SOME SET THEORY WITHOUT MUCH CHOICE (I)
DAVID ASPERÓ
Notation: Given a set X and a cardinal µ, [X]µ is the collation of all
subset of X of cardinality µ. Similarly we define [X]µ and [X]<µ .
Definition 0.1. Let X be a set and let µ be a cardinal. A function
cl : [X]µ ! [X]µ
is a closure operator i↵ for all x, y 2 [X]µ ,
(a) x ✓ cl(x),
(b) if x ✓ y, then cl(x) ✓ cl(y), and
(c) cl(cl(x)) = cl(x).
We say that x 2 [X]µ is closed if x = cl(x).
We say that cl is well–founded if there is no sequence (xn )n<! such that
for all n,
(i) xn is closed and
(ii) xn+1 is a strictly contained in xn .
Given a set X, L(X)
S is defined like L but starting from X rather than ;.
Specifically, L(X) = ↵2Ord L↵ (X), where
(a) L0 (X) = X,
(b) for all ↵, L↵+1 (X) is the set of all subsets of L↵ (X) definable over
(L↵ (X), 2) from parameters in L↵ (X), and S
(c) for every nonzero limit ordinal , L (X) = ↵< L↵ (X).
L(X) is the ✓–minimal transitive model M of ZF such that Ord[X ✓ M .
On the other hand, L(X) need not satisfy the Axiom of Choice (AC). For
example, most of the time L(R) satisfies the Axiom of Determinacy, which
implies that AC fails, and adding uncountably many Cohen reals to the
universe always gives rise to a model in which L(R) |= ¬AC.
Recall: The principle of Dependent Choices (DC) is the following weak
form of AC: Suppose R is a nonempty relation such that range(R) ✓
dom(R). Then there is a sequence (an )n<! such thet an R an+1 for all n. It
is a standard fact that if the AC holds in V, then every inner model of the
form L(X) inherits DC. We will be using the foliowing well–known fact.
Fact 0.2. (ZF + DC) Given a relation R, the following are equivalent.
(i) R is well–founded.
(ii) there is no (an )n2! such that an+1 R an for all n 2 !.
The following theorem is from Shelah’s [Sh:835] (“PCF without choice”).
1
2
D. ASPERÓ
Theorem 0.3. (Shelah) (ZF + DC) Suppose  is an uncountable regular
cardinal, µ , is an ordinal and there is a well–founded closure operator
cl : []µ ! []µ .1Then there is a family Fp (for p 2 H, where H =
P(P( (µ+ )))) such that
(1) for every p 2 H there is some Z ✓  such that Fp is a collection of
functions from Z into ,
(2) for every p 2 H, Fp is well–orderable, and
(3) for every S
function f :  ! there is (Zn )n<! such that
 = n Zn , and
for every n there is pn 2 H such that f Zn 2 Fpn .
Proof. This is one of those proofs which are trivial once we are given the
right definition of some object. But first let us see some standard preliminary
notions and facts.
Given a –complete filter F on a set X and a function f : X ! Ord,
let us define the F –rank of f recursively by setting
rankF (f ) = {sup(g) + 1 : g <F f }
where g <F f means {x 2 X : g(x) < f (x)} 2 F .
Note that <F is well–founded by DC, and that therefore the F –rank of
f is well–defined in the present situation.
Given a filter F on a set X and given a set Y ✓ X, let F Y be the
collection of all Z ✓ X such that (X \ Y ) [ Z 2 F (we can alternatively
view F Y as the filter of all Z ✓ Y such that Z [ (X ⇥ Y ) 2 F ).
Definition 0.4. Let F be a –complete filter on a set X, let f : X ! Ord
be a function, and let ⇢ = rankF (f ). Let J(F, f ) be the filter on X –
generated by
F [ {Y ✓ X : rankF (X\Y ) (f ) > ⇢}
[The notation J(F, f ) is sometimes used to denote the dual ideal of the
filter I have defined here.]
Fact 0.5. If F is a proper (i.e., ; 2
/ F)
also proper.
–complete filter, then J(F, f ) is
⇤
Proof. By DC (easy exercise).
f0
Lemma 0.6. Suppose F is a –complete filter on X, f ,
: X ! Ord,
0
0
rankF (f ) = rankF (f ), and J(F, f ) = J(F, f ). Then f =F f 0 (i.e., {x 2
X : f (x) = f 0 (x)} 2 F ).
Proof. Let ⇢ = rankF (f ) = rankF (f 0 ). Assume, without loss of generality, that Y = {x 2 X : f (x) < f 0 (x)} 2 J + (F, f ) = J + (F, f 0 ). Then
rankF Y (f 0 ) > rankF Y (f ) = ⇢, where the equality follows from Y being
J(F, f )–positive. But this means that X \ Y is in J(F, f 0 ), and therefore
(X \ Y ) \ Y = ; 2 J(F, f 0 ), which is impossible by Fact 0.5.
⇤
1Typically, , µ and H will be much smaller than
.
SOME SET THEORY WITHOUT MUCH CHOICE (I)
3
We are now ready for the main definition.2
Definition 0.7. Let Z ✓ , F0 ✓ F1 –complete filters on Z, ⇢ 2 Ord,
and h : Z ! µ+ . Let ⌃(F0 , F1 , Z, h, ⇢) be the collection of all functions
f : Z ! such that
(a) rankF0 (f ) = ⇢,
(b) F1 = J(F0 , f ),
(c) there is no Z 0 ✓ Z such that Z 0 2 F1 and cl(f “Z 0 ) is strictly contained in cl(range(f )), and
(d) for all ↵ 2 Z and all , f (↵) = if and only if 2 cl(range(f )) and
.
ot(cl(range(f ) \ )) = h(↵)
Claim 0.8. For all F0 , F1 , Z, h and ⇢ as in the above definition,
|⌃(F0 , F1 , Z, h, ⇢)|  1
Proof. This follows easily from Lemma 0.6: Suppose f , g are both in the
family ⌃(F0 , F1 , Z, h, ⇢). By Lemma 0.6, Z ⇤ = {↵ 2 Z : f (↵) = g(↵)} 2 F1 .
But then,
cl(range(f )) = cl(f “Z ⇤ ) = cl(g“Z ⇤ ) = cl(range(g))
where the first equality holds by Z ⇤ 2 F1 together with Definition 0.7 (c)
applied to the fact that f 2 ⌃(F0 , F1 , Z, h, ⇢), the second equality holds
from the definition of Z ⇤ , and the third equality holds by (again) Z ⇤ 2 F1 ,
together with Definition 0.7 (c) applied to g 2 ⌃(F0 , F1 , Z, h, ⇢). Now, from
cl(range(f )) = cl(range(g)) we immediately get that f = g using Definition
0.7 (d).
⇤
Given F0 , F1 , Z, h and ⇢ as in Definition 0.7, let fF0 ,F1 ,Z,h,⇢ be the unique
(by Claim 0.8) member of ⌃(F0 , F1 , Z, h, ⇢) if ⌃(F0 , F1 , Z, h, ⇢) 6= ;. Also,
given p = (F0 , F1 , Z, h) as in Definiton 0.7, let
Fp = {fF0 ,F1 ,Z,h,⇢ : ⇢ 2 Ord, ⌃(F0 , F1 , Z, h, ⇢) 6= ;}
and note that p can indeed be seen as a member of P(P( (µ+ ))). For all p
as above, Fp is obviously well–orderable. It follows that in order to prove
the theorem it suffices to show the following:
Claim 0.9. Let f :  ! . Then there is (F0n , F1n , Z n , hn , ⇢n ) (for n < !)
such that
S
(a)  = n Z n , and
(b) for every n < !, f Zn 2 ⌃(F0n , F1n , Z n , hn , ⇢n ).
Proof. Let F0 be the filter on  –generated by the collection of Z ✓  such
that there are –complete filters F0 ✓ F1 on  \ Z, a function h : Z ! µ+ ,
and an ordinal ⇢ such that f ( \ Z) 2 ⌃(F0 , F1 ,  \ Z, h, ⇢).
2As we will see, the proof of the theorem is “coded” in this definition.
4
D. ASPERÓ
Now comes the cool part of the proof. Suppose the conclusion of the
claim fails. Then F0 is a proper filter on . This is the crucial point. Let
⇢ = rankF0 (f ) and let F1 = J(F0 , f ). Let Z 2 F1 be such that there is
no Z 0 ✓ Z in F1 such that cl(f “Z 0 ) is strictly contained in cl(range(f ))
(since cl is well–founded, such a Z can be easily found using DC). Now let
h : Z ! µ+ be the function defined by letting
h(↵) = ot(cl(range(f
Z) \ f (↵))
Now it is easy to check that f Z is in ⌃(F0 , F1 , Z, h, ⇢). But this means
that  \ Z 2 F0 ✓ F1 , and therefore ; = ( \ Z) \ Z 2 F1 , contrary to the
fact that F1 is proper.
⇤
This finishes the proof of Theorem 0.3.
⇤
The following theorem shows that the existence of a well–founded closure operator cl : [ ]µ ! [ ]µ , for suitable µ, follows from the well–
orderability of [ ]@0 .
Theorem 0.10. (Shelah) (ZF + DC) Let
be an ordinal and suppose
@
0
there is a well–order of [ ] . Then there is a well–founded closure operator
cl : [ ]µ ! [ ]µ for µ = sup <!1 µ , where (µ ) <!1 is the continuous
sequence of cardinals given by
( ) µ0 =  and
S
( ) µ +1 is the supremum of all cardinals of the form | range( )|,
where
: [Y ]@0 ! [ ]@0 is a function, for some Y ✓ with
|Y |  µ , such that (x) ◆ x for all x 2 [Y ]@0 .
Proof. To start with, note that there is an sequence (u⇠ )⇠2 , for some ordinal
, enumerating all of [ ]@0 , and such that for every ⇠ <
there are no
⇠0 , . . . ⇠n 1 below ⇠ such that u⇠ is contained in u⇠0 [ . . . [ u⇠n 1 mod. finite.
Given x 2 [ ]@0 , let (⇠ix )i<nx , for some nx < !, be the longest sequence
such that
S
(i) x ✓f in u⇠0x [ ⇣2w u⇣ for some finite w ✓ ⇠0x , and
S
S
(ii) for every i > 0 such i < nx , x \ j<i u⇠jx ✓f in u⇠ix [ ⇣2w u⇣ for some
finite w ✓ ⇠ix .
Note that nx < ! since the sequenceSof the ⇠ix is strictly descending.
Now, given x 2 [ ]@0 , let F (x) = i<nx u⇠ix , and given z 2 [ ]µ , let cl(z)
be the ✓–minimal set y such that
(a) z ✓ y and
(b) for every x 2 [y]@0 , F (x) ✓ y.
S
It is easy to check that cl(z) =
=
<!1 y , where y0 = z and y
S
S
S
@
0
F “[ 0 < y 0 ] for all > 0 (note that every countable subset
0< y 0 [
S
of
y
is
in y for some < !1 by the regularity
of !1 under DC),
<!1
S
and that if z can be written as ✓–increasing union
<!1 Z such that each
Z has cardinality at most µ , then cl(z) can also be written in this manner
(this uses the fact that  !1 is regular). Hence, the range of cl is contained
SOME SET THEORY WITHOUT MUCH CHOICE (I)
5
in [ ]µ . Now it is straightforward to check that cl : [ ]µ ! [ ]µ is a
closure operator.
Finally, to see that cl is well–founded, suppose (zn )n<! is a sequence of
closed sets such that zn+1 is strictly contained in zn for all n, and let (↵n )↵<!
be such that ↵n = min(zn \ zn+1 ) for all n. Note now that x ✓f in F (x)
for every x 2 [ ]@0 and that F (x) = F (x0 ) for every x 2 [ ]@0 and every
x0 ✓ x such that x \ x0 is finite. It follows that, if n < ! is such that
{↵m : m n} ✓ F ({↵m : m < !}), then ↵n 2 F ({↵m : m > n}) ✓ zn+1 ,
which is impossible since ↵n 2 zn \ zn+1 .
⇤
The following result is a corollary from the proof of Theorem 0.3, together
with Theorem 0.10.
Corollary 0.11. (ZFC) Suppose
is a singular strong limit cardinal of
uncountable cofinality. Then L(V +1 ) |= ZFC.
Proof. Since is a strong limit we may fix a bijection
:
! V , and
2 V +1 ✓ L(V +1 ).
Let  = cof ( ). Given X 2 V , let fX :  ! be the function given by
fX (↵) =
1
(X \ V↵ )
Note that if X and Y are distinct subsets of V , then fX (↵) 6= fY (↵) for all
↵ on a tail of .
We know that L(V +1 ) |= DC. Also, in L(V +1 ) there is a well–order of
[ ]@0 since [ ]@0 ✓ V and is a strong limit. Hence, by Theorem 0.10, all
hypotheses of Theorem 0.3 hold in L(V +1 ), and therefore its conclusion is
also true there.
Now, given X ✓ V look at the first, relative to a well–order of H given
by , pair (pX , ZX ) such that pX 2 H, ZX ✓  is unbounded in , and such
fX ZX 2 FpX , and look at the index of fX ZX in the natural well–order
of FpX (These objects exist since the conclusion of Theorem 0.3 holds in
L(V +1 ). Also, there is some ZX as above unbounded since  is not the
union of countably many sets bounded in .)
Having drawn your attention to the above objects, it should now be clear
to you that a well–order of P(V ) = L(V +1 ) can be defined in L(V +1 )
from (since fX ZX and fY ZY will be distinct functions whenever X,
Y ✓ V are distinct).
⇤
Recall that an I0 embedding is an elementary embedding j : L(V +1 ) !
L(V +1 ) such that j 6= id and crit(j) < (see A. Kanamori’s “The Higher
Infinite”). If ZFC holds in V , then has necessarily cofinality !, and is
the least fixed point of j above crit(j). The existence of an I0 embedding
is one of the strongest large cardinal assumptions people have considered
not known to be incompatible with ZFC.3 It is a well–known – and easy to
3There is a very interesting theory for L(V
+1 ) under the assumption that there is an I0
embedding j : L(V +1 ) ! L(V +1 ). This theory has been developed mostly by Woodin.
6
D. ASPERÓ
prove – fact that if there is an I0 embedding j : L(V +1 ) ! L(V +1 ), then
L(V +1 ) |= ¬AC.4
Let T be the theory ZFC + “There is a strong limit cardinal of countable
cofinality such that L(V +1 ) |= ¬AC”.
By the above observation, the existence of an I0 embedding gives an
upper bound for the consistency strength of T . A lower bound is given by
the following:
Proposition 0.12. (ZFC) Suppose is a singular strong limit cardinal. If
L(V +1 ) |= ¬AC, then X ] exists for every set of ordinals X bounded below
.
Proof. Let X be a set of ordinals bounded in whose sharp does not exist.
Then X 2 L(V +1 ) and L(V +1 ) thinks (correctly) that X ] does not exist.
By Jensen’s Covering Lemma in L(V +1 ) it holds there that every set of
ordinals Y is covered by a set of ordinals Z in L[X] such that |Z| = |Y | +
@1 + |X|.5 But now, given a countable subset Y of , look at the <L[X] –first
ZY 2 L[X] of minimal size such that Y ✓ ZY . Since in L(V +1 ) there is
a well–order of P(Z Y ), where Z Y is the transitive collapse of ZY (because
|ZY |  @1 + |X|), there is a well–order of all subsets of ZY in L(V +1 ).
From this it follows easily that there is in L(V +1 ) a well–order of P( ),
and therefore there is a well–order of V +1 in L(V +1 ) as in the proof of
Corollary 0.11.
⇤
By a similar argument as in the proof of the proposition using Dodd–
Jensen’s covering lemma it can be shown that the existence of some strong
limit such that L(V +1 ) |= ¬AC implies the existence, for every X ✓
bounded in , of an inner model containing both X and a measurable
cardinal.
I don’t know the answer to the following.
Question 0.13. What is the consistency strength of T ?
Magidor conjectures that the theory T is equiconsistent with ZFC together with the existence of a measurable cardinal  of Mitchell order ++ .
David Asperó, School of Mathematics, University of East Anglia, Norwich
NR4 7TJ, UK
E-mail address: [email protected]
4Otherwise one can reach a contradiction by arguing as in Kunen’s proof of the nonexistence of a nontrivial elementary embeddingg j : V ! V in ZFC.
5Even if the Axiom of Choice fails, the Covering Lemma is true since it is true in the
ZFC model L[A], where A is any set of ordinals coding both X and Y . Also, recall that
L[A], with is defined like L except that L↵+1 [A] is the collection of all subsets of L↵ [A]
definable in the structure (L↵ [A], 2, A \ L↵ [A]) from parameters, is always a model of
Choice, and in fact there is in L[A] a well–order <L[A] of its universe definable exactly as
the canonical well–order of L.