Saharon Shelah- On Nice Equivalence Relations on ^λ-2

8/3/2019 Saharon Shelah- On Nice Equivalence Relations on ^-2

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ON NICE EQUIVALENCE

RELATIONS ON 2

SH724

Saharon Shelah

The Hebrew University of JerusalemEinstein Institute of Mathematics

Edmond J. Safra Campus, Givat Ram Jerusalem 91904, Israel

Department of Mathematics Hill Center-Busch CampusRutgers, The State University of New Jersey

Piscataway, NJ 08854-8019 USA

Abstract. Let E be an equivalence relation on the powerset of an uncountableset, which is reasonably definable. We assume that any two subsets with symmetricdifference of size exactly 1 are not equivalent. We investigate whether for E thereare many pairwise non-equivalent sets.

2000 Mathematics Subject Classification. (2000); FILL!! 03E47,03E35; 20K94.Key words and phrases. set theory; definable equivalence relations, generalizing descriptive set

theory to uncountable cardinals; perfect sets of pairwise non-equivalent; abelian groups, Ent.

I would like to thank Alice Leonhardt for the beautiful typingWritten 99/11First Typed - 2000/Feb/12; (revised with proofreading for the journal and little more)Latest Revision - 03/Jan/29

Typeset by AMS-TEX

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Annotated content

0 Introduction

1 Dichotomical results on nice equivalence relations

[Assume E is a 11[]-equivalence relation on2 such that , are not E-

equivalent whenever they differ in exactly one place. Assume further thatthis holds even after adding a -Cohen subset of . If = 0. We find on quite nice equivalencerelations for which the parallel of the results of 1 fail badly. If is stronglimit, we can use 2.]

3 Countable cofinality: positive results

[Assume that > cf() = 0 and is the limit of measurables, or justa related property (which consistently holds for = ) is satisfied. Weprove the parallel of the result in 1 on .]

4 The countable cofinality case: negative results

[We show that if our universe is far enough from large cardinals (and closeto L) then we can build counterexamples as in 2.]

5 On rp(Ext(G, Z))

[We return to the p-rank of the abelian group Ext(G, Z) where G is tor-sion free abelian group (1-free, without loss of generality see [Fu](xxx)).

We show that if is compact, strong limit (singular) cardinal > andrp(Ext(G, Z)) then rp(Ext(G, Z)) 2. This is preserved by adding Cohens, super-compact. If GCH holds above we have a completecharacterization of {Ext(G, Z) : G}.]


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ON NICE EQUIVALENCE RELATIONS ON 2 SH724 3

0 Introduction

The main topic here is the possible generalizations of the following theorem from[Sh 273] on simple equivalence relation on 2 to higher cardinals.

0.1 Theorem. 1) Assume that

(a) E is a Borel 2-place relation on 2

(b) E is an equivalence relation

(c) if , 2 and (!n)((n) = (n)), then , are not E-equivalent.

Then there is a perfect subset of 2 of pairwise non E-equivalent members.2) Instead of E is Borel, E is analytic (or even a Borel combination of analyticrelations) is enough.3) If E is a 12 relation which is an equivalence relation satisfying clauses (b) +(c) also in VCohen, then the conclusion of (1) holds.

In [Sh 273], Theorem 0.1 was used to prove a result on the homotopy group: if Xis a Hausdorff metric topological space which is compact, separable, arc-connected,and locally arc-connected, and the homotopy group is not finitely generated thenit has the cardinality of the continuum; the proof of 0.1 used forcing in [Sh 273],see [PaSr98] without the forcing.

We may restrict E to be like the natural equivalence relation in presentingrp(Ext(G, Z)) or just closer to group theory as in Grossberg Shelah [GrSh 302],[GrSh 302a], Mekler-Roslanowski-Shelah [MRSh 314], [Sh 664]. In 5 we say some-what more. We here continue [Sh 664] but do not rely on it.

Turning to 2 the problem split according to the character of and the simplic-ity ofE. IfE is 11 and =


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0.2 Definition. For a cardinal let B be2 (or or cf()); we write B for

such set.1) For a logic L we say that E is a L-nice, (say 2-place for simplicity), relation

on B if there is a model M with universe and finite vocabulary , and unaryfunction symbols F1, F2 / (denoting possibly partial unary functions), such thatletting + = {F1, F2}, for some sentence = (F1, F2) in L(

+) we have

for any 1, 2 B letting M1,2 = (M, 1, 2) be the +-model expanding

M with FM1,2 = for = 1, 2 we have

1E2 (M, 1, 2) |= .We may write M |= [1, 2] and [1, 2, M] or (x,y ,M) or write a coding M instead of M.

2) E is a 11-relation onB means that above we allow to be of the form (X )

where is first order or even in inductive logic (i.e., we have variables on sets andare allowed to form the first fix point for formula (x, X) such that (x, X1) &X1 X2 (x, X2)); if we allow just first order we say strictly if we allowformulas from L we say L-strictly. Similarly 11,

12, projective; writing nice

means L is L(induction) i.e. first order + definition by induction. We may writeE nice(B),

11[B] etc, and may replace B by if this holds for every B =

2.We write very nice for L-nice when L is L first order logic.

0.3 Notation:(i < ) means for every large enough i < .Jbd is the ideal of bounded subsets of .L denotes a logic, L() denotes the language (i.e, a set of formulas, for the logic

L in the vocabulary ), L denotes first order logic, L, denotes the extension of

L by allowing

0 then if R is L(induction)-nice relation on B then R is strictly11-nice (hence being

11 is equivalent to being strictly

11).


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Proof. 1) The quantification on X can code the satisfaction relation for anysubformula.2) Easy.

3) It is well known that a linear order 2) (>2) ordered naturally, i.e., (0, 1) (0, 1) iff0 0 & 1 1. Nowdo we have V[0, 1] |= 0E1? If so, then for some (p0, p1) (>2) (>2) wehave (p0, p1) Q

0E

1, let < be > g(p0), g(p1) and by clause (c)

+() in

V[0, 1] we can find 1

2 such that 1 = 1 , and for some (, ), 1

[, ) = 1 [, ), (here = + 1 is O.K. but not so in some generalizations) andV[0, 1] |= (1E1).


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So V[0, 1] |= (0E1) (again as in V[0, 1], E is an equivalence relationby clause (c)+ and we are assuming for the time being that V[0, 1] |= 0E1).But also (0,

1) is generic over V for (

>2) (>2) with (p0, p1) in the generic set

and V[0, 1] = V[0, 1] so we get a contradiction to (p0, p1) (0E1). Hence

1 (>2)(>2) (0E

1).

Stage B:

Let be large enough and let N (H(), ) be such that N = , N


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8 SAHARON SHELAH

1.4 Remark. A tree T as in clause (a) of 1.3(2) is called a -Kurepa tree and muchis known on its existence (and non existence). E.g. if is strong limit then suchT exists.

Proof. 1) Let P : < be such that P P(), |P| < , and for everyX the set { : X P} is stationary. So by coding we can find P

{(0, 1) : 0, 1 2} of cardinality < such that for every 0, 1 2 the set{ < : (0 , 1 ) P } is stationary. Lastly, let I : < list the denseopen subsets of (>2) (>2) which belong to N. Now we define by induction on < , : 2 such that:

(a) >2

(b) < g()

(c)

(d) if is a limit ordinal and (0, 1) P , 0 < 2, 1 < 2 and 00 = 11

then (00, 11)

I.

There is no problem to carry the definition (using |P | < = cf()) and {


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2) In claims 1.1, 1.2 and in 1.5(1), for any we can replace E byE : < ,each E satisfying clauses (b) and (c), (c)

+, (c), (c) there respectively and westrengthen the conclusion:

() there is a -perfect setQ such that

() Q = { : 2} and

() if 1 = 2 are from 2 then 1 = 2 and < (1E2)

() for 2 the set {g( ) : 2\{}} is a closed unbounded

subset of .

3) In 1.2, 1.5(1),(2) we can weaken (c)+ or (c) to

() for a stationary set of N [H(+)] there is (in V) 2 which is

Cohen over N such that 11[] sentences are absolute from N[] to V (for11[]-sentences this is necessarily true) and clause (c) (or (c)

) holds.

Proof. 1), 2) The same as the proof of 1.1.3) The only place it makes a difference is in Stage A of the proof of Claim 1.1. Wechoose N, as in () of 1.5(3), and let = (2 + ) : < in N[] = N[0, 1]instead of working with V[

0,

1]. 1.5

Now we would like not to restrict ourselves to 11[]-equivalence relations.

1.6 Claim. 1) Assume

(a) = 2, ), i.e. is a Cohen sequence over Vthen in V[], clause (c) still holds(note that for 1, 2 (2)V anyhow V |= 1E2 V[] |= 1E2)

(d) for every A and > 2 there are N, : < such that(i) N (H(), ), N2 >2)

(iv) 11[] formulas are preserved from N[, ] to V for < < .


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Then E has equivalence classes.

2) We can replace by perfect in the conclusion if in (d), { : < }

2is perfect [see 0.6].3) We can replace >2 by a subtree T >2 such that forcing withT adds nobounded subset to .

Proof. By [Sh 664, 2.2t].

1.7 Definition. Clause (d) of 1.6 is called is [, )-weakly Cohen-absolute:[, )-w.c.a., in short (as in [Sh 664, 2.1t]s notation).

1.8 Claim. We can strengthen 1.6 just as 1.5 strenghthens 1.1.

We may wonder when does clause (d) of 1.6 hold.

1.9 Claim. 1) Assume

(i) = 2>2)VP

and the forcing P/(P + +

) is -complete (or at least -strategically

complete).

Then is (, )-w.c.a. (see 1.7) in the universe VP.

Proof. Straightforward.


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2 Singulars of uncountable cofinality

In this section we show that the natural generalization of 0.1 usually provablyfails badly for cf(), singular of uncountable cofinality.

2.1 Claim. Assume

(a) > = cf() > 0

(b) 2 +


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12 SAHARON SHELAH

of () () is very nice where A is closed when: if (, ) andfor every finite u for some (, ) A we have u = u & u = uthen (, ) A.

6) If =


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Proof of 2.1. We choose = i : i < , nondecreasing, i.e. i < j i jwith limit , (e.g. i = which is the case stated in the claim) let j =

i


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14 SAHARON SHELAH

and so u = ui(0,1) is a neighbor of (0, 1) and by the

definition of B() we have u E0 = hence u E1 = (b) for B = B : < , B

let B = { i


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() E1 has j equivalence classes for any j <

[why? let i 0

(b) 2 +


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1 lim(T) 2 lim(T) & (1 lim(T) 1 = 2).

2.3

2.4 Claim. In Claim 2.1 we can replace clauses (), () by

()1 E is very nice, moreover is the union of closed sets minus the union of closed sets

()1 for every , the set { : = modJbd } is a set of representa-

tives for the family of E-equivalence classes.

Proof. Let be as there but < + i. Let Ki be a group, with universe

i and unit 0Ki . Let 0

2 if G is a torsion free abelian group, then rp(Ext(G, Z)), if not finite, hasthe form 2

3 in (2) rp(Ext(G, Z)) = 2fr-rk[p](G), see below.

5.12 Definition. For a prime p.1) Let Kp = {G : G is a torsion free abelian group such that even if we add |G|

+

Cohen reals still rp(Ext(G, Z)) = 0}.2) For a torsion free abelian group G let

fr-rk[p](G) = Min{rk(G) :G is a pure subgroup of

G and G/G Kp}.

Proof. Essentially by [MkSh 418].


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48 SAHARON SHELAH

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[Fu] Laszlo Fuchs. Infinite Abelian Groups, volume I, II. Academic Press,New York, 1970, 1973.

[GrSh 302] Rami Grossberg and Saharon Shelah. On the structure of Extp(G, Z).Journal of Algebra, 121:117128, 1989. See also [GrSh:302a] below.

[GrSh 302a] Rami Grossberg and Saharon Shelah. On cardinalities in quotients ofinverse limits of groups. Mathematica Japonica, 47(2):189197, 1998.math.LO/9911225.

[J] Thomas Jech. Set theory. Springer Monographs in Mathematics.Springer-Verlag, Berlin, 2003. The third millennium edition, revisedand expanded.

[JMMP] Thomas Jech, Menachem Magidor, William Mitchell, and KarelPrikry. On precipitous ideals. J. of Symb. Logic, 45:18, 1980.

[MRSh 314] Alan H. Mekler, Andrzej Roslanowski, and Saharon Shelah. On thep-rank of Ext. Israel Journal of Mathematics, 112:327356, 1999.math.LO/9806165.

[MkSh 418] Alan H. Mekler and Saharon Shelah. Every coseparable groupmay be free. Israel Journal of Mathematics, 81:161178, 1993.math.LO/9305205.

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[SgSh 148] Gershon Sageev and Saharon Shelah. Weak compactness and thestructure of Ext(A, Z). In Abelian group theory (Oberwolfach, 1981),volume 874 of Lecture Notes in Mathematics, pages 8792. Springer,Berlin-New York, 1981. ed. Goebel, R. and Walker, A.E.

[Sh 124] Saharon Shelah. may have a strong partition relation. Israel Jour-nal of Mathematics, 38:283288, 1981.


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[Sh 273] Saharon Shelah. Can the fundamental (homotopy) group of a spacebe the rationals? Proceedings of the American Mathematical Society,103:627632, 1988.

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[Sh 460] Saharon Shelah. The Generalized Continuum Hypothesis revisited.Israel Journal of Mathematics, 116:285321, 2000. math.LO/9809200.

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Saharon Shelah- On Nice Equivalence Relations on ^λ-2

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