol),lbFl = w l .
We shall use the following lemma to obtain extensions having exactly the same ordinals as the original model in many situations. LEMMA C. Let % be a model of ZF, % < 23, and let C = {b E B: For some c1 E A, 23 I. c1 is an ordinal ~b E R(c1)). Then if 6 is the model with universe C,
138
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I39
PROOF.We show that for every formula cp(uo . . . u,) and all cl, . . ., c,
E c,
.
If 23 k (3uo)p[cl . . c,] then for some co E C,B k cp[co . . . c,]. (1) It will then follow that 0.i 23. Since also B i 23 and 'Z c Q, we can conclude from (1) that the required formula B i 0.i2' 3 holds. For some a E A , 23 k (a is an ordinal), and
B C c1 E R(Ix)A.. . A c, E R(a). By the reflection principle in 8 there exists fl E A such that 8 k (fl is an ordinal) A a 5 /?,and both the formulas cp, (3uo)cp are absolute for B and the R(P) of B;that is, 8 k u o , . . ., u, E R(B) + (cp(uo . . . u,)
c-)
cpR'B)(~o. . . on)),
(2)
and
B t. u l , . . ., u, E R(B) -+
((3uo)cp(u . . . u,,)c-)(3uo E R(/3))cpR'B'(uo. . . u,)). (3)
Since 'Z i 23 the universal closures of (2) and (3) also hold in 8.Since a
IP,
23 k c1 E R(fl)A . . . A c, E R(P).
(4)
Thus (2) and (3) hold for 23 with u l , . . ., u, replaced by cl,. . ., c,. By assumption, 23 k (3uo)cp[c . c,]. Hence by (3) there exists co E B such that 8 C co E R(P)A rpR(')[c0 . . . c,].
..
Using (2),
23 C c0 E R ( / ~ ) A ~ [.C* cn]. ~.
Therefore co E C and (1) holds for 'p. -I COROLLARY D. Let B be a countable model of ZF plus 'every union of a well-orderable set of well-orderable sets is well-orderable'. Then there is a model 23 > B such that (i) B has exactly the same ordinals as B. (ii) For every b E B such that 23 C (b is not well-orderable), we have lbFl
=
01.
Moreover, if
B is well-founded so is B.
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SHORT, UNCOUNTABLE MODELS OF SET THEORY
REMARK. The conclusion (i) implies that every well-orderable u E A is fixed, and also if b E B and 8 k ( b is well-orderable), then lbFl 5 w. Conclusion (ii) implies that every non well-orderable a E A is enlarged. Examples of well-founded models M satisfying the hypotheses of the corollary and in which not every set is well-orderable (i.e. choice fails) have been given, e.g., by MATHIAS [1970]. PROOFOF COROLLARY D. Let q ( u ) be the formula ‘ u is not well-orderable’ and $ ( u ) be ‘ u is well-orderable’. It then follows from the hypotheses that q(M) is regular over $(a). Hence by Theorem 36 there exists 8’> CU such that every ordinal of M is fixed and for all non-well-orderable b E B’, IbPl = ol.By Lemma c the submodel B of all b E 23’ such that 93’.It follows that for some CI E A , B’ k b E R(a), is such that 9l < B i 8 has exactly the same ordinals as a, i.e. (i) holds. Since bF = bF,for all b E B, (ii) also holds. The ‘moreover’ clause follows at once from (i). -I Our next aim is to generalize the last corollary to arbitrary models of ZF. LEMMA E. There is a formula W(u) of ZF such that in every model % of ZF, W(M) is the least class in such that: (i) Every singleton of % belongs to W(M). (ii) W(M) is closed under unions of well-orderable subsets, that is, if uE c W(M) and % C (a is well-orderable), then % k W ( u a). PROOF.For each ordinal a, define W, to be the least subset of R(a+ 1) which contains all singletons in R(a+ 1) and is closed under unions of well-orderable subsets. This definition can be expressed in ZF. If a < j, then W, n R ( a + 1) is obviously closed under unions of well-orderable subsets and contains all singletons in R(a + l), whence W, c W,. Define W ( v )c, (3a)(a is an ordinal A v E W,). Then W(u)clearly satisfies (i) and (ii). Let M be a model of ZF and suppose a class q(M) satisfies (i) and (ii). Since each R(a+ 1) is closed under unions of subsets, each set q(%) n R(a+ 1) is closed under unions of well-orderable subsets. Therefore each W, c cp(%), whence W(%) c q(%). This shows that W(M) is the least class satisfying (i) and (ii). i
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LEMMA F. There is a formula W"(u)of ZF with variables c(, u, such that in ZF: (i) Wo((u)t)u is either empty or a singleton. (ii) If a is a limit ordinal, W"(u)c-f (3p cc)Ws(v),intuitively W" = U B < a Ws(iii) For any ordinal a, W"+'(u)c-f u = x for some well-orderable x such that (Vy E x)W"(y), intuitively W a + l = x: x c W" and x is well-orderable}. (iv) W ( u )c,For some ordinal a, W"(u).
-=
u
{u
The proof of (i)-(iii) is by restricting to R(P+ 1) as in Lemma E, and then (iv) follows at once. G . For any model W of ZF, the class A - W(W) is regular over the LEMMA class of ordinals of W.
PROOF. If x
=
us<=y s and each ys
E
W ( a ) ,then x E W(94).i
COROLLARY H. Let W be a countable model of ZF. Then there exists an elementary extension 23 > W such that 23 has exactly the same ordinals as '21 and for all b E B - W(23),lbFl = wl.
PROOF. By Theorem 36 there exists 23' > W such that each ordinal of &L is fixed and for all b E B'- W ( B ' ) ,IbF.l = w1 (using Lemma G ) . Let 8 be the submodel of 23' such that B
=
(b E B': For some ordinal a of
2,23' b b E R(a)}.
By Lemma c, W < 23 < 23'. It is easy to see that 23 has exactly the same ordinals as W and that for all b E B, bF = bF'. The result follows. i COROLLARY I. Let W be a well-founded model of ZF and let 23 > W have exactly the same ordinals as W. Then every U E W(W) is fixed. Thus if 9 k ZFC or even W k (Vu)W(u), then we must have 23 = W.
PROOF. By induction on the ordinals a of W, show that every a E W"(W) is fixed. i A set d is said to be Dedekind iff neither Id] < w nor w 5 (dl. That is, d is neither finite nor infinite. In ZFC it can be proved that there are no
142
SHORT, UNCOUNTABLE MODELS OF SET THEORY
125
Dedekind sets. Halpern and Levy, and later Mathias, have given examples of countable €-models of ZF in which every set can be linearly ordered but the set of real numbers has a Dedekind subset. These models satisfy the hypotheses of the next corollary. COROLLARY J. Let % be a countable model of ZF in which the following holds: If 0 < Ix,I < o for all n < o,then the sequence (xn: n < o) has a choice function. Then there is a model 23 > % which has exactly the same ordinals as 9.l and such that for all b E B, if 23 C b is Dedekind then lbFl = ol.Thus Dedekind sets are ‘much larger’ than the class of all ordinals when % is viewed from the outside. PROOF.In view of our previous arguments it suffices to show that the class cp(%) of all Dedekind sets in 2 is regular over the class I)(%) of all ordinals of a. Let x E cp(%), y E I)(%), and
fl C (fis a function and x
=
u
{ f ( z ) :z E y } ) .
Suppose that for all zEy, % k If(z)l < o.Since 2 C ( y is an ordinal), we may assume without loss of generality that
% C ( z E z‘
Ey
+ f ( z ) , f ( z ’ )are disjoint and non-empty).
For we may replace f ( z ’ ) by f ( z ’ ) - u z s z , f ( z )and ‘delete’ z’s with f ( z ) empty. If k y < o,then % C 1x1 < o,contradicting % C x is Dedekind. Thus 2 C y 2 w . Using our choice assumption,
8 C (3g)(g is a choice function for ( f ( n ) :n < 0)). Let % C u = { g ( n ) :n < w}. Since the f ( n ) ’ s are disjoint, % k IuI = to A u c x . But this contradicts 2 I= (x is Dedekind). We cannot have Yl k If(z)l 2 w. Hence for some zEy, % C ( f ( z ) is Dedekind), i.e. ‘!I k cp(f(z)).-I Corollaries H and I can be explained in terms of transitive E-models. If \u = ( A , E) is a transitive €-model of ZF, then by ord(%) we mean the
251
SHORT, UNCOUNTABLE MODELS OF SET THEORY
143
least ordinal not in A , which coincides with the set of all ordinals u E A . Using Corollaries H and I we can draw the following conclusions: Let T be a complete extension of ZF. Then T has a transitive ernode123 of power u1such that ord(23) is countable if and only if T has a countable transitive €-model and T k (3u) W(u). T has a transitive €-model 23 of power o1with ord(23) = a, (a a given countable ordinal), if and only if T has a countable transitive model 91 with ord(9l) = u and T b (30) 1W(u). Theorem 36 above appeared in KEISLER [1970], Corollaries A and B appeared earlier in KEISLER and MORLEY [1968]. The models mentioned in connection with Corollary J are constructed in HALPERN and LEVY [1970] and MATHIAS [1970]. The first examples of transitive €-models 23 of ZF such that ord(23) is countable but 93 has power w1 were constructed by Cohen and were not published. His proof used the forcing construction. His method was refined by Easton, Solovay, and Sacks to obtain transitive €-models 23 of ZF such that ord(23)is countable and 23 has power 2" (also unpublished). They evidently did not settle exactly which complete extensions of ZF have such models, as in Corollaries H and r above. The precise connection between the models of this kind constructed via forcing and via the Omitting Types Theorem has not been worked out. The question of whether Corollary H holds with 2" in place of o1is open.