Chapter 8 Partition Cardinals and Model Theory: Silver's Results

CHAPTER 8

PARTITION CARDINALS AND MODEL THEORY: SILVER'S RESULTS $1. Indiscernibles in a structure The strongest results have been obtained from partition properties, not by looking at sets alone, but by looking at more general structures. We have already made one application in this direction in ch. 6 $4.4; we want to make several more. We shall make heavy use of the notion of indiscernibles for a structure: 1.1. Definition. Let 'u be a structure and X a subset of A , the universe of 'u, which is linearly ordered by < Then (X,< ) is a set of indiscernibles for 'u iff for each n, any two n-element subsets of X , taken in the order given by <, satisfy exactly the same formulas in 'u, i.e., if xo < . . . < x ~ -xb~ <, . . . < xk- 1, are any two n-element subsets of X taken in order, and is any formula of 9% with at most v,,, u l , . . ., v,free, then 'u k +(xo, . . ., x,) iff 'u i=+(x;, . . ., xk). (Here the relation < ordering X may be one of the relations of the structure '21, but it does not have to be.) This definition is really only of interest when X is infinite. We shall be interested in the connection between partition properties and indiscernibles. We first give a use of Ramsey's theorem in the following result, due to Ehrenfeucht and Mostowski. We shall want to use the methods of this proof again.

.

1.2. Theorem (ZFC). Let Yl be an infinite structure, and let ( X , < ) be any ordered set, where X is disjoint from A , the universe of 'u. Then there is an elementary extension 2S of 'u in which X can be embedded, in such a way that ( X , <) is a set of indiscernibles f o r 2S. If 3 has a relation which linearly orders an infinite subset of A , then we can take the embedding to be such that i in 2l' extends the order < . Proof. First we note from the definitions in ch. 3 $4 that we shall have 'u i 9" iff 2l' can be regarded as a model of the theory of (3, (a)aeA),which must be such that the individual in 'u' interpreting

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31

the constant symbol for a is identified with a. (W will actually be the reduct of such a model which leaves out the distinguished constants (a),,,). So we take the language for the structure
. ., C,J *

++

4(c,;,

. . *,

c,;),

where 4 is a sentence of 2'involving only the constants c, shown, and xo < . . . < x, and xb < . . . < xi are from X;together with the sentences c, < c: for each x < x' from X,where < is the relation symbol denoting the relation of 'u which is to be extended to include <. (We omit these last sentences if no such relation is involved.) Write Th('u, (a),,,) for the theory of (a, (a),€,); we shall show that T u Th('u, (a),,,) is consistent. This will suffice for the theorem; for suppose 23 is any model of this set of sentences. If we identify the individual of 23 interpreting the constant symbol for a, with a itself (for each a in A ) , and then restrict to the language of 2l, we shall have a structure 2l' which is an elementary extension of 2 ; and if we also embed X in this structure by identifying x E X with the individual of B interpreting c,, then since B satisfies T, ( X , < ) will be a set of indiscernibles for 91', and < will be the restriction of < to X . So we must show that T U Th(2l, (a),€,) is consistent. We do this by relying on the compactness theorem, quoted in ch. 1 $2.5, and we show that every finite subset is consistent. We do this by showing that for every finite T' c T, T' U Th(8, (a),,,) is consistent; we show that by suitably interpreting the constants c, which occur in T' as elements of 2l. we can make '8 into a model of T' u Th('U, (a),eA>. We can assume that ( X , < ) has no last element (by extending X if necessary), and then we can take m large enough so that we can assume that the sentences of T' are the sentences

. . ., c,,) ++ +,(c,;, . . ., c;,) for i = 1, . . ., 1, where xo < . . . < x, and xb < . . . < xk (we may have to add +i(czo,

redundant constants to make all sentences of T' have the same number of constants c,, but if X has no last element we can do that), or are of the form c, < c,,. Now let B be the infinite subset of A ordered by <'. We define a partition of [B]"'+'by:

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$11

INDISCERNIBLES IN A STRUCTURE

. .

237

if b, <' . . . <' b , and b; <' . . <' bk, then {b,, . . ., b,} {bh, . . ., bk} iff for each i = 1, . . ., 1, 3 k 4i(b07 . ., bm) iff '2I k +i(bh, . . ., bk). N

This partitions [B]"'+' into at most 2' sets, and so since I is finite and B is infinite, we can apply Ramsey's theorem and find an infinite subset B, = B which is homogeneous for this partition. Now let Y = X be the set of all x E X such that c, occurs in T'. Since B, is infinite, and linearly ordered by <', and Y is finite and linearly ordered by <, there must be an order preserving map from Y into B,; suppose h : Y + B, is order preserving. Then we claim that the structure
* *,

C,J

-

4i(C,b7

*

*,

c,',)

is in T', then xo <, . . ., < x , and x: < . . . < xk from Y, so h(x,) <' . . . <' h(x,) and h(xh) <' . . . <'h(xb) from B,, and so since B, is homogeneous, (h(xo),. . ., h(x,)) (h(x;), . . ., h(xk)},in particular 'u k +@(x,), . . ., h(x,)) iff % C g$(h(xh), . . ., h(xk)). Hence (%, (a)aEA, ( A ( X ) ) ~k T', ~ ~ and ) we have the result. 0

-

1.3. Exercises (1) Models with few element types. The n-element type of an n-tuple a,, . . ., a,,-l of elements of a structure % is the set of formulas with at free which hold at a,, . . ., in 1 ' 1. It is clear that most uo, . . ., if ( X , <) is indiscernibIe in '2I then the n-element type of any n-tuple from X will be the same if the order is fixed. Use Theorem 1.2 to show that if % is infinite, then Th % has models which are arbitrarily large, but which realize at most I distinct n-element types, where 1 is the length of %. [Take % as being tidy and consider the elementary substructure of '3' generated by X,where % and X are as in 1.2.1

(2) Show that for any theory Q, with a definable ordering <, any structure % which is a model for Q such that <' is infinite has an elementary extension 'u' in which < " is not well-founded.

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$2

[Use Theorem 1.2 to embed a non-well-ordered ordering (Y,<) as a set of indiscernibles in %'.I Deduce that no set of first-order axioms can ensure that a given relation is a well-ordering in every model of the axioms, unless it implies that the given relation is finite. (Note that it is easy to give second-order axioms to say that a given relation is a well-ordering; this was done for well-founded in ch. 5 37.10.) S2.

K -+

(a)'" and indiscernibles

We shall not continue with general applications of partition properties to models but restrict our attention to the special case of relations of the form K ( x ) ' " . First we can give a model-theoretic equivalent of -+

K -9

(cC)l~":

2.1. Theorem (ZFC). condition:

(*I

If A

is inJinite,

K

-+

is equivalent to the

For every structure 2l of length A, which has a subset X of its universe which is ordered in order type K by a relation <, there is a subset Y c X with order type cc under < such that ( Y , <) is a set of indiscerniblesfor 2l.

Proof. We show first that K -+ (.)if implies (*). (This is almost a repetition of part of the proof of ch. 6 Theorem 4.4.) We take 2l to be a structure of length A, and we shall take X to be K itself. We define a partition of [ K ] < " by the relation

{Xl, . . xn} a,

6,., xh}

(where x1 < . . . < x,, x',< with at most vl, . . ., v , free,

*

. . . < xh)

iff for all formulas

#J

of 9%

% ' b +(xl, . . ., x,) iff 3 b +(xi, . . ., xh).

Since there are only il formulas in 2ZX, this partitions [ K ] < ~into at most 2" sets. So since K + ( K ) ; ~ " , we must have a subset Y c K of order type cc which is homogeneous for this partition. But then ( Y , < > is a set of indiscernibles for X,by Definition 1.1. Now we show that (*) implies that K + (IX),'~. Suppose that

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$21

(a)<

K i

239

AND INDISCERNIBLES

f:[ K ] < @ -i '2 is given. We form the structure 'u as foilows: the universe of 2l is K . For each n < cu, ,4 < 1, we add the relation RnSB given by

Rn,B(xl,.. ., x,) iff x1 < . . . < x, < K andf(x,, . . ., x,)(P) = 1 .

This will give a structure of length 3, (since il is infinite), and so by (*), taking X = K , we have a set Y of indiscernibles, of order type M. But then for each x1 < . . . < x,, x', < . . . < x: from Y,and each i3 < il, we have Rn.B(X1,

-

* -9

xn) iff Rn.B(XL . .

.?

i.e. f(x1, . . ., xn)(P) = f ( d , . . ., x:)(P). But this means thatf(x,, . . ., x,) =f(xi,. . ., xh), and Y is homogeneous forf, and we have shown that K -i (u),',w. We can use this theorem to finish the job which was started in ch. 7 Theorem 4.8 :

2.2. Corollary (ZFC). r f u is a limit ordinal and K is ~ ( a )i.e., , the least cardinal such that K -i ( N ) : ~ , then for any il < K , we have K -i (u):". Proof. Since il < K , there must be a function g : [A]< -i 2 such that there is no subset of ilof order type u homogeneous for g . Now suppose ~ il parts; we form the thatf: [ K ] < O -i il is any partition of [ K ] < into structure with universe K , and for each n, m < w, the relations +, y, (of 2n places) and xm., (of m . n places), where for x1 < . . . < x2, we have +n(X1, . . Xzn) holds iff f(x1, * * * > xn) = f(xn+ 1, * . .>~ z n ) , and Y n ( X 1 , * XZn) holds iff f(x1, * * xn) < f(xn+1, * Xd; and for x1 < . . . < xm.,we have @

-3

-

-3

*?

$5

X,,,.,(X~, . . ., xm.,) holds iff g(al,. . ., uk) = 1, where pi = f ( ~ * . , , +. ~ . .,, x({+~),) for i = 0, 1, . . ., m - 1, and a1 < . . . < uk are the ordinals Po, . . ., Pm-l in order. We also add the relation < on K . This gives a structure 2l = ( K , <, y,, x ~ . , ) ~ . ,of < countable ~ length. Now by Theorem 4.8 of ch. 7, we know that K -+ (u);:, and so by Theorem 2.1,2l must have a set of indiscernibles Y ordered in type u by < . We show that Y must be homogeneous for$ Given n < w, suppose

+,,

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PARTITION CARDINALS AND MODEL THEORY

[CH.8,

-

$2

that ~1 < * . . < Y z n from Y. If f b l , * * *,Yn)= f ( ~ n + l , * *,YZn), then this must happen for all such 2n-tuples from Y (since Y is indiscernible for the relation +,J. So if y1 < . . . < y n and y i < . . . < yk from Y, since Y has no last element (as u is a limit ordinal), we can pick Y , , + ~< . . . < y z n from Y with y , < Y , , + ~and yk < Y , , + ~ .But thenf(y1, ., Yn) = f(Yn+l, * -,~ 2 n = ) f ( y ' l , * . .,~ h ) and , Y is homogeneous for f on n-tuples. We show that this must in fact happen for each n. For suppose not, for some n < w . Then we can find a sequence of n-tuples yf < . . . < yfl from Y, for j3 < u, such that for 9, < j3' < u, we have yft < y i . (This needs only that u is a limit ordinal, since then n . u = u for ordinal multiplication.) Now put

-

yo

-

=f(r!,. . .,A>,for B

< u.

Since Y is indiscernible for the relation +,,, and we have assumed that f ( ~ l , . . ., yn) # f ( y n + l , . ., yzn) for Some 2n-tuple yi < . . . < . ~ 2 a from Y, we must have yB # yB,for ,b # @' from a, and since also Yis indiscernible for the relation y,,, we must also have yB < yBt for 13 < p' < u (since the opposite assumption would give an infinite descending sequence of ordinals, since u is infinite). So we set I' = {yB I j3 < a}, and T is a subset of /I of order type u. Now we use the fact that Y is indiscernible for the relations x,,,.,,for each m < w . By definition of x,,,,,if j3,, < . . . < ,Bm-l < u, we shall have

-

;s,,,(yp,

. . ., yfto, y p , . . . . . . . . .,

y$-1)

holding iff g(yBo,. . ., yB,-l) = 1

(since we have seen that the ordinals yso, . . ., yB,-l will all be different and will be in order). So since Y is indiscernible for x,,,.~,g must be constant on m-tuples from I: But this is true for each m c UJ, hence is homogeneous for g , and this contradicts the assumption that there is no homogeneous set for g of order type u. So the corollary is proved. 0

r

2.3. We now want to look at indiscernibles in models of set theory. where a is a Suppose that K is the first cardinal such that K + limit ordinal > w . We shall work in the rest of this section from the assumption that such a K exists. Then K is strongly inaccessible by ch. 7 $4.7, so we know that V, is a model of ZFC (really (V,, E(V,)>).

CH.8,

$21

K

24 1

AND INDISCERNIBLES

-+

It contains the definable well-ordering ( K , <), and so by 2.1 we know there is a set of ordinals X c K , of order type at least a, which is a set of indiscernibles for V,. (Since L, c V, and is definable in V,, X will also be a set of indiscernibles for &.) Now let T be the set of terms (involving free variables) which are definable in V,, i.e., those abstraction terms {x I d(x, vo, . . ., u , , - ~ ) } (which we shall write as t(vo, . . ., v , , - ~ )for ) which J', Vvo,

*

-

*?

v n - 1 3 ~(Y = {X

I +(x, ~ 0 ,

* * -3

vn-d}),

and so are always legitimate in V,. We want to discuss the values of these terms in V,, when the free variables are assigned values from X , and we shall use the following notation. We write x',$,,2, etc., for arbitrary finite sequences xo,. ., x,-~from X , of any length n, always taken in order (so xo < . . < x,-~). We write x' < $ for: every term of the sequence x' precedes every term of the sequence $ (i.e., if 1 has length n, x' < $ iff x,-~ < yo). For a sequence of length one we may write, e.g., y instead of $; and we indicate concatenation of sequences by juxtaposition, so that we shall write, e.g., I , y , 2 to stand for a single sequence (with the implication that x' < y < 2). Now if t is a term in T with vo, . . ., free, we shall write t ( I ) for the value assigned to t in V,, when xo,. . ., x , - ~ are assigned to u0, . . ., u , , - ~ , respectively; we shall always assume that 1 is a long enough sequence for this purpose. Given this notation, we can prove the first facts about X :

.

.

2.4. Lemma. There is a set of indiscernibles X of order type tl such that f o r each term t in T: (i) ift(2)is an ordinal, then t(x') < y for any I , y from X with 2 < y ; (ii) ift(2, y , 2) < y , then t(2, y , 2) = t(1, y , 2') for any ?'from Xwith y < z'; (iii) if y < t(1, 2), where 1 < y < ?from X , then zo < t(I,z') (where zo is theJirst member of 2). Proof. Since K is the least ordinal such that K 4 we know that for each p < K there is a function g o : [p]' -+ 2 with no homogeneous set of order type a. For each n < u), let G,@, 2) = go($) for increasing sequences 2 of length n from p, and take G,(B, 2) = 0 if 2 is not of this form. We let B, be the structure (V,, E(V,), (G,)),,<@;since this has length w, by 2.1 there is a set of indiscernibles X c K for BK,of order

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type a, and any such X will be a set of indiscernibles for V,. Suppose X is enumerated in order as (xv),<%;we shall pick such a set of indiscernibles X for which x, is as small as possible, and show that for this set X , (i)-(iii) will hold. (In fact (i) and (ii) will hold for any set of indiscernibles for B,, but (iii) needs the condition on xu-note that we need tc > w for x, to exist.) For (i), suppose that t(2) B y for some 2 < y from X . Then since X is a set of indiscernibles we must have t(2) 2 Sup(X). (If K = a, this is already a contradiction, since then Sup(X) = K 4 V , ; we proceed with the case tc < K . ) So let t(2') = 0,where 2' is the sequence xo,xl,.. ., x,-~ of the first n members of X ; and using the fact that X is indiscernible for each G,, n < w , we must have Gn(t(ZO), u") = Gn(t(lo),21,

for all 3, 2 with go < $ and go < 2 (remember that each G, only takes the values 0 and 1). But by definition of G,, this means that the set X - {xo,. . ., xn-.l}is homogeneous for the function g , ; since M 2 co, the order type of X - {xo,. . ., x,-~} is also a, and this contradicts the assumption about go. So (i) is proved. The proof of (ii) is very similar: Suppose that t(2,y , 2) 6 y and that t(2,y , 2) # t(2,y , 2') for some 2' from X with y < 2'. Then for all 2, y , 2,z"' from X with 2 < y < P < P', we must have either (a) t(2,y , 2) < t(2,y , 2'), or (b) t(2,y , 2) > t(2,y , 2'). (b) would give an infinite descending sequence of ordinals since X has no last member and we could choose (.'"),,< ,from X so that y < 2 O < 2l < . . . So we must have (a). Now choose go < y o as small as possible from X,and an increasing sequence of n-tuples (2"),
-3

TVmJ

= Gm(yO7 r7&,

.

-3

r&-J.

So since T,, 6 yo for q < u, the set { r , I q < a} is a homogeneous set for gyo,and its order type (by (a)) is a, which contradicts the assumption about gyo.So (ii) is proved. (Note that again if K = M, there will be

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CONSTRUCTING MODELS USING INDISCERNIBLES

243

no need to use G,, and every set of indiscernibles for V, of order type K must satisfy (i) and (ii).) To prove (iii), we use the further assumption that x, is as small as possible. Suppose that y < t(x",z') < zo, where 2 < y < z'and zo is the first member of z';let I have length n, and z'have length m. First we note that if I < z' < Y , we must have t(I,2) < t(x',z"). For if not, we should always have either t(2,z') > t(I,z")(which would lead to an infinite descending sequence of ordinals, as in (b) above, and so is impossible), or t(2,z') = t(I,Y); and then if 2" starts with y , since I < y, we shall have y < t(2,2) = t($, 2") < y by the assumption on t , which is impossible. So let Yo be the first n elements from X,and y o the (n + 1)st element. be an enumeration of the rn-tuples from X after yo, so that Let (."')),<, y o < 2' < z'l < . . . and chosen as small as possible from X , so that we shall have < x , for q < w , and 2, = x,, . . ., Now set T~ = t(x", 2") for q < a,and let Y be {T, I q < a]. We claim that Y is also a set of indiscernibles for DK,for if 4 is a formula of the language for Dx,and qo < . . . < q n < a, then C $ ( T ~ ~., . .,T,,) is C$(t(IO,

z'"),

. . .)t ( 9 , z'"),

and writing this last formula as x(2O, 5no, . . ., Z'"n), we see that it will hold in 23, independently of the choice of qo < . . . < q n from u, since X i s indiscernible for 'BK. But now Y has order type u, and T , = t(I", 2,) < x,, which contradicts the choice of X as having x , as small as possible.

$3. Constructing models using indiscernibles We want to see in this section how we can use a set of indiscernibles for one structure in order to get many other structures. We shall start with L,, and show that these new structures will also be models of ZFC V = L, and if K satisfies K + (u)'" for an u 2 q,we show that the new structures will be well-founded. In the next sections we look at the consequences for the constructible universe. We shall really be extending the ideas used in the proof of the Ehrenfeucht-Mostowski theorem 1.2.

+

3.1. Definition. Suppose that 'u is any structure, and that ( X , <) is an infinite set of indiscernibles for 'u. Form the structure {a, (x),,~),

244

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[CH.8,

$3

and let 9; be the language for this, with new constants {cz I x E X } , c, denoting x. We write K(%, X ) for the set of sentences of 2'; which hold in ('u, (x),,~);i.e., r$(czo,. . ., czn-,) belongs to K(%, X ) iff where in writing +(czo, . . ., czn-,) we shall always % +(xo,. . ., x,assume that xo < . . . < xnelfrom X and that czo, . . ., czfl-, are all the new constants which appear in 4. Note that K(%, X) is complete, in that if +(uo, . ., un-J is any formula without new constants, then either +(czo,. . ., czn-,) is in K('u, X ) for all xo < . . . < x , - ~ from X , or +(c,~,. ., cZfl-,) is in K(%, X ) for all xo < . . . < x,-~ from X ; and also the theory of 2l, Th 'u, is included in K(%, X ) . Now given any set K(%, X ) , and any other infinite ordered set ( Y , <), we write K(%, Y ) for the set

.

.

7

IYO

- -

< * < Y n - 1 from Y , and for some xo < . . . < x,-~from X , +(CZO, * ., C Zn- ,) E K('u, XI>. These will be sentences of a language 9; with new constants c, for y E Y . The proof of Theorem 1.2 showed that for any infinite structure 'u and any ordered set ( Y , <), some consistent set K(%, Y ) could be found; similarly here the same proof shows that if ( X , < ) is an infinite set of indiscernibles for 'u, then the set K(%, Y ) given as above from K(%, X) is consistent for any ordered set ( Y , <). Now we shall be interested in models of K(%, Y ) , and the case we shall be concerned with will be when the language of 'u has plenty of terms available, so that these terms can be used to build models. This will be true when % is the structure L,, since < t is a definable wellordering, and terms giving Skolem functions are definable. So let % be ( L K E(L,)), , and let To be the set of terms definable in 'u, with free variables, and suppose that ( X , <) is a set of indiscernibles for a. Since To contains a set of Skolem functions for 'u, the closure of any subset of L, under To (or really under the operations on L , defined by the terms in To)will be an elementary substructure of 'u, and we can define : (+(cy,,

* * *,

cyn-,)

3.2. Definition. % ( X ) for the elementary substructure of 21 generated by X using T o ; similarly for Y c X , g / ( Y ) is the elementary substructure of '%I generated by Y . Since 2 ( X ) < X, we can see that ((#A'), (x),,,) is a model of K(%, X ) , and similarly {Z(Y ) , ( J I ) ~ ~is~ )a model of K ( 3 , Y ) . We shall also have 8(Y ) < Z ( X ) .

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3 31

245

CONSTRUCTING MODELS USING INDISCERNIBLES

Also iff : Z -+ X is an order-preserving embedding, we can define a structure S ( Z ) .We can take this to be Xcf ” (Z)),and (P(Z), (z),,~> will be a model of K(%, Z ) , and we shall have an elementary embedding of S ( Z ) into 2 ( X ) extending f (where an embedding is elementary iff its range is an elementary substructure). We want to extend all this so that we have (at any rate up to isoY ) for every infinite ordered set Y such that morphism) a structure S( (A?( Y ) ,( Y ) , , ~ ) is a model of K(%, Y ) ;and further iff: Y Y’ is an order-preserving embedding, then we wantfto extend to an elementary embedding of 2(Y ) into #( Y’). We can do this using To,with the following definitions:

3.3. Definition. Given an ordered set ( Y , <), let To(Y ) be the set of terms t(cy0, . . ., cy,-,) for y o < . . . < y a - l from Y , where t E To and Let . has at most vo, . . ., v , , - ~free (these will be closed terms of 9;) be the equivalence relation on To(Y ) given by N

t(cyo, *

*

*,

CYnJ

tYcy;,

-

., c,;-,)

iff the sentence t(cy,, . . ., cy,-,) is in K(%, Y ) ,

= t’(cy;,

. . ., cY;-,)

and let E be the relation on To(Y ) given by * * -,CYnJ E t’(CYb, * * * Y Cy;J iff the formula t(cy0, . ., cy,-,) is in K(%, Y).

t(Cyo’

.

. . ., cy;-

E tr(cy;,

-

,)

Now since K(%, Y ) includes the axioms of equality, the relation E will factor by the equivalence relation ,and we set: #( Y )is the structure whose universe is the set of equivalence classes of To(Y ) under , with the relation which is the quotient of E under .Let Y be the equivalence class of the term c, under ,then we shall have:

-

N

N

3.4. Lemma. (2( Y ) , (jj),Ey> is a mudel ufK(%, Y ) . for Proof. By induction on the construction of sentences of 9;; atomic formulas, this is just the definition of and E. The induction steps for the cases -7, v are trivial, since K(%, Y ) is complete. For quantifiers, we use the fact that To contains terms giving Skolem functions, and the sentences giving their properties will be in K(%, Y ) . 0 N

Note that in view of this, Y can be embedded in X ( Y ) so that

( Y, < ) is a set of indiscernibles for A?( Y ) ;and each element of #( Y )

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PARTITION CARDINALS AND MODEL THEORY

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$3

is given from members of Y (under this embedding) by operations defined by the terms of To. Hence if Y c X , then the structure Z ( Y ) defined in 3.3 is isomorphic to the structure 2 ( Y ) defined in 3.2. For einbeddings, i f f : Y -+ 2 is an order-preserving map between ordered sets Y and 2,we define the embedding f’:#( Y ) -+ Z(2) extending f in the obvious way; for t(cYo,.. ., cy,-,) in To(Y), set f(t(cyo,. . ., c ~ , - ~ ) )= t ( ~ ~ (. ~. .,, c,(~~-,)), ), and let f’ be the quotient of this under the respective equivalence relations; this will be well defined in view of the definitions of K(%, Y ) and K(%,2).In exactly the same way as Lemma 3.4, we now can prove:

3.5. Lemma. I f f : Y -+ 2 is order-preserving, then f’:#( Y ) + Z ( Z ) is an elementary embedding. 0 So Definition 3.3 has given us many more models of set theory if we have an infinite set of indiscernibles for L, to start from. As we noted earlier, we shall show that these are well-founded in some cases (it is clear that they will not always be well-founded, since the ordering ( Y , < ) will be embedded in the ordinals of 2(Y ) , and so at the very least ( Y, <) must be a well-ordering if 3(Y ) is to be well-founded). But we shall first show some other properties of these models. Note that the properties of indiscernibles discussed in Lemma 2.4 can all be cast in terms of properties of K(%, X);we want to see the effect of these properties on the structures Z(Y ) . 3.6. Definition. Let K(%, X)be as in 3.1. We say that K(%, X ) preserves segments if, for any two infinite ordered sets Y, 2 such that Y is a n initial segment of 2 with no last element, the map f:X( Y ) + iP(Z), obtained as above by extending the injection Y c 2, takes the ordinals of X ( Y ) onto an initial segment of the ordinals of 2 ( 2 ) . We want to recast this property as a syntactical property of K(%, X ) (i.e., certain sentences belong to K(%, X)),and we shall adapt the notation of 2.3 to express this. We shall again write 2,y’, etc., for sequences from X,to be taken in order, but now we shall write t(X!), etc., to stand for t(cZo,. . ., czn-l)y and we shall write y instead of c, in writing formulas of 2;for any Y, so that if y is an element of Y, then y is the constant This . makes for simpler writing. Using this symbol denoting it in 9; notation, we have:

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3.7. Lemma. K(%, X ) preserves segments iff it contains all sentences of the form (i) Ord(t(z)) -+ t ( g ) < y-, for 1,y f r o m X with 2 < y , and (ii) t ( 1 , ~JZ, < 2 + t ( $ , i , 1)= t(2, z, z’),for 1,y , 2, ?from X with 9 < y < 2’, where t is in To. Proof. First suppose that the sentences (i) and (ii) are in K(%, X ) , and that Y is an initial segment of Z with no last element. Then any ordinal of S( Y ) is of the form t(1) for some 1 from Y, and if y E Y is such that 1 < y , then by (i) the sentence t @ ) < y is in K(%, Y ) and so t(2) < y holds in .8( Y).So all ordinals of X ( occur before some Y )can be taken as an elementary substructure of member of Y ; since S( 2 Z ) , this will be true in 2 ( Z ) also. Now suppose that we have an ordinal of &(Z) which occurs before y in X ( Z ) ,where y E Y.Suppose this ordinal is t(2, y , 2), where 2 < y < 2 are from 2.We must show that it is in fact in X ( Y ) ; but this is immediate by (ii), since the formula t(g, y , 2) = t(2, y , 1’)will be in K(%, Z), where 2’ is from Y with y < ,?,-and t(2, y,z”) is in #( Y ) . (Such a 2’ will exist since Y has no last member.) Now suppose that for one pair Y,Z with Y a proper initial segment of Z and both with no last element, the ordinals of X ( Y )form an initial segment of 2 ( Z ) under the induced embedding. Let z E Z - Y,then for 9 from Y and any term t in To,if t ( 9 )is an ordinal of#( Y )and t($) = z, then we would have t ( 3 ) = z in K(%, Z), and hence t($) = z for all z E Z with $ < z, which is impossible. So z $2( Y ) , and since the ordinals of 2“( Y ) are by assumption an initial segment of the ordinals of %(Z), we must have t($) < z. Hence the formula t ( $ ) < z is in K(%,Z ) and so all sentences of the form (i) must be in K(%, X ) . Similarly, if t(2, y , 2) 6 y for 1 < y from Y, 2 from Z - Y, then t(3, y , 2) must be in 2(Y ) and so t(2, y , 2) = t‘($’) for some t’ E To, 9’ from Y. But then t(2, y , ?) = t’(jY) also for any z‘ from Z - Y,since then $‘ < 2’. So t ( 5 y , 2) 6 v_ -+ t(g, y, 2) = t ( g , y , 2’) is in K(%,Z), and hence all sentences of the form (iirmust be in%(%, X ) . 0 For the third property of Lemma 2.4, we use the following:

r)

3.8. Definition. Let K(%, X ) be as in 3.1. We say that K(%, X ) preserves limits if whenever y E Y is Sup Z for Z c Y, we have also that y is Sup(2) in 2(Y). We shall have, continuing with the same notation:

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3.9. Lemma. Suppose that K(%, X ) preserves segments, then K(%, X ) preserves limits fi it contains all sentences of the form (iii) y < t(2,g) -+ zo Q t(2, g), where 2 < y < i are from X,zo is thejirsimember of z, and t is in To. Proof. First suppose that the sentences (iii) are in K(%, X ) , and that y o = Sup(Z) in the ordering ( Y , <), for Z c Y. Since K(%, X ) preserves segments, we may suppose that Z is the initial segment of Y determined by yo, and also that Z has no greatest element (otherwise the property is trivial). Then we want to show that yo is the first ordinal in Z'(Y ) which is not in Z ( Z ) , since the ordinals of S ( Z ) form an initial segment of the ordinals of X ( Y ) . So suppose that t ( 3 ) is an ordinal of X ( Y ) not in Z ( Z ) ; we can write y' as i , 2 ,where 2 is from Z and 1 is from Y - Z , and for each y from Z we shall have y < t(2,2). S o , then, using (iii) we have xo < t(2, 2), where xo is the first member of 2, and since 2 is from Y - Z, yo Q xo. Hence yo is the first ordinal of X ( Y ) not in Z ( Z ) . Now suppose that for one zo, zo is Sup(Z) for a subset Z with no last element in a linearly ordered set Y,also with no last element, and that zo is also Sup(2)in X ( Y ) .We show that the sentences (iii) must be in K(%, X ) . Let t be any term in To,and 2 < y < i from Y, such that y < t(x',2) (in 2(Y ) ) . Since Y is a set of indiscernibles for 2@( Y ) , we can choose any x' and y from Z with 2 < y , and z'with first element z,; we shall still have y < t ( 2 , i ) . But y can be arbitrarily large in Z , since Z has no last element, and so zo = Sup(2) < t(2,2) for this 2 , i . Hence we must always have y o Q t(2,p), where yo is the first element of y'; in other words,

2 < @, Z) + go will hold in

< @, 1)

Z( Y ) . Hence all sentences (iii) must be in K(%, X). 0

The main property which we want of the model S(Y ) concerns wellfoundedness, and we make the definition: 3.10. Definition. Let K(%, X) be as in 3.1. We say that K('u, X ) preserves well-foundedness iff, whenever Y is a well-ordering, the model A?( Y ) is well-founded. Note that since we shall only consider cases where 'u is a model of ZFC, it will be equivalent to ask that the ordinals of S(Y ) be wellordered, and this is what we shall really consider. For this property of

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K(U, X ) , it is clear that syntactical conditions cannot be sufficient to make K(U, X ) preserve well-foundedness. But we can go some way towards simplifying it: 3.11. Lemma. Let K(U, X ) be as in 3.1. Then the following are equivalent: (i) K(U, X ) preserves well-foundedness; (ii) for one uncountable well-ordering Yo,%(Yo) is well-founded; (iii) for each countable well-ordering Y, X ( Y ) is well-founded. Proof. (i) implies (ii) trivially, and (ii) implies (iii) since if Y is countable and Yo is uncountable, then Y can be embedded in Yo,and this induces an embedding of #( Y ) into %(Yo),so that if X ( Yo)is well-founded, so is X ( Y ) . We show that (iii) implies (i). Suppose that Z ( Z ) is not well-founded for some well-ordered Z ; then there will be a sequence of terms tl (i < w ) from To and of sequences 2' (i < w ) from Z such that to(io) >

tl(i1)

>

fZ(22)

>

. . .,

since there is to be an infinite descending sequence of ordinals in X ( Z ) . But now let Z o be the sub-ordering of Z containing just the members of each 2' for i < w . Zomust be countable (each Z"l is finite) and each of the ordinals ti(,?') is in X(Zo);since this is elementarily embedded in A?(Z),this means that the infinite descending sequence of ordinals is in A?(Zo),and %(Z0)is not well-founded. But this contradicts (iii). These properties of K(U, X ) , namely preserving segments, limits, and well-foundedness, are the three we have been aiming for; the last is the most powerful. Putting together all the lemmas, we get:

3.12. Theorem (ZFC). If there is a cardinal K f o r which K -+ (U)J<~, then there is a set of formulas K(U, X ) which preserves segments, limits, V = L. and well-foundedness, and which contains all theorems of ZF Proof. We saw in 2.4 that if K is the first cardinal for which K -+ (wl)
+

+

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3.13. Exercise (1) Show that the constructions of 3.1-3.5 can be paralleled starting with any structure '%I which has a sufficient number of definable terms (in particular if 'u is tidy in the sense of ch. 6 $4.1) such that there is an infinite set (X,< ) of indiscernibles for 3 ; and that if 2I has a definable relation which is a linear ordering which extends (X,< ), then 3.6-3.11 will also go over for the parallel construction. 94. Implications for the constructible universe

We want to look now at the structuresYf(a) produced in $3, where

cc is an ordinal (i.e., the well-ordering (u, <)). Suppose that K(%, X)

preserves segments, limits and well-foundedness, and contains Z F + V = L, as in Theorem 3.12. Then *(a) is a well-founded model of ZF + V = L, and so is isomorphic to a unique transitive model for some ordinal /I(.). We also of ZF + V = L ; this must be have elementary embeddings from 2 ( u ) to Z ( 6 ) if u < 6. We make the definitions: 4.1. Definition. For each ordinal u 2 w , define fa to be the unique

collapsing map from2"(u) onto a transitive set, and let /I(.) be the index of this transitive set. Then we shall have f a : X ( u )-+ Also define ea,d for tc < 6 to be the elementary embedding ea,6:2P(u)4Z ( 6 ) given in Lemma 3.5. Since K(%, A') preserves segments, eor,6will take the ordinals of 2 ( u ) onto an initial segment of the ordinals of Z ( 6 ) , if u is a limit ordinal < 6. We use this to show that the maps fa and ea,*fit together as one would expect, at any rate for limit ordinals:

4.2. Lemma. If cc < 6 and u is a limit ordinal, then for x E*(U), f(x> = fd(e".o(X.)). Proof. Let E" be the membership relation for Z ( K(as ) given in 3.3) and Ed for Z ( 6 ) . We prove the result by induction on x in %'(a), i.e., on the relation E", which is well-founded since we have assumed that K(X, X)preserves well-foundedness. So suppose that for all y in =@(a)with y E" x,fa(y)=f,(ea,d(y)). Then

I

I

&(XI = { f ( Y > Y E" x> = {h(ea*d(y)) Y E" XI? fn(e,.n(x)) = (h(4 Ed~",&>-

I

So we must show that for zin=@(d),z Edea,6(x)iff z in ~ ( I xwith ) y E" x.

= e,&)

for some y

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One way is immediate. Since ea,6 is an elementary embedding from X ( a ) into %(6), then y E" x implies ea.d(y)Edea,d(x).So we must show This that if z Edea,d(x),then for some y E %(a), y E" x and z = ea,d(y). needs two facts: First, that ecc,dmaps the ordinals of %(a) onto an

initial segment of the ordinals of Z ( 6 ) (as noted above), and secondly V = L and so has a that each of *(a), Z ( 6 ) is a model of ZF definable well-ordering. So suppose that in Z ( a ) , g is a 1-1 mapping from an ordinal w of %(a) onto x. Then since ea.6 is an elementary embedding, in #(6), ea,d(g)is a 1-1 mapping from ea,d(w)(which is an ordinal ofZ(6)) onto ea,6(x).So since z Ed eaSd(x), z must be (in %(a)) the image under e,,,(g) of an ordinal u, i.e., z = (ea,d(g))(u)(in s ( 6 ) ) . But then u Ed ea,d(w), and so using the initial segment property, u = ea&') for an ordinal u' of &'(a). Hence using again that ea.a is an elementary embedding, z = ea,d(g(u')),and g(u') is in #(a) as required; the result follows. 0

+

4.3. The property which was needed in the above proof between Z ( 6 ) and the image of %(a) under ea,6 is often called the end-extension

property. If ( A , EA) = 2l c 23 = ( B , EB), and both are models of ZF (or any other theory involving E ) , we say that 23 is an end-extension of 2l if, for every a from A , if b EBa for b in B, then b is in A. In other words, members of A get no new members in the extension from 2l to 23. For transitive €-models, it should be clear that there are no other sorts of extensions; but for other structures (even if they are E-structures) extensions need not be end-extensions. One reason why the structures 2 ( a ) are so useful is that they are well behaved in the sense that they can be regarded as being end-extensions of all 2 ( p ) for limit ordinals B < a.

-

Lemma 4.2 can be expressed by saying that the following diagram commutes when a is a limit < S: %(a)

b

LB(a)

In these cases we shall often ignore the mapping eaSdand consider &'(a) as a subset of S ( 8 ) for a < 6, and fd to be an extension of fa.

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Also we shall consider a, 6 to be subsets of #(a) and #(S) respectively (they will be indiscernibles for these structures). The maps f a take the ordinals < a, as members of #(a), onto ordinals in L4(a)which will be indiscernible in These indiscernibles are very important, and we define : 4.4. Definition. For limit ordinals a, set Xa = f a ” (a) (Le., Xa = = {fa(?) I y < a ) will be the set of indiscernibles for L4(,)). Now Lemma 4.2 gives the corollary:

4.5. Corollary. I f a < 6 and both are limit ordinals, then Xa is an initial segment qf X,, and both X , and X , are relatively closed in the order topology (i.e., if Y c X , and Y is bounded in X,,then Sup( Y ) E Xa). Proof. By Lemma 4.2, if y < a, then f a ( y ) =f,(e,,,(y)) =f6(y), (where we are using y as a member of both #(m) and Z ( 6 ) ) . Since each f, is increasing on ordinals, the first result follows. The second result is a restatement of the fact that K(%, X ) preserves limits, for by Definition 3.8, if y is a limit ordinal < a, we havef a ( y ) = Sup,,,( fa( 5)). 0 For cardinals > w , we can say more about the functions t9 andf: 4.6. Lemma. If K > w is a cardinal, then P(K) = K and f,maps # ( K ) onto L,. Proof. First we note that the set of terms Toused in building # ( K ) is countable, and that X is embedded in #(X) for any X. Hence card(%(rc)) = K, and so also card&(,)) = K , i.e., Card(p(K)) = K . But also we shall have card(#(m)) = B for any a 2 w ,and hence for a < K, P(N) < K ; and if X E % ( K ) , then x ~ # ( a ) for some a < K , where since K > w we can take a as a limit ordinal. (We should write x = e,,,(y) for some y ~ & ( a ) ,but we are ignoring this difference.) But now f,(x) = f,(x) E L4(=)c L,, and so LB(,)c L,, i.e., B(K) < K , which gives the result. 0

4.7. Corollary. For cardinals K > w , L, has a set of indiscernibles X, of cardinal K, and i f w < K < I ( I also a cardinal), we have L, 4 LA. Proof. By Lemmas 4.2 and 4.6. 0 We are now in a position to draw conclusions about the constructible universe from our hypothesis, which is that there is a cardinal K such < w . We gather these results together as one theorem. Note that K + (q) that by ch. 6 $3.6, a measurable cardinal satisfies K + ( K ) < ~and so a fortiori K + ( ~ 0 ~ )So~ these ~ . results are extensions of the results of

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Rowbottom given in ch. 6 $4. Some of them were already known to Rowbottom and Gaifman, but our way of deriving them is due to Silver.

If there is a cardinal K such that K -+ (wl) < w, then there is a proper class X of ordinals such that the following hold: (i) X is closed in the order topology, and for any uncountable cardinal A, X , = X n A is of cardinal A (and hence unbounded in A); (ii) if a,/?E X with u < 0, then L, < L,, and ( X n a, <) is a set of indiscerniblesfor La; (iii) every set in L is deBnable in L from parameters in X ; (iv) if a E X , then La < L, and X , < ) is a class of indiscerniblesfor L ; (v) truth in L is definable, i.e., the relation L 1 +(yo,. . ., Y , , - ~ )for yo, . . .,Y , , - ~in L can be expressed by a single formula of ZF with two parameters r+l and ( y o , . . .,y n F 1 ) ;and (vi) every set definable in L (without parameters) is countable (so that certainly V # L). Proof. We take Xa as in Definition 4.4 and set x = U(X, 1 Iim(a) A u > 01. Then (i) follows by 4.5. (ii) follows from 4.2. If a,/? are limit points of X (i.e. a,B E X and a = Sup(X n a), /3 = Sup(X n P)), then a = fs(y) for some limit ordinal y < 6, where a E X,. But then we must have u = B(y); similarly, B = /?(y‘), and by 4.2, LB(y) < LB(,,).Also X n a = X,by 4.5, and so (Xn a, < ) is a set of indiscernibles for La. For other a‘, B‘ E X with a‘ < B’, choose limit points a <3!, < y of X with B’ < y. Then a, /3, a’, B‘ E X n y , a set of indiscernibles for L,; and L , k L, < L,, so also L , k La. < L,,. Also X n a’ = X n La,, and the full result follows by absoluteness. For (iii), suppose Y E L ;then y € L a for some a which is a limit point of X . Then La is the image under fd of Zi(6) for some limit ordinal 6, hence y = &(t(i)),where t is a term of To and z’is an n-tuple from 6. The terms of To are all definable in L,; but by (i) and (ii) we have either L , < La or L, < L , or K = a (where K is the first cardinal such that K + ( w , ) < ~ ) ,and so y is definable in L, using the image of 2 under&, which will be an n-tuple from X,. Since a is also in X , y is definable in L using M. also. For (iv), we are really quoting the theorem (see ch. 3 §4.15(6)) that the union of an elementary chain is an elementary extension of each of the members of the chain. Then since L = u { L aI a E XI,the result follows. 4.8. Theorem.

<

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We can, however, also give a proof which sticks more closely to ZF by using the reflection principle Ro. Given a formula (b and xl,. . ., x, E L , find an tl such that V, reflects the formulas @(xl, . . ., x,), 37 (x E L,) and “X is unbounded”, and such that xl, . . ., x, E V,. (Note that X i s definable from one set of indiscernibles for Lx.)Then we shall have : u is a limit point of X , v, nL = L,,

. . ., xn)++

+L,(~l,

+‘(xl,

. . ., Xn).

But then since +La(xl, . . ., x,) holds iff L, k +(x17. . ., x,), we can use (ii) to see that we can take u to be any member of X with xl, . . ., x, EL,, and we still have +‘(xl, . . ., x,) holds iff La 1 (b(xl,. . ., x,);

i.e., L, < L. (The final statement must still be regarded as a metatheorem, not as a statement in ZFC.) That ( X , <) forms a class of indiscernibles for L now follows from (ii). Now for (v), we use the formula, expressible in ZF, 3 ~(a. E X A yo, . . ., Y n - 1 E L a A L, k +(yo, . . ., ~n-1)).

Within ZFC, we can prove that this formula satisfies the recursive clauses of the definition of L k +(yo, . . ., Y , - ~ ) ,and hence this formula defines truth in L. We shall not write out any of these steps, of course; far more convincing is the argument from outside L, which says that since L is a union of an elementary chain, truth in L is determined by truth in all sufficiently large members of the chain. (Note that now we can take L, < L as a statement of ZF.) For (vi) we notice that since X A col is of cardinal ol,there are certainly countable members of X . Suppose that u is the first member of X.Then L, < L, by (iv), and any set x definable in L (with no parameters) must be in La;since a is countable, so is x. 0 As examples of the power of (vi), notice that it implies that not only HI;,Hi,. . . are countable (since they are definable in L), but even the first inaccessible cardinal of L, the first Mahlo cardinal of L, . . . are all countable. We shall go even further in ch. 9. The class X is in fact completely determined by the properties (i), (ii) and (iii) of Theorem 4.8. A proof of this is sketched in Exercise

4.10(1).

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4.9. Relativizations of this construction The construction of $3 began with a set of indiscernibles for the struc~ . could equally have started with the ture L,, where K -+ ( w ~ ) < We structure L,(x) of sets constructible from a transitive set x, or a set x such that x c L;we only need card(L,(x)) = K. We shall look at this construction for x c o;the same ideas go through with very little change. We shall regard x as a one-place relation (really the relation y E x ) as well as a set, and begin by looking at the structure
E V r A x, (G,),<,)

(where (G,),,, is as in 2.4), and finding a set of indiscernibles for this, X" say, with wth member as small as possible. Then if a" = (L,(x), E(L,(x)), x ) , we can define structures%" ( Y ) from this as in 3.3, and K(21z, X") will preserve segments, limits and wellfoundedness, just as in 3.6-3.11. In particular we shall have wellfounded structures %"(.) for ordinals u > w , and when these are collapsed to transitive structures, these must be of the form Ld(x),since each will contain x ( x will not alter in the collapsing since every member of x is a natural number and so definable). As in 4 . 2 , the elementary embeddings between the %"(a) (for limit a) and the collapsing maps will all fit together, and we shall have the analogue of Theorem 4 . 8 . We shall set C" to be the class of ordinals indiscernible over L(x) obtained in this way for x c w . Note that X, the class given in 4.8 can be taken as Co, since L(0) = L ; indeed, if x E L , then L(x) = L and C" = Co, using Exercise 4.10(1). Cardinals > w will be limit points of C" for each x , just as for X , and we can deduce that any set definable in L(x), for X c w , is also countable.

4.10. Exercises (1) Show that properties (i), (ii) and (iii) of Theorem 4.8 uniquely determine the class X. [Show first that for any two classes satisfying (i), (ii) and (iii), X and x',say, the intersection X n X ' is also closed, unbounded, and use that to show that the set of formulas satisfied by increasing n-tuples is the same. Then follow the construction of the class X from this set of formulas, to show that X and X' must be the same.J (2) Show that the conclusions of 4.8 cannot be drawn from the hypothesis of a cardinal K such that K -+ ( E ) ' " , where u is countable in L, nor from the hypothesis that V u < w'; ( K -+ (u)'"). But the

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conclusion V # L does follow from the hypothesis that there is a cardinal K such that K -+ (w';)<". [Use ch. 7 34.9 ex. (2) to show that K + (a)." for a < oi and V M < 0'; ( K (a)<")will be true in L if they are true in V, and hence they cannot lead to conclusions contradicting V = L unless ZF is 3~ ( K -+ (of;)' ") is clearly inconsistent since inconsistent. But V = L V = L implies oi = ol.Note that K + (wi)'" does not seem to lead to all the conclusions of Theorem 4.8; the obstacle to deriving the conclusions of 4.8 from K + (a)<" for M < w';, or any countable a, lie in proving that K(%, X)preserves well-foundedness for the set of indiscernibles X which arises.] --f

+

$5. A A: non-constructible set In $$3 and 4 the set of formulas K(%, X) was given as a set of which has constants c, for x in X . But sentences of the language 9; it is clear that the properties of the set were relatively independent of the set X ; we substituted any infinite ordered set Y for X i n 3.3, and the further development could equally well have started with K(%, Y ) . The essential properties of K(%, X ) are already possessed by the simplest such form, which will be K(%, o),so we shall look effectively at this. But we can make this even simpler by taking free variables vo, vl, . . . rather than constants c, cl.. . We can then describe K(%, o) in a simple way as follows: Let 9 be the language of set theory (with no constants), and take the set of formulas of 9 satisfied in L by the to v , for assignment (wl, coz, ma,. . .,con,. . .),,<", i.e., assign n < w. Since the cardinals wl, w 2 , . . . are in the class X of indiscernibles for L, it is clear that this will effectively be K(%, w ) , and that the construction of 33 can be built from this set. We want to show that in terms of the analytical hierarchy (as in ch. 5 $7) this is a A: set. For this, we must, of course, translate the formulas into natural numbers, so we shall suppose some simple system of Godel-numbering has been chosen, and for each formula rp of 9, let r#l be its Godelnumber. [We could use the hereditarily finite sets for this purpose as we did in ch. 3 $5; but the existence of simple maps from the hereditarily finite sets to the natural numbers means that the two methods are equivalent, and in the present context it is more usual to think of sets of natural numbers in connection with the analytical hierarchy. For

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definiteness we can assume that the new ‘4’ is the image of the r$l of ch. 3 $5.1 under the function from V , to w given in ch. 2 $4.10(2).] We shall also take 2’‘to be the language 2’with a further one-place relation symbol x added, and assume we have Godel-numbers for formulas of 2” also. With these preliminaries, we make the following definitions, due to Solovay : 5.1. Definition. If there is a cardinal K such that

0’ for and for x

= w,

x‘ for

{‘4’ I (L, E(L)) b 4 If]>, where,f(n)

=

K

-, ( w ~ )
for n < o ;

Y4l I (L(x), E(L(x)),x> 14 m>,

(with the samef). (As in 4.9 we are using x both for the set and for the one-place relation Y x.) We could have given this definition without the hypothesis, so that Ob and x p would always exist, but we are really only interested in them if they give rise to the results of Theorem 4.8 (i.e., if they code the formulas of K(%, w) and K(%”,w)), and so we have put in the hypothesis which we know ensures this. Now we can prove the result, due to Solovay and Silver: f

5.2. Theorem. 0’ is A:, and x’ is A ; in the parameter x . Proof. We shall show that X = 0’isu;, and that X = xfti s n ; in the parameter x . (Then, e.g., n E 0’ will be both 3 X ( X = 0% A n E X) and V X ( X = 0%-,n E X ) , and so is A;.) Now we use the results of $J3 and 4, to say that 0% is uniquely characterized by the following five facts: (1) X is a set of formulas of L which is complete and consistent; (2) Xincludes all theorems of ZF + V = L ; (3) X includes the formulas saying uo < u1 < u2 < . . ., and that uo, v l , v2, . . . form a set of indiscernibles; (4) X contains all formulas given by (i), (ii) and (iii) of 3.7 and 3.9 (where here we take Toto be all terms, and add to each such formula the hypothesis saying that the term involved is legitimate) ; (5) X preserves well-foundedness. (Of course we should say: Godel-number of the formulas, each time, rather than the formulas themselves, but we shall ignore the difference

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[CH.8, $ 5

here). (l), (2) and (3) will ensure that Xcodes a set K(%, Y ) for a model ’2l of ZF V = L,with a set of indiscernibles Y of order type w , so that the constructions of $3 can be carried out. Using Lemmas 3.7 and 3.9, this set will preserve segments and limits, so with (5) the constructions of $4 can be carried out and Theorem 4.8 proved, and so if X satisfies (1)-(5), it is the set O# defined in 5.1. We shall show that (1)-(4) are 0:conditions on X , and that (5) isI7;; so the conjunction will be a IIi characterization of X = O#, as required. Conditions (2), (3) and (4) are all of the form: certain formulas belong to X.The formulas involved can be simply described, and when translated to Godel-numbers they will give a A : set of numbers which must be in X.So (2)-(4) are A:. (1) is also d i. The whole condition can be written using only number quantifiers, in a form which says: for every formula, X contains either it or its negation, and not both; and Xis closed under deduction. (We must use the fact that deductions can be simply described here, which is the completeness theorem for first-order logic.) For condition (9, we use the equivalent form from Lemma 3.11, which says that X preserves well-foundedness iff for every countable ordinal u,%(u) is well-founded. If u is countable, so is %(a), and so is isomorphic to ( w , R) for some binary relation R ; and ( w , R ) will be isomorphic to A?’(.) for some countable u iff for some I c w , (a) 1 is a set of indiscernibles for ( w , R) which generates o; (b) the set of formulas satisfied in (0, R) by finite sequences from I , increasing in the sense of R, is X; (c) the relation < of ( w , R) is in fact a well-ordering of I. If the order type of I in (c) is u, then (a)-(c) imply that ( w , R) is isomorphic to the structure %(a) formed from cc using the formulas of X , as in 3.3. So condition ( 5 ) on X i s equivalent to

+

VR VI((a)

A

(b)

A

(c) -+ ( w , R) is well-founded).

Now (a) and (b) can be written as A: formulas in R, I and X , since all they involve is saying that certain formulas hold in ( w , R), and by ch. 5 $7.7, “truth in ( w , R)” is A:. (c) can in fact be simplified to ‘‘J is well-ordered by R”, since the statement that members of I are ordinals in the sense of (0,R) is included in (b). In this form it can be written a s a n : formula (as in ch. 5 $7.10). Similarly, “ ( w , R) is well-founded” is simply R Wf w , and can also be written as a I7:

CH.8,

$51

A

NON-CONSTRUCTIBLE SET

259

formula. So to compute the place of (5) in the analytical hierarchy we get (5)

V R v I ( ( d : A di

A

na -tn:).

R and I are second-order variables, and so by the methods of ch. 5 $7.2, this i sI l i . Putting these together, we get X = 0' as a n , ' formula, as required. For x p , the only changes in describing X will be in (2) and (5). (2) must become: (2)' Xincludes all the theorems of ZF V = L(x), and the formulas Vu, (x(uo) -+ u,, E w), and x(nJ for n E x and -x@) for m $ x , for each n , m < w. (Here _n, m are terms of 9describing n, m.)This will be A : in the parameter x . (5) must still say: X preserves well-foundedness. But now when translated in terms of o-models, (a) and (b) must be rewritten as: (a)' I is a set of indiscernibles for (LO,R, no) which generates o ; (b)' the set of formulas satisfied in ( w , R, no) by finite sequences from I, increasing in the sense of R, is X . (Here no is an integer chosen to represent x . ) These will still be A : , and the computation will give, as before, that X = x* is 17: in the parameter x , as required. 0

+

5.3. Corollary. If there is a cardinal K such that K -+ (ol) < w , then there is a A t non-constructible subset of w. Proof. By 5.2 since 0' cannot be in L, since its existence implies the conclusions of 4.8, and these would be derivable in L if 0' E L .

Note that the description of 0' in 5.2 will be absolute for L by Shoenfield's absoluteness theorem, ch. 5 $7.15, so that if O* were in L , it would still satisfy the description in L ; and since the same result showed that 2; a n d n ; sets must be in L, we deduce that we cannot hope to reduce the definition of 0' to any simpler form. 5.4. Exercise

(1) Show that every constructible subset of o can be effectively computed from ,'O and that every subset of o in L(x) can be effectively computed from x#. [If Y = w is in L, then Y EH(a) for large enough a, so Y is t(2)

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PARTITION CARDINALS AND MODEL THEORY

ECH.8,

$6

for some term t and 2 from a. Show that Y = {n E cc) I ‘g E t(v,, . . ., u , , - ~ ) ’ E O y ) and similarly for x ~ , ]

$6. Further properties of L” We want to apply some of the methods we have been using on sets of indiscernibles to show that in L” there is a A ; well-ordering of the reals. (Silver [1971a] gives more results in this line.) We will have to extend the method used to prove Theorem 5.3 of ch. 6; we have left this till now because it requires some of the techniques we have developed in $02 and 3 for examining and using indiscernibles. We shall assume in this section that K is a fixed measurable cardinal and we need : 6.1. Definition. V for the characteristic function of the closed unbounded filter on K (thus for x c K, V ( x ) = 1 iff x contains some closed unbounded subset of K ) . Now the first result shows that which normal measure is used in constructing L” is not really very important; for stages at which new subsets of cardinals less than K are appearing, V would do just as well. (Note that by ch. 6 §3.7(3), if V(x) = 1, then p(x) = 1 for any x = K and any normal measure p on K , so that V c p.) We suppose (as in ch. 6 $5) that p is some normal measure on K, and we have:

6.2. Lemma. Suppose that X c 1 < K and X E L ” , and that cr is the jirst ordinal such that X E LE. Then Lt = L:. Proof. We prove this by looking more closely at the way we proved Theorem 5.3 of ch. 6. We took an elementary substructure ‘u < < (L”, E(L@),X , p, (a),Gr> of cardinal K such that K c A = l’ul, and applied Rowbottom’s theorem (ch. 6 Theorem 4.4) to find an elementary substructure 123 < ‘u with p ( B n K ) = 1, where B = 11231, and with the Rowbottom property that subsets of B definable in 23 have cardinal K or
CH.8,

$61

FURTHER PROPERTIES OF L”

26 1

that Z is a set of indiscernibles for 92, and also that it is unbounded in K (since q is 1-1 and p( Y ) = 1). So we must show that it is closed in K ; suppose not, and that u c Z is a non-empty subset of Z bounded below K , with u u $ 2(so u must have no last member). Now since N is transitive, u u E N , and so is q(y) for some y in B ; since y E B, y must be t‘(ul, . . ., a), for some term t of the language for ‘u and u1 < . . < un from Y. Let u = q ” w, where w c Y, and let uo be the first member of Y above w ;since u u $2,we must have y < uo, and also u < y for u E w. So suppose u1 < . . . < urnare the members of ul,. . ., u, before a,; and that IS’E w is such that urn< u‘ (since w has no last member, there must be such a a’). Then we have u’ < ?‘(alt. . ., a,) < um+l (*I (since uo < urn+l).But exactly as in 2.4 we can assume that Y was chosen so that this cannot happen; in fact in this case we can see that the choice of Y as having p( Y ) = 1 will have excluded this from the start. Take T~ < . . . < 7, = T as any m members of Y, and given u E Y, u > 7 , let T , + ~ , . . ., T , be the n - m lowest members of Y - u (i.e., u = T , + ~ ) . Set h(u) = ?‘(T~, . . .,T,); thenf,(u) < u on a set of p-measure 1 (by (*) and indiscernibility of Y ) . Since p is normal, fT must be constant on a set of p-measure 1, and hence by indiscernibility again it is constant for u > T . Let g(T) =fz(a) for any u > T ; by (*), g(T) > u’ for some u‘ > T , and this implies g(T) 2 Sup( Y ) = K , which is impossible since we assumed q ( y ) < K . Now given that 2 is a closed unbounded subset of K , indiscernible for ‘3, the lemma follows. Suppose that W EN and W c K . Then W = = q(ty(al, . . ., u,,)), and so we shall have either for all u E Y, cr > u,, q(u) E W, or for all such u, q(u) $ W. But this says that either W contains a final segment of Z , or K - W does; and since q(p) = p n N (as was shown in ch. 6 $5.3), we shall have q(p)( W ) = 1 iff W contains a closed unbounded subset of K , i.e., q(u) = p n N = %?n N . But also N = L: for some T , and LI: = LJ‘LnLt’, so that we have L: = Ly.Since X E Lt,we also have L: = Lr for the first u such that X E L:. 0

.

We can now use this to prove another property of Lp, again due to Silver: 6.3. Theorem. If V = L”, then there is a A: well-ordering of the reals. Proof. We shall show that the ordering < L p of Y(o) nL’ can be described by a .Zi formula #(x, y). Then if V = L”, we shall have

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PARTITION CARDINALS AND MODEL THEORY

+

[CH.8,

86

+(y, x) for x,y c w , and so will be A:. The description is almost a repetition of the method of $5. We take any structure 'u. of cardinal K which is a model of ZFC of the form (Lt, E(Lt),K , p n L;, x , y ) , where x, y c w and K < T ; and using the fact that this has a set of indiscernibles of p-measure 1, we form the set of formulas K(%, w ) . In forming this, we shall suppose that U,C, a and b are the symbols denoting K , p, x and y, respectively; we take them all as constant symbols, and let 9be the language resulting. In forming K('u, w ) , we shall let the free variables vo, v l , u2, . . . denote the indiscernibles (as we did in forming 0' in 5.1). We want to show two things, and the theorem will follow from these: (i) " X = K('u, o)for some 'u of the form above" can (when coded as was Os in $5) be expressed by a formula. (ii) For x,y c w , x < L p y(and also x , y EL^), iff there is an X c w such that X = K(%, 0)and in X,a represents x, b represents y , and X includes (the Godel number of) the sentence a < L~ b. (Here a represents x in X iff for each n < w , g E a is in X if n E X and 11 $ a is in X if n $ x,where g is a term representing n.) To see (i) we show that Xis K(X, o)for some 'u of the required form, iff (a) X is a set of (Godel numbers of) sentences of 9; (b) X i s complete, and contains the axioms of ZF and the sentences V = Lc, C is a normal measure on U. (c) v,,, v l , u2, . . . are indiscernibles in U, i.e., vo E U is in X,and if no < . . . < n, and $(vno,.. ., v,,,) is in X, and n; < . . . < ni, then #(vnb,. . ., vn;)is in X. (d) X preserves segments, limits and well-foundedness. To show that if 'u. is of the required form, then K(%, w ) satisfies (a)-(d) is straightforward, using 6 . 2 ; to see (d), we use the results of $3. Now suppose that X satisfies (a)-(d); then we can take the set of formulas coded by X and use these formulas to build a structure Z(K), as in 3.3. Since X preserves well-foundedness, this will be a wellfounded structure, and so is isomorphic to L," for some 3, T , by (b). Since the language is countable, and X preserves segments and limits, we shall have U*(K) = K , and the indiscernibles will be closed, unbounded in K . B is C#(K)and will be a normal measure on K in L:. So applying the argument in 6.2, B must be % A L:, and hence also p n Lr, and Z ( Kis)of the form required; since X is K(%(K), w ) , we have the characterization by (a)-(d).

-+(x, y ) iff x = y v

CH.8,

$61

FURTHER PROPERTIRS OF L”

263

Now to see that (a)-(d) can be written as a L7; condition on X is almost a straight repetition of 5.2; as there, the only part which is not A: is to say in (d) that X preserves well-foundedness, and as there, this is I?,. To see (ii) we only need the absoluteness of the formula x < L~ y for structures 2l of the form considered, since if there is an X such that X is K(B,o),and in X,a represents x, b represents y, and a < Lc b is in X , then we must have x, y EL:, and in L:, x < L; y ; so if x < L~ y is absolute, x < Lu y will hold in L’ , and similarly for the converse. But we showed in ch. 5 56 that x < L~ y was dFFif V = L’, and so (ii) follows. The theorem is now immediate: for x,y EL^ ng ( w ) , x < L~ y iff 3 X (Xis K(B, o)for some 2l of the form above, and in X , a represents x , b represents y, and a < Lp b is in X ) .

By (i) this is L’i.

0

6.4. Exercises (1) Extend the language 9 to include constants g for each ordinal a < A, where A is a cardinal, and code this language by ordinals

o,then “R is well-founded” is 0: in this language. [Use the formulation: R is well-founded iff for each a < A, there is a map of a into I taking R a into < , i.e., into E(I).] Hence show that for cardinals I < K , with cf(A) > w , there is a wellordering of 9 ( A ) n L’ which is 2: in this language, and so if V = L’, there is a well-ordering of 9 ( I ) which is A:. (Silver [1971a].)

r

(2) Rowbottom cardinals. A cardinal K is called a Rowbottom cardinal if, whenever B = ( A , U,. . , ) is a structure of countable length with d = K , U c A and 6 < K , then 2l has an elementary substructure 23 = (ByU‘, . . .) with = K , and < No.Show that any Ramsey cardinal must be a Rowbottom cardinal. [Show that in fact a Ramsey cardinal must satisfy Theorem 4.3 of ch. 6. Consider the substructure of 2l generated by a set of K indiscernibles for Byas in 52, and show that this must have the property as in Theorem 4.3 of ch. 6, that any definable subset must have power K or < KO,using essentially the same proof.]

c‘

264

PARTITION CARDINALS AND MODEL THEORY

ICH.8

[Prikry [1970]has shown that if it is consistent to assume the existence of a measurable cardinal, then it is consistent to assume that there is a Rowbottom cardinal of cofinality w ; so that the converse of this result cannot be proved. However, Kunen [1970] shows that in the model L O of this section, all Rowbottom cardinals are Ramsey, so that the converse is independent of ZFC.J (3) Show that if there is a Rowbottom cardinal, then

card(Y(w) n L)

=

KO,

and further that w1 (i.e., the real wl, as in V ) must be inaccessiblein L. [Apply the definition to the structure (LK,9 ( a ) n L, (p)8Gcr),where K is a Rowbottom cardinal and K < wl, to see that P(K) n L iscountable and hence (ol # ( K + ) ~ ; note that since the GCH holds in L, weak and strong inaccessibility are the same in L.] (4) Show that K is a Rowbottom cardinal iff for anyf:[~]"" + 1, where 1 < K , there is a set X c K with = K such that card(f" [XI'") < KO. [Given f :[K]'" + 1, apply the definition to the structure
r

Notes to Chapter 8 The work in §$l-5 was the main part of Silver's thesis, Silver [1966], since published as Silver [1971b]. Many of the implications of $4 had previously been obtained by Gaifman 119671; he used the method of iterated ultrapowers, which has since been exploited by Kunen and by Paris, see Kunen [1970]. Silver's contribution was primarily to show that the results were bound up with the idea of indiscernibles in a structure,

CH.81

NOTES

265

as introduced in Theorem 1.2 by Ehrenfeucht and Mostowski [1956]. This gave the relatively simple A ; set 0’ as in $ 5 ; Solovay had earlier shown the existence of a A : non-constructible set by different methods (see Solovay [1967J).Note that using forcing methods, the existence of a d; non-constructible set has been shown consistent relative to the consistency of ZF alone (Jensen [1970]). The results of $6 are from Silver [1971a]; further properties of L” are proved in Kunen [1970]. The phrase “O* exists” has come to be used in the literature to mean that there is a set of natural numbers satisfying then; definition X = 0’ given in 5.2. Many other conditions, besides K + ( o ~ )