Download Properties of the real line and weak forms of the Axiom of Choice

Math. Log. Quart. 51, No. 6, 598 – 609 (2005) / DOI 10.1002/malq.200410052 / www.mlq-journal.org
Properties of the real line and weak forms of
the Axiom of Choice
Omar De la Cruz∗1 , Eric Hall∗∗2 , Paul Howard∗∗∗3 , Kyriakos Keremedis†4 , and
Eleftherios Tachtsis‡5
1
2
3
4
5
Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA
Department of Mathematics and Statistics, University of Missouri, Kansas City, MO 64110, USA
Department of Mathematics, Eastern Michigan University, Ypsilanti, MI 48197, USA
Department of Mathematics, University of the Aegean, Karlovassi 83200, Samos, Greece
Department of Statistics and Actuarial-Financial Mathematics, University of the Aegean,
Karlovassi 83200, Samos, Greece
Received 27 November 2004, revised 29 July 2005, accepted 2 August 2005
Published online 1 October 2005
Key words Axiom of Choice, weak axioms of choice, real line.
MSC (2000) 03E25, 03E35
We investigate, within the framework of Zermelo-Fraenkel set theory ZF, the interrelations between weak forms
of the Axiom of Choice AC restricted to sets of reals.
c 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1 Introduction and some preliminary results
This paper is a continuation of [6] and [10] and is concerned with Paul Howard’s and Jean E. Rubin’s project
“Consequences of the Axiom of Choice” [7]. Our aim is to clarify the interrelations between weak forms of the
Axiom of Choice restricted to the real line R. We use the notations established in [6, 7, 10]. Furthermore, any
statement “Form x” has been considered in [7] where all known implications between these forms are given
in Table 1, see http://www.math.purdue.edu/˜jer/Papers/conseq.html.
Below we list the statements we will be dealing with in the sequel.
Form 1, AC: Every family of non-empty sets has a choice function.
Form 85, C(∞, ℵ0 ): Every family of denumerable (i. e. countably infinite) sets has a choice function.
Form 203: Every partition of ℘(ω) into non-empty subsets has a choice function.
Form 79, AC(R): AC restricted to families of subsets of R.
Form 212: AC(R) restricted to families of size |R|.
ACω (R): AC(R) restricted to families of countable subsets of R.
Form 5, CCω (R): ACω (R) restricted to countable families.
PCCω (R): Every countable family of countable subsets of R has an infinite subfamily with a choice function.
In [6] it is shown that PCCω (R) is equivalent to CCω (R).
ACω (2ℵ0 , R): ACω (R) restricted to |R| sized families.
AC(ω1 , R): AC(R) restricted to ω1 sized families.
Form 94, CC(R): AC(R) restricted to countable families.
∗
e-mail: [email protected]
e-mail: [email protected]
∗∗∗ e-mail: [email protected]
† Corresponding author: e-mail: [email protected]
‡ e-mail: [email protected]
∗∗
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ω-CC(R): For every family A = {Ai : i ∈ ω} of non-empty subsets of R there is a family B = {Bi : i ∈ ω}
of non-empty countable sets such that for all i ∈ ω, Bi ⊂ Ai .
CH, Continuum Hypothesis: 2ℵ0 = ℵ1 .
Form 170: ℵ1 ≤ 2ℵ0 .
Form 9: Every infinite set is Dedekind-infinite, i. e. it has a countably infinite subset.
Form 13: Every infinite subset of R is Dedekind-infinite.
Form 31, CUC: A countable union of countable sets is countable.
Form 6, CUC(R): A countable union of countable subsets of R is countable.
Form 38: R can not be written as a countable union of countable sets.
Form 26: The union of denumerably many sets, each of power 2ℵ0 , has power 2ℵ0 .
Form 151, UT(WO, ℵ0 , WO): A well-ordered union of countable sets is well-orderable.
Form 35: The union of countably many meager subsets of R is meager.
Form 34: ℵ1 is a regular cardinal.
Form 56: H(2ℵ0 ) = ℵω , where H(2ℵ0 ) is Hartog’s ℵ, the least ℵ not less or equal to 2ℵ0 .
Form 74: Every sequentially compact subset of R is compact.
Form 169: There is an uncountable subset of R without a perfect subset (i. e. closed without isolated points).
Form 211: If R ⊆ R × R satisfies (∀x ∈ R) (∃y ∈ R) (xRy), then there exists a sequence (xn )n∈ω of real
numbers such that (∀n ∈ ω) (xn Rxn+1 ).
Form 368: The set of all denumerable subsets of R has power 2ℵ0 .
Form 369: If R is partitioned into two sets, at least one of them has cardinality 2ℵ0 .
Proposition 1.1
(1) CCω (R), CUC(R), Form 38 do not imply Form 74, Form 169.
(2) Form 13 does not imply Form 34, Form 35, Form 38, Form 169.
(3) Form 34 does not imply Form 74 and Form 35 does not imply Form 74, Form 169.
(4) Form 74 does not imply CCω (R), CUC(R), Form 34, Form 35, Form 38, CC(R), Form 169, Form 211,
Form 212.
(5) Form 74 implies Form 13.
(6) CC(R) does not imply Form 169, Form 212.
(7) Form 170 inplies Form 169.
(8) Form 170 does not imply Form 74, Form 368.
(9) Form 169 does not imply Form 13, Form 74, CC(R), Form 211, Form 212, Form 369.
(10) Form 203 implies Form 169, Form 212.
(11) Form 211 does not imply Form 169, Form 212.
(12) Form 212 does not imply AC(R), Form 203 and Form 368.
(13) Form 368 does not imply Form 13, Form 74, AC(R), CC(R), Form 203, Form 211, Form 212, Form 369.
(14) Form 368 implies Form 169.
(15) ¬Form 169 implies Form 369. Hence ¬Form 369 implies Form 169 and Form 169 does not imply
Form 369.
P r o o f.
(1) In Cohen’s original model, model M1 in [7], CUC(R) and consequently CCω (R) and Form 38 are true.
On the other hand, in [4] it is shown that Form 74 implies Form 13 and since Form 13 is false in M1 for the set A
of the added Cohen reals, the independency follows. Now in Truss’ model M12(ℵ) in [7] (see also [22]), where
ℵ is a singular cardinal, CUC(R) is true, whereas every set of reals has a perfect subset in this model. Thus,
Form 169 fails in M12(ℵ).
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(2) In the Feferman-Lévy model M9 in [7], R is expressed as a countable union of countable sets. Hence
each one of Form 35 and Form 38 fails in that model. In [4] it is shown that Form 74 is true of M9. Hence
Form 13 holds in that model. In Truss’ model M12(ℵ), Form 13 is true because every uncountable subset of the
reals has a perfect subset, hence a subset of size |R| (in [22], it is shown that in ZF every perfect subset of R has
power 2ℵ0 ). Thus, Form 169 fails in that model, and since ℵ1 is singular in M12(ℵ), Form 34 fails, too.
(3) Form 34, Form 35 hold in M1, whereas Form 74 fails since Form 13 fails. Form 35 holds in Solovay’s
model M5(ℵ) in [7], whereas Form 169 fails, see [7].
(4) Form 74 holds in M9. Since CCω (R), CUC(R), Form 34, Form 35, Form 38, Form 211 and Form 212
all fail in M9, the independence result follows. On the other hand, Form 74 holds in M5(ℵ), whereas Form 169
fails, see [7].
(5) This has been proved in [4] and [12].
(6) In M5(ℵ), CC(R) is true, whereas Form 169 fails, see [7]. The second assertion of (6) has been proved
in [10, Proposition 8].
(7) Assume on the contrary that every uncountable set of reals has a perfect subset. By Form 170, this implies
that ℵ1 has a subset of size |R|. But then |2ℵ0 | = |ℵ1 |, hence R is well-ordered. It is well-known that this implies
Form 169. This is a contradiction.
(8) In [6] it is shown that Form 170 is true in the model M1. By (1) we have that Form 74 fails in that model.
The second assertion of (8) is proved in [10, Proposition 9].
(9) As mentioned in (8), Form 170 holds in the model M1, and so Form 169 holds in M1 by (7). On the
other hand, Form 74 fails in M1 (as explained in (1)), and it is known (see [7]) that Form 13, CC(R), Form 211,
Form 212, and Form 369 are false in M1.
(10) In [6] we showed that Form 203 implies Form 170, and by (8) we deduce that Form 203 implies Form 169.
The implication “Form 203 ⇒ Form 212” has been proved in [10].
(11) Form 211 holds in M5(ℵ), whereas Form 169 fails, see [7]. The second assertion has been proved
in [10, Proposition 8].
(12) In [21] it is shown that Form 212 does not imply either of AC(R), Form 203, and in [10] it is shown that
Form 212 does not imply Form 368.
(13) In [10] we showed that Form 368 is true in M1. Since Form 13, Form 74, AC(R), CC(R), Form 203,
Form 211, Form 212 and Form 369 all fail in M1, the independence result follows.
(14) Tarski [19] has shown that Form 368 implies Form 170. The conclusion now follows from (8).
(15) Assume the contrary and let R = A ∪ B with |A| < |R| and |B| < |R|. Since the union of two countable
sets is countable, it follows that either A or B is uncountable. Thus, in view of ¬Form 169 it follows that either A
has the power of R or B has the power of R.
Definition 1.2 Let X, K be two sets.
1. X K denotes the set of all functions from K into X.
2. [X]K denotes the set of all subsets of X of cardinality |K|.
3. X <K denotes the set of all functions from subsets of K of cardinality less than |K| into X.
4. [X]<K denotes the set of all subsets of X of cardinality less than |K|.
Since |[R]<ω | = |R| in ZF1), it seems reasonable to expect that the statement “for every infinite subset A of R,
|[A]<ω | = |A|” is a theorem of ZF also. We show in Theorem 2.3 that this is not the case.
1) Using the fact that each finite subset of R is well-ordered under the usual ordering of the reals, it is easy to construct for each n ∈ ω
an injection gn : [R]n −→ Rn . For X ∈ [R]n , define gn (X)(i) = min(X \ {gn (X)(j) : j < i}). Then |[R]<ω | ≤ |R<ω | under the
injection f defined by f (X) = gn (X) for X ∈ [R]n , n ∈ ω. Now |R<ω | ≤ |Rω | since the function
(x1 , . . . , xn ) −→ (x1 , . . . , xn , n, n, . . .)
is one-to-one. Furthermore, as Rω with the Tychonoff product topology is a separable metrizable space, it follows that |Rω | = |R| and
consequently |[R]<ω | ≤ |R|. On the other hand, it is evident that |R| ≤ |[R]<ω |, hence by the Cantor-Bernstein theorem (if |A| ≤ |B| and
|B| ≤ |A|, then |A| = |B|), which is provable in ZF (see [9]), it follows that |[R]<ω | = |R|.
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It can be readily verified in ZFC (= ZF + AC) that:
Proposition 1.3 (ZFC) For every uncountable subset X ⊂ R, |[X]≤ω | = |X| if and only if CH.
Since CH implies AC(R), one does not expect that Proposition 1.3 holds in the absence of AC. We show in
Corollary 2.4 that this is not the case and Proposition 1.3 is a theorem of ZF. This means that some form of
choice is incorporated in the statement “for every uncountable set X, |[X]≤ω | = |X|”. In particular, we show in
Corollary 2.4 that the latter statement implies CUC (Form 31).
Definition 1.4 If k is an ordinal number, 0 < k ≤ ω, then kX stands for the statement “for every infinite
set X, |k × X| = |X|”. Likewise, k X stands for “for every infinite set X, |X k | = |X|”.
Tarski [20] has shown that 2 X is equivalent to AC. On the other hand, the statement 2X is known to be
strictly weaker than AC in ZF, see Sageev [17]. Clearly kX implies that every infinite set X has a subset of
size k (if f : k × X −→ X is a one-to-one function, then {f ((i, a)) : i ∈ k} is the required subset of X of
size k). In particular for k = ω, ωX implies that every infinite set X is Dedekind-infinite (hence ωX implies
Form 9). Halpern and Howard [5] have shown that ωX is equivalent to 2X, hence 2X also implies Form 9.
In Theorem 2.5(i) we give a direct proof that 2X implies Form 9 (see also Lévy [15]). Now, it is true that
|2 × R| = |R| but the statement
2X|R :
“for every infinite subset X of R, |2 × X| = |X|”
fails to be a theorem of ZF since it implies Form 13.
Similarly, |R × R| = |R| but the statement
2
X|R :
“for every infinite subset X of R, |X × X| = |X|”
is not a theorem of ZF. Indeed, 2 X|R implies 2X|R since if a, b are distinct elements of an infinite subset X of R,
then |2 × X| ≤ |{a, b} × X| ≤ |X × X| = |X|. Hence 2 X|R is unprovable in ZF. Moreover, in contrast to the
fact that 2 X is equivalent to AC, 2 X|R is strictly weaker than AC(R) as Theorem 2.5(ii) clarifies.
2 Main results
Theorem 2.1 C(∞, ℵ0 ) (Form 85) implies UT(WO, ℵ0 , WO) (Form 151). In particular, C(∞, ℵ0 )+Form 34
implies CUC (Form 31), hence in every permutation model, C(∞, ℵ0 ) implies CUC.
P r o o f. Let A = {Ai : i ∈ ℵ}, ℵ a well-ordered cardinal, be a family of countable sets. Clearly
C = {℘(Ai ) : i ∈ ℵ}
is a family of countable sets. Fix by C(∞, ℵ0 ) a choice function f for the family C. For every i ∈ ℵ we construct
via a transfinite induction on ℵ1 a well-ordering of Ai of type vi ∈ ℵ1 .
For j = 0, put ai0 = f (Ai ), for j = v + 1, let aij = f (Ai \ {aiu : u ≤ v}) if Ai \ {aiu : u ≤ v} = ∅, and
for j a limit ordinal of ℵ1 , we let aij = f (Ai \ {aiu : u < j}) if Ai \ {aiu : u < j} = ∅.
Since Ai is countable, it follows that for some j ∈ ℵ1 , Ai \ {aiu : u ∈ j} = ∅. Let
vi = min{j ∈ ℵ1 : Ai \ {aiu : u ∈ j} = ∅}.
It follows that A = {{aij : j ∈ vi } : i ∈ ℵ}, being a well-ordered union of uniformly well-ordered sets, is
well-ordered as required.
34 implies CUC, fix a family
To see that C(∞, ℵ0 )+Form
of countable sets A = {Ai : i ∈ ω}. By C(∞,ℵ0 ),
A is well-ordered. If A is uncountable, then ℵ1 ⊂ A and ℵ1 can be expressed as a countable union of
countable sets. Thus, ℵ1 is singular and Form 34 fails. This is a contradiction.
The last assertion follows from the fact that Form 34 holds in every permutation model, see [7].
Working as in the last theorem one can readily verify the following corollary.
Corollary 2.2 C(∞, ℵ) (if ℵ is a well-ordered cardinal, then every family of sets of size ≤ ℵ has a choice
function) implies UT(WO, ℵ, WO) (if ℵ is a well-ordered cardinal, then the well-ordered union of sets of size
less than or equal to ℵ is well-orderable).
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Theorem 2.3
(i) “For every uncountable set X, |[X]≤ω | = |X|” implies
CH + “for every uncountable set A, every uncountable subset X of A has an uncountable
well-ordered subset”.
In particular, AC does not imply “for every uncountable set X, |[X]≤ω | = |X|”.
(ii) “For every infinite set X, |[X]<ω | = |X|” implies Form 9. In particular, “for every infinite set X ⊂ R,
|[X]<ω | = |X|” implies Form 13.
P r o o f.
(i) It suffices to show “|[X]≤ω | = |X| implies that X has an uncountable well-ordered subset”. Fix such a
set X and let f : [X]≤ω −→ X be a one-to-one function. Let x = f ({x1 , x2 }), x1 , x2 ∈ X, x1 = x2 . We shall
construct recursively a set A = {ai : i ∈ ℵ1 } ⊂ X as follows: For i = 0, put a0 = x. For i = v + 1, a non-limit
/ {aj : j ≤ v}. For i a limit ordinal
ordinal of ℵ1 , put ai = f ({av }). Since f is one-to-one, it follows that ai ∈
of ℵ1 , put ai = f ({aj : j < i}). As in the non-limit case, ai ∈
/ {aj : j < i}, terminating the induction.
The second assertion, i. e. the fact that CH holds, follows from |R| = |℘(ω)| = |[ω]≤ω | ≤ |[ℵ1 ]≤ω | = |ℵ1 |.
(ii) This can be proved as in (i).
Corollary 2.4
(i) The statement “for every infinite subset A of R, |[A]<ω | = |A|” is unprovable in ZF.
(ii) “For every uncountable set X, |[X]≤ω | = |X|” implies CUC.
(iii) “For every infinite set X, |[X]<3 | = |X|” implies Form 9. In particular, “for every infinite set X ⊂ R,
|[X]<3 | = |X|” implies Form 13. Furthermore, for every infinite set X ⊂ R, |[X]<3 | = |X| if and only if
|X × X| = |X|.
(iv) If |[X]<|X| | = |X|, then X is well-orderable. In particular, |[R]<|R| | = |R| implies AC(R) (Form 79).
(v) “For every uncountable subset X ⊂ R, |[X]≤ω | = |X|” if and only if CH.
(vi) Form 368 implies Form 170.
P r o o f.
(i) It is known (see [7]) that Form 13 fails in the basic Cohen model, M1 in [7], for the set A of the countably
many added Cohen reals. Thus, our statement fails in M1.
(ii) Our hypothesis implies CH which in turn implies Form 34,i. e. ℵ1 is regular (CH ⇒ AC(R) ⇒ Form 34).
Now let A = {Ai : i ∈ ω} be a family of countable sets. If A is uncountable, then in view of the proof
of Theorem 2.3(i), A has a subset of size ℵ1 . Hence ℵ1 is singular, a contradiction. This completes the proof
of (ii).
(iii) For the first two assertions, notice that the proof of Theorem 2.3(ii) may be carried out using only
|[X]<3 | = |X|. Now fix an infinite set X of reals and suppose that there is an injective function g : [X]<3 −→ X.
Define f : X × X −→ X by requiring f ((x, y)) = g({g({x}), g({x, y})}) for all x, y ∈ X. It can be readily
verified that f is injective, hence |X × X| = |X|. The converse can be established similarly to the discussion
following Definition 1.2.
(iv) Mimic the proof of Theorem 2.3(i) in order to define a well-ordered subset Y of X such that Y ∈
/ [X]<|X| .
(v), (vi) These, in view of Theorem 2.3, are straightforward.
Theorem 2.5
(i) 2X implies Form 9. In particular, 2X|R implies Form 13.
(ii) 2X|R and 2 X|R do not imply AC(R).
(iii) AC(R) if and only if “for every infinite set X, if |X ≤ω | = |X|, then X has a well-orderable subset of
size |R|”. In particular, “for every infinite set X, if |X ≤ω | = |X|, then X has a well-orderable subset of size |R|”
is true in every Fraenkel-Mostowski permutation model.
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P r o o f.
(i) Fix an infinite set X satisfying 2X and let f : 2 × X −→ X be a one-to-one and onto function. Via an
easy induction we construct a set {xi : i ∈ ω} as follows: For i = 0, let x0 = f ((0, a)), where a ∈ X is fixed
and f ((0, a)) = a. (If f ((0, a)) = a, take x0 = f ((1, a)). Then x0 = a since f is one-to-one.) For i = v + 1,
let xi = f ((1, xv )). Since f is one-to-one, it is straightforward to verify that xi = xj for all i, j ∈ ω, i = j.
(ii) In Solovay’s model M5(ℵ), Form 169 is false. Thus, AC(R) fails in this model. Now if X ∈ M5(ℵ)
is an uncountable subset of R, then |X| = |R| (in M5(ℵ), every uncountable set of reals has a perfect subset,
see [7], and in ZF, every perfect subset of R has power 2ℵ0 , see [22]) and consequently |2 × X| = |X|. Hence
2X|R is true in M5(ℵ). The same argument applies for 2 X|R .
(iii) (⇒) Fix an infinite set X. Clearly |2ω | ≤ |X ≤ω | = |X| which means that X has a continuum sized
subset Y . Since AC(R) is equivalent to the statement “R is well-orderable”, we conclude that Y is well-orderable.
(⇐) This is evident since |R≤ω | = |R|.
Proposition 2.6 The statement
“for every infinite subset X of R, if X is Dedekind-infinite, then |X × X| = |X|”
implies Form 13. Hence it is not provable in ZF.
P r o o f. Let Y be an infinite set of reals. Assume that Y is Dedekind-finite (i. e. it has no denumerable subsets)
and without loss of generality suppose that Y ∩ ω = ∅. Put X = Y ∪ ω. From our hypothesis, it follows that
there exists a bijection f : X × X −→ X. There are two cases:
1. There exists an element x ∈ Y such that for all n ∈ ω, f ((x, n)) ∈ Y . Since f is one-to-one, it follows
that ℵ0 ≤ |Y | which contradicts our assumption on Y .
2. For every x ∈ Y , there exists n ∈ ω such that f ((x, n)) ∈ ω. Define the function h on Y by letting
h(x) = f ((x, nx )), where nx = min{n ∈ ω : f ((x, n)) ∈ ω}. Then h is clearly one-to-one, hence |Y | ≤ ℵ0 .
This is again a contradiction.
The proof of the proposition is complete.
Theorem 2.7 The statement “if X is a Dedekind-infinite subset of R, then |2 × X| = |X|” is not provable
in ZF.
P r o o f. We present a symmetric model similar to the basic Cohen model M1 in [7], but we add an uncountable set of Cohen reals and use countable supports. Let M be a countable transitive model of ZF + V = L, and
P = Fn(ω1 × ω, 2) the set of finite partial functions p with dom(p) ⊆ ω1 × ω and ran(p) ⊆ 2 = {0, 1}. Order P
by reverse inclusion, i. e. p ≤ q if and only if p ⊇ q. Then (P, ≤) is a c. c. c. partially ordered set in M (every
antichain in P is countable) having the empty function ∅ as its largest element 1. Thus, P preserves cofinalities,
hence cardinals (see [14, Theorem 5.10, p. 207]). Let G be a P-generic set over M and M[G] the corresponding
generic extension of M. In M[G], define the following sets:
(1)
ai = {j ∈ ω : (∃p ∈ G) (p((i, j)) = 1)},
(2)
A = {ai : i ∈ ω1 }.
for i ∈ ω1 ,
G is the group of all permutations on ω1 . The effect of a permutation π ∈ G on an element p ∈ P is as follows:
πp((π(i), n)) = p((i, n)). That is, πp is produced by p by changing the first coordinate of the triples in p according to the dictates of π. The effect of π on an element of M is defined by ∈-recursion:
π(∅) = ∅
and π(x) = {π(y) : y ∈ x}.
Let F be the (normal) filter generated by {fix(E) : E ∈ [ω1 ]<ω1 }, where
fix(x) = {π ∈ G : (∀y ∈ x) (π(y) = y)}.
A set s ∈ M is called symmetric if and only if sym(x) ∈ F, where sym(x) = {π ∈ G : π(x) = x}. s is
called hereditarily symmetric if and only if every element z ∈ TC({s}) is symmetric, where TC(x) is the transitive closure of x. Let HS be the set of all hereditarily symmetric names of M and let N = {τG : τ ∈ HS},
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where τG is the value of the name τ given in [14, Definition 2.7, p. 189]. As in [8, Theorem 5.14], it can be
verified that N is an almost universal, transitive class of M[G] which is closed under the eight Gödel operations.
Hence (N , ∈) is a model of ZF.
Claim 2.7.1 The sets ai , i ∈ ω1 , and A defined by (1) and (2), respectively, belong to N .
P r o o f. Define the following names: ai = {(ǰ, p) : j ∈ ω ∧ p ∈ P ∧ p((i, j)) = 1}, A = {(ai , 1) : i ∈ ω1 }.
Clearly ai and A are names for ai and A respectively. Since their elements are hereditarily symmetric, it suffices to show that these names are symmetric. It is straightforward to verify that fix({i}) ⊆ sym(ai ), so ai is
(Claim 2.7.1)
symmetric. Finally, it is evident that sym(A) = G.
Claim 2.7.2 A is Dedekind-infinite in N and A does not have any subsets of size ℵ1 in N .
P r o o f. Clearly F = {(op(ň, an ), 1) : n ∈ ω} is a name for the enumeration f = {(n, an ) : n ∈ ω}
which is hereditarily symmetric since fix(ω) ⊆ sym(F ). The second assertion can be proved similarly to the fact
that the set of the added generic reals in the basic Cohen model (model M1 in [7]) does not have a denumerable
subset (see [2]).
(Claim 2.7.2)
Claim 2.7.3 In N , |A| < |2 × A|.
P r o o f. Assume on the contrary that there exists a bijection f : 2 × A −→ A. Since the sets f [{0} × A] and
f [{1} × A] are disjoint and both of size |A|, it can be readily verified that the set
Y = {j ∈ ω1 : (∃i = j) (f ((0, ai )) = aj )}
has cardinality ω1 . Let F be a hereditarily symmetric name of f with support E. Then there is a condition p ∈ G
such that
p “F is a one-to-one function from 2̌ × A onto A”.
Since p is finite, E is countable and |Y | = ω1 , it follows that there is an ordinal j ∈ Y \ (E ∪ dom(dom(p))).
Let i ∈ ω1 \ {j} be such that f ((0, ai )) = aj and let q ∈ G, q ≤ p be such that
(3)
q op(op(0̌, ai ), aj ) ∈ F.
Fix now an ordinal k ∈ ω1 \ (E ∪ {i, j}) such that (k, n) ∈
/ dom(q) for all n ∈ ω and consider the transposition
π = (j, k). Clearly π ∈ fix(E), hence π(F ) = F . Moreover, π(aj ) = ak and the conditions q and πq are
compatible. Thus, g = q ∪ πq is a well-defined extension of q, hence of p. From (3) we deduce that
g [op(op(0̌, ai ), aj ) ∈ F ] ∧ [op(op(0̌, ai ), ak ) ∈ F ].
This clearly yields a contradiction. Thus, (|A| < |2 × A|)N and the proof of Claim 2.7.3 and the theorem is
complete.
(Claim 2.7.3)
Theorem 2.8(i) below shows that unlike the result of Proposition 2.6 the statement “if X is a Dedekind-infinite
subset of R, then |X <ω × X <ω | = |X <ω |” is a theorem of ZF.
Theorem 2.8
(i) Form 13 implies the statement “if X is an infinite subset of R, then |X <ω × X <ω | = |X <ω | (hence
|2 × X <ω | = |X <ω |)”.
(ii) For every infinite subset X of R, |Inj(X <ω )| = |X| if and only if |[X]<ω | = |X|, where
Inj(X <ω ) = {f ∈ X <ω : f is an injection}.
(iii) Form 13 if and only if “for every infinite subset X of R, |Inj(X <ω ) × Inj(X <ω )| = |Inj(X <ω )|”.
(iv) The statement “for every infinite subset X of R, if |X × X| = |X|, then |X <ω × X <ω | = |X <ω |” is
provable in ZF. However, “for every infinite subset X of R, if |X <ω × X <ω | = |X <ω |, then |X × X| = |X|”
implies Form 13, hence it is not provable in ZF.
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605
P r o o f.
(i) Let X be an infinite set of reals and {xn : n ∈ ω} a denumerable subset of X.
Claim 2.8.1 |ω × X <ω | = |X <ω |.
P r o o f. It suffices to show that |ω × X <ω | ≤ |X <ω |. Define the function h : ω × X <ω −→ X <ω by letting
for each n ∈ ω and each sequence f ∈ X <ω , h(n, f ) = xn f , i. e. the concatenation of xn with f . h is
one-to-one. Let n, f = m, g. If n = m, then xn = xm , hence xn f = xm g. If f = g, then again
(Claim 2.8.1)
xn f = xm g. Thus, |ω × X <ω | = |X <ω | as required.
n
<ω
<ω
<ω
<ω
<ω
= {X × X
: n ∈ ω} and ω × X
= {{n} × X
: n ∈ ω}. For each n ∈ ω,
Now X × X
define a function hn : X n × X <ω −→ {n} × X <ω by requiring for all f ∈ X n and g ∈ X <ω
hn (f, g) = n, f g.
Working similarly to the proof of Claim 2.8.1 one can verify that hn is one-to-one, so this is left as an easy
exercise. Consequently, we have that
|X <ω × X <ω | = | {X n × X <ω : n ∈ ω}|
≤ | {{n} × X <ω : n ∈ ω}|
= |ω × X <ω |
(by Claim 2.8.1).
= |X <ω |
The proof of (i) is now complete.
(ii) Fix an infinite subset X of R and suppose that |Inj(X <ω )| = |X|. Since |[X]<ω | ≤ |Inj(X <ω )| (see footnote 1) on page 600), it follows that |[X]<ω | = |X| as required. For the converse, suppose that |[X]<ω | = |X|.
Thus, we may view each finite subset of X as an element of X. It follows that |Inj(X <ω )| ≤ |[X]<ω |. Indeed,
consider the function f which maps each element x1 , x2 , . . . , xn ∈ Inj(X <ω ) to
{{x1 }, {x1 , x2 }, . . . , {x1 , . . . , xn−1 }, {x1 , . . . , xn }}.
Since the elements of Inj(X <ω ) are injective functions, it can be readily verified that f is an injection. Thus,
|Inj(X <ω )| = |[X]<ω | and consequently |Inj(X <ω )| = |X|.
(iii) (⇒) This can be proved as in (i). Simply replace each occurrence of X <ω by Inj(X <ω ) and each
occurrence of X n by Inj(X n ) = {f ∈ X n : f is an injection}.
(⇐) Fix an infinite subset X of R. Our hypothesis implies that Inj(X <ω ) has a denumerable subset
{fn : n ∈ ω}.
Since the fn ’s are one-to-one functions, we have that the set Y = {ran(fn ) : n ∈ ω} is infinite. Hence Y , being a denumerable union of finite subsets of R, is a denumerable subset of X.
(iv) If |X × X| = |X|, then X has a denumerable subset, hence by (i), |X <ω × X <ω | = |X <ω |. The second
assertion of (iv) can be established similarly to the proof of Proposition 2.6.
Let (D) stand for the proposition “every uncountable subset of R has an uncountable well-orderable subset”.
Since (D) clearly implies Form 13 and Form 368 holds in M1 (see [10]) but Form 13 fails in M1, we cannot
expect that Form 368 implies (D). We show next that the conjunction of Form 368 with Form 212 implies (D).
Theorem 2.9
(i) Form 212 + Form 368 implies (D).
(ii) ACω (2ℵ0 , R) + Form 368 implies “a well-ordered union of countable subsets of R can be well-ordered”.
P r o o f.
(i) Fix A, an uncountable subset of R. Since [A]≤ω ⊂ [R]≤ω , we have by Form 212 and Form 368 that
|[A]≤ω | = |R| (by Form 212, A has a countably infinite subset and, in ZF, |[ω]ω | = |R|)). Let, by Form 212,
f be a choice function of A = {Ax : x ∈ R}, the family of all cocountable subsets of A. On the basis of f , and
following ideas from the proof of Theorem 2.1, define recursively an uncountable subset {ai : i ∈ ℵ1 } of A.
(ii) This can be proved as in Theorem 2.1.
Corollary 2.10
(i) Form 203 implies (D).
(ii) Form 203 implies “a well-ordered union of countable subsets of R can be well-ordered”.
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O. De la Cruz, E. Hall, P. Howard, K. Keremedis, and E. Tachtsis: Properties of the real line and Axiom of Choice
P r o o f. Form 203 implies Form 212 + Form 368 + ACω (2ℵ0 , R) (see [6] and [10]).
Theorem 2.11
(i) C(∞, ℵ0 ) + AC(ω1 , R) implies “for every infinite set X, |[X]≤ω | ≤ |X ω × ω1 |”.
(ii) ACω (R) + AC(ω1 , R) implies Form 368.
P r o o f.
(i) Let f be a choice function of [X]≤ω . As in the proof of Theorem 2.1, for every Y ∈ [X]≤ω we can
define recursively a bijection hY : iY −→ Y , where iY is some countable ordinal. Then F : [X]≤ω −→ X <ω1
≤ω
<ω1
given by F (Y ) = hY is one-to-one, meaning that |[X]
|. In order to complete the proof, it suffices
| v≤ |X
<ω1
ω
<ω1
to show that |X
| = |X × ω1 |. Clearly X
= {X : v ∈ ω1 }.
In [6] we showed that AC(WO, R), i. e. the Axiom of Choice for well-ordered families of non-empty
subsets of R, implies Form 170. The proof readily adapts to show that AC(ω1 , R) implies Form 170. Let
A = {Av : v ∈ ω1 }, where Av = {f ∈ v ω : f is a bijection}. In view of Form 170 we may consider ω1 ⊂ R,
ω
|R|, we may view A as a family of subsets of R. Let, by AC(ω1 , R),
hence v ω ⊆ Rω . Furthermore, since |R
|=
h be a choice function of A. Let F : {X v : v ∈ ω1 } −→ X ω × ω1 be the function given by
for v ∈ ω1 , f ∈ X v .
It can be readily verified that F is one-to-one. Hence | {X v : v ∈ ω1 }| ≤ |X ω × ω1 | and consequently
|[X]≤ω | ≤ |X ω × ω1 | as required.
(ii) From part (i) of this theorem we have |[R]≤ω | ≤ |Rω × ω1 |. Since |Rω | = |R|, |R × R| = |R| and, by
Form 170, ω1 ⊂ R, we see that |Rω × ω1 | = |R| and the desired result follows.
F (f ) = (f ◦ h(v), v),
In ZF0 (ZF minus the axiom of foundation), Form 38 + Form 368 does not imply Form 26. Indeed, in the
permutation model N 55 in [7], see also [11] (in this model, the set of atoms is a set of circles, each one with
center at the origin and of radius 1/n for each n ∈ N, permutations are rotations around the circles, and supports
are finite), Form 38 + Form
368 is true but the countable family A of circles has no choice. Furthermore, each
circle has size |R|. Thus, A cannot have size |R| for otherwise A would have to have a choice set. We show
next that Form 26 implies Form 38.
Theorem 2.12
(i) Form
368 implies “the power set of every set Y ⊂ R which is expressible as a disjoint union of countable
sets Y = {Ai ⊂ R : i ∈ k} has size less than or equal to |Rk |”. In particular, Form 368 implies Form 38.
(ii) For every ordinal number u, if ℵu+1 ≤ |ℵℵuu |, then (ℵℵuu )ℵu cannot be written as a union of a family A
such that |A| ≤ ℵu and |x| ≤ ℵu for all x ∈ A. In particular, Form 170 implies Form 38 (see also [18]).
(iii) Form 26 implies Form 38.
P r o o f.
ω
(i) Fix A = {An : n ∈ k}, a disjoint family of countable subsets of R. Let, by Form 368, f : [R]
−→ R be
k
a one-to-one function. It can be readily verified that the function H : ℘(Y ) −→ R , where Y = A, given by
H(X)(i) = f (X ∩ Ai ), is a one-to-one function. Thus, |℘(Y )| ≤ |Rk | as required.
The second assertion follows from the fact that |Rω | = |R| and Cantor’s theorem that |X| < |℘(X)|.
ℵu
(ii) Fix an ordinal number u and
without loss of generality assume that ℵu+1 ⊂ ℵu . Towards a contraℵu ℵu
diction, suppose that (ℵu ) = {Ai : i ∈ ℵu }, where for each i ∈ ℵu , |Ai | ≤ ℵu . For every i ∈ ℵu , put
Bi = {f (i) : f ∈ Ai }. It is evident that |Bi | ≤ |Ai | ≤ ℵu since Ai can be mapped onto Bi and Ai is well
orderable. Define now the function f : ℵu −→ ℵℵuu by requiring for each i ∈ ℵu that
f (i) = min{k ∈ ℵu+1 : k ∈
/ Bi }.
For every i ∈ ℵu , f (i) is definable since |Bi | ≤ ℵu < ℵu+1 . But by the definition of f it follows that
f ∈ {Ai : i ∈ ℵu } = (ℵℵuu )ℵu ,
a contradiction.
The particular case for R follows from the fact that in ZF, |(ℵℵ0 0 )ℵ0 | = |Rℵ0 | = |R|.
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607
(iii) Assume the contrary and fix a disjoint family of countable sets A = {Ai : i ∈ ω} covering R.
have | {[An ]ω : n ∈ ω}| = |R|. Thus, we
Clearly for every n ∈ ω, |[An ]ω | = |R|. Hence by Form
26 we
ω
ω
may view each
[An ] as a copy of R, hence | n∈ω [An ] | = |R|. Continue now as in (i) to show
that |[R]ω | ≤ | n∈ω [An ]ω |. By (i), this contradicts our assumption and completes the proof of the theorem.
Theorem 2.13
(i) CCω (R), CUC(R), Form 13, Form 34, Form 35, Form 38, Form 74, CC(R), Form 169, Form 170,
Form 211, Form 212 do not imply Form 56.
(ii) Form 56 does not imply CCω (R), CUC(R), Form 34, Form 35, Form 38, AC(R), CC(R), Form 203,
Form 211, Form 212, Form 368. In particular, ¬Form 56 implies Form 170.
P r o o f.
(i) Derrick and Drake [3] have constructed a forcing model, M10 in [7], such that
(∀n ∈ ω) (ℵn ≤ 2ℵ0 ∧ ℵω ≤ 2ℵ0 ).
Thus, in M10, Form 170 is true and H(2ℵ0 ) = ℵω , meaning that Form 56 fails in this model. Since Form 170
implies both Form 38 and Form 169 (see Theorem 2.12(ii) and Proposition 1.1(7)), it follows that the latter
two are also true in M10. Furthermore, M10 extends the model M2ω2 of [7] and similarly to the proof
of [21, Theorem 4] which states that in M2ω2 every continuum sized family of non-empty sets has a choice
function, it can be shown that Form 212 also holds in M10. Therefore, each one of CCω (R), CUC(R), Form 13,
Form 34, Form 35, Form 38, Form 74, CC(R), and Form 211 holds in this model.
(ii) In the Feferman-Lévy model, M9 in [7], R is written as a countable union of countable sets, thus Form 38
fails in M9. Each one of the remaining forms listed in (ii), being stronger than Form 38, fails in M10. Now,
since the only well-orderable subsets of R are the countable sets (see Cohen [2, p. 146]), it follows that in M9,
H(2ℵ0 ) = ℵ1 = ℵω and consequently Form 56 is true in that model.
Theorem 2.14
(i) ¬Form 169 implies CUC(R) (Form 6).
(ii) Form 38 + “for every infinite subset A of R, either |A| = ℵ0 or |A| = 2ℵ0 ” implies CUC(R).
(iii) ω-CC(R) + ¬Form 169 implies CC(R) (Form 94).
(iv) ω-CC(R) + Form 169 does not imply Form 26, Form 368.
(v) Form 169 does not imply CCω (R), CUC(R), Form 34, Form 35, Form 38.
P r o o f.
(i) First we show that ¬Form 169 implies CCω (R) (Form 5). Fix A = {Ai : i ∈ ω}, a disjoint family of nonempty countable subsets of R. Without loss of generality we may assume that for every i ∈ ω, Ai ⊂ (i, i + 1).
As mentioned on page 598, CCω (R) is equivalent to PCCω (R), therefore
it suffices to show that A has a partial
choice function. If A has no partial choice function, then A = A is uncountable. Thus, A has a perfect
subset K. Then |K| = |R| and consequently there are infinitely many members of A meeting K non-trivially.
Since for every i ∈ ω, K ∩ Ai ⊂ Ai and the family of all closed subsets of R has a choice function (see [11]
and [16]), itfollows that A has a partial choice, which is a contradiction.
Now if A is uncountable, then ¬Form 169 implies that A has the power of the continuum. Thus, R is
written
as a countable union of countable sets. This contradicts the fact that CCω (R) implies Form 38 (see [6]).
Hence A is countable.
(ii), (iii) are evident.
(iv) It is known (see [4]) that ω-CC(R) holds in the model M9. Moreover, Form 169 holds in M9, as
otherwise by (i) we would have that CCω (R) holds and consequently Form 38 would also hold in that model,
which is a contradiction. On the other hand, in view of parts (i) and (iii) of Theorem 2.12, we deduce that
Form 368 and Form 26 fail in M9.
(v) Form 169 is true of M9 (as explained in (iv)), whereas CCω (R), CUC(R), Form 34, Form 35, and Form 38
are false in that model (each one of CCω (R), CUC(R), Form 34, and Form 35 implies Form 38, and the latter is
false in M9).
c 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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3 Summary
The meaning of the numerals in the entries of the following table is as follows: A “0” in the square at row n,
column m means that it is unknown whether statement Form n implies statement Form m. A “1” means that
Form n implies Form m and a “3” means that there is a Cohen model in which Form n is true but Form m is false.
The boldface entries of the table are original results of the paper. The slanted entries are either known or
easy results whose proof is based on previous works (see Proposition 1.1). However, their status is indicated as
unknown in [7], so we incorporate these results for completeness sake. For the positive and independence results
whose numeral is in normal typeface the reader is referred to [7].
5
6
13
34
35
38
56
74
79
94
5
1
0
3
3
0
1
3
3
3
3
3
3
3
3
3
3
3
6
1
1
3
3
0
1
3
3
3
3
3
3
3
3
3
3
3
13
3
3
1
3
3
3
3
0
3
3
3
3
3
3
3
3
0
34
3
3
3
1
0
1
3
3
3
3
3
3
3
3
3
3
3
35
0
0
3
0
1
1
3
3
3
3
3
3
3
3
3
3
3
38
3
3
3
3
0
1
3
3
3
3
3
3
3
3
3
3
3
56
3
3
0
3
3
3
1
0
3
3
0
3
3
3
3
3
0
74
3
3
1
3
3
3
3
1
3
3
3
3
3
3
3
3
0
79
1
1
1
1
1
1
0
1
1
1
1
1
1
1
1
1
1
94
1
1
1
1
1
1
3
1
3
1
3
3
3
0
3
3
0
169
3
3
3
3
3
3
3
3
3
1
3
3
3
3
3
3
170
3
3
3
3
0
1
3
3
3
3
1
1
3
3
3
203
1
1
1
1
1
1
0
1
0
1
1
1
1
1
1
1
1
211
1
1
1
1
1
1
3
1
3
1
3
3
3
1
3
3
0
212
1
1
1
1
1
1
3
1
3
1
0
0
3
1
1
368
0
0
3
0
0
1
0
3
3
3
1
1
3
3
3
1
3
369
0
0
0
3
0
0
0
0
3
3
3
3
3
3
3
3
1
3
169 170 203 211 212 368 369
3
3
3
1
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
A. Church, Alternatives to Zermelo’s assumption. Trans. Amer. Math. Soc. 29, 178 – 208 (1927).
P. J. Cohen, Set Theory and the Continuum Hypothesis (Benjamin, New York 1966).
J. Derrick and F. Drake, A model of Z-F set theory with ℵ(2ω ) = ℵω . J. Symbolic Logic 32, 559 – 559 (1967).
G. Gutierres, Sequential topological conditions in R in the absence of the axiom of choice. Math. Logic Quart. 49,
293 – 298 (2003).
J. D. Halpern and P. E. Howard, Cardinals m such that 2m = m. Proc. Amer. Math. Soc. 26, 487 – 490 (1970).
P. Howard, K. Keremedis, J. E. Rubin, A. Stanley, and E. Tachtsis, Non-constructive properties of the real line.
Math. Logic Quart. 47, 423 – 431 (2001).
P. Howard and J. E. Rubin, Consequences of the Axiom of Choice. Math. Surveys and Monographs 59 (AMS,
Providence 1998).
T. Jech, The Axiom of Choice (North-Holland, Amsterdam 1973).
T. Jech, Set theory (Academic Press, New York 1978).
K. Keremedis and E. Tachtsis, Some weak forms of the axiom of choice restricted to the real line. Math. Logic Quart. 47,
413 – 422 (2001).
c 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Math. Log. Quart. 51, No. 6 (2005) / www.mlq-journal.org
609
[11] K. Keremedis and E. Tachtsis, On Loeb and weakly Loeb Hausdorff spaces. Scientiae Mathematicae Japonicae 53,
247 – 251 (2001).
[12] K. Keremedis and E. Tachtsis, On sequentially compact subspaces of R without the axiom of choice. Notre Dame J. Formal Logic 44 (3), 175 – 184 (2003).
[13] K. Keremedis and E. Tachtsis, Choice principles for special subsets of the real line. Math. Logic Quart. 49, 444 – 454
(2003).
[14] K. Kunen, Set Theory: An Introduction to Independence Proofs (North-Holland, Amsterdam 1983).
[15] A. Lévy, The independence of various definitions of finiteness. Fund. Math. 46, 1 – 13 (1958).
[16] P. A. Loeb, A new proof of the Tychonoff theorem. Amer. Math. Monthly 72, 711 – 717 (1965).
[17] G. Sageev, An independence result concerning the axiom of choice. Annals Math. Logic 8, 1 – 184 (1975).
[18] E. Specker, Zur Axiomatik der Mengenlehre (Fundierungs- und Auswahlaxiom). Zeitschrift Math. Logik Grundlagen
Mathematik 3, 173 – 210 (1957).
[19] A. Tarski, On well-ordered subsets of any set. Fund. Math. 32, 176 – 183 (1939).
[20] A. Tarski, Sur quelques théorèmes qui équivalent à l’axiome du choix. Fund. Math. 5, 147 – 154 (1924).
[21] J. Truss, The axiom of choice for linearly ordered families. Fund. Math. 99, 133 – 139 (1978).
[22] J. Truss, Models of set theory containing many perfect sets. Annals Math. Logic 7, 197 – 219 (1974).
c 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim