Non-constructive Properties of the Real Numbers

Math. Log. Quart. 47 (2001) 3, 423 – 431

Mathematical LogicQuarterly

c© WILEY-VCH Verlag Berlin GmbH 2001

Non-constructive Properties of the Real Numbers

Paul Howarda, Kyriakos Keremedisb, Jean E. Rubinc, Adrienne Stanleyd,and Eleftherios Tatchtsisb

a Department of Mathematics, Eastern Michigan University,Ypsilanti, MI 48197, U. S.A.1)

b Department of Mathematics, University of the Aegean,Karlovasi, Samos 83200, Greece2)

c Department of Mathematics, Purdue University,West Lafayette, IN 47907, U. S.A.3)

d Department of Mathematics, University of Northern Iowa,Cedar Falls, IA 50614, U. S.A.4)

Abstract. We study the relationship between various properties of the real numbers andweak choice principles.

Mathematics Subject Classification: 03E25, 04A25.

Keywords: Real numbers, Axiom of choice, Weak choice principles.

1 Introduction

It is a well known result of ZFC (Zermelo-Fraenkel set theory with the axiom ofchoice AC) that ℵ1 is a regular cardinal (Form 34 in [3]), i. e., ℵ1 is not the limit of anincreasing sequence (an)n∈ω of ordinals in ℵ1). On the other hand, this statement isnot provable in ZF (Zermelo-Fraenkel set theory without AC). Indeed, Form 34 impliesthat the real line R cannot be written as a countable union of countable subsets (Form38 in [3]). For a proof of this fact see [4, p. 148] and the latter statement is not validin the Feferman/Levy model M9 in [3]. A recent work concerning the cofinalityof ℵ1 is due to C. Good and I.Tree [2]. In particular, in [2] is showed that Form 34is implied by each one of the following statements:

1. ℵ1 with the order topology is not paracompact.2. ℵ1 with the order topology has the property that every infinite subset has a limit

point in ℵ1.

3. ℵ1 with the order topology does not contain a countable discrete family of opensubsets.

4. ℵ1 with the order topology is not Lindelof.

1)e-mail: [email protected])e-mail: {kker, itah}@aegean.gr3)e-mail: [email protected])e-mail: [email protected]

424 P. Howard, K. Keremedis, J. E. Rubin, A. Stanley, and E. Tachtsis

In addition, P. Howard and J. E. Rubin [3] prove (see Note 107 in [3]) that Form34 implies each one of the latter statements. In this paper we study weak forms ofthe axiom of choice and their relationships to each other. We are mainly concernedwith properties of the real numbers that require some form of choice in their proof.

Before we set out with results, let us state some of the well known weak choiceprinciples we are going to use.

1. AC(R) (Form 79 in [3]) is the proposition: For every family A = {Ai : i ∈ k}of non empty subsets of R there exists a set c = {ci : i ∈ k} such that for all i ∈ k,ci ∈ Ai. (AC(R) is equivalent to the statement that R can be well ordered.)

2. AC(WO, R) is the proposition: For every family A = {Ai : i ∈ µ}, µ an ordinalnumber, of non-empty subsets of R there exists a set c = {ci : i ∈ µ} such that for alli ∈ µ, ci ∈ Ai.

3. AC(WO, LO) is the proposition: Every well ordered family A = {Ai : i ∈ k},k an ordinal number, of sets such that

⋃A is linearly ordered has a choice set.

4. CAC(R) (Form 94 in [3]) is AC(R) restricted to countable families.5. CACω(R) (Form 5 in [3]) is CAC(R) restricted to countable families of countable

subsets of R.6. CUC(R) (Form 6 in [3]) is the proposition: The union of a countable family of

countable subsets of R is countable.

7. CMC, the Countable Multiple Choice Axiom (Form 126 in [3]), is the proposi-tion: For every family A = {Ai : i ∈ ω} of disjoint non-empty sets there exists a setF = {Fi : i ∈ ω} of finite non-empty sets such that for all i ∈ ω, Fi ⊆ Ai.

8. DMC, the Dependent Multiple Choice Axiom (Form 106 in [3]), is the proposi-tion: IfR is a binary relation on a non-empty set E such that (∀x ∈ E)(∃y ∈ E)(xR y),then there exists a sequence (Fn)n∈ω of non-empty finite subsets of E such that(∀n ∈ ω)(∀x ∈ Fn)(∃y ∈ Fn+1)(xRy).

9. DCR, Dependent Choice for Relations on R (Form 211 in [3]), is the proposition:If R is a binary relation on R such that (∀x ∈ R)(∃y ∈ R)(xR y), then there exists asequence (xn)n∈ω of real numbers such that (∀n ∈ ω)(xn Rxn+1).10. PCAC(R) is the proposition: Every countable family of non-empty subsets of R

has an infinite subfamily with a choice set.

11. PCACω(R) is PCAC(R) for countable families of countable subsets of R.

Other statements we use from [3] include the following: (Forms 34 and 38 arementioned above.)

12. Form 34: ℵ1 is regular.

13. Form 35: The union of countably many meager subsets of R is meager. (A setis meager if it is the union of a countable family of nowhere dense sets.)14. Form 36: If A ⊆ R

n and A⋂B is countable for every bounded B then A is

countable.

15. Form 38: R is not the union of a countable family of countable sets.

16. Form 51: Every linear ordering has a cofinal sub well ordering.

17. Form 170: ℵ1 ≤ 2ℵ0 .

Non-constructive Properties of the Real Numbers 425

18. Form 130: P(R) is well orderable.

19. Form 203: Every partition of P(ω) into non-empty subsets has a choice function.

20. Form 212: If R is a relation on R such that (∀x ∈ R)(∃y ∈ R)(xR y), then thereis a function f : R −→ R such that (∀x ∈ R)(xR f(x)).21. Form 368: The set of all denumerable subsets of R has cardinality 2ℵ0 .

22. Form 369: If R is partitioned into two sets, at least one of them has cardi-nality 2ℵ0.

In the next section we shall derive relationships between these statements andother properties of the real numbers.

2 Results

L emma 1. CAC(R) implies CUC(R).P r o o f . Let A = {Ai : i ∈ ω} be a countable family of countable subsets of R.

For each i ∈ ω, put Si = {f ∈ (Ai)ω : f is a bijection}. Clearly Si ⊆ Rω. Since

|Rω| = |R|, identify Rω with R. By CAC(R), let {fi : i ∈ ω} be a choice set for

{Si : i ∈ ω}. On the basis of the fi’s one obtains easily a bijection f : ω −→ ⋃A. ✷

Th e o r em 1. (i) CAC(R) implies Form 35. (ii) Form 35 implies Form 38.P r o o f . For (i) fix a family M = {Mn : n ∈ ω} of meager subsets of R. We will

show that⋃M can be expressed as a countable union of nowhere dense sets of R. It

is a well known fact that the set of all open subsets of R has power 2ω. Thus, theset of all open dense subsets of R and consequently the set NR of all closed nowheredense subsets of R has size at most 2ω. Therefore we can view NR as a subset of R.Consider now the family E = {En : n ∈ ω}, where

En = {f ∈ NRω : Mn ⊆

⋃k∈ω f(k)}.

We assert that each En is a non-empty set. Indeed, let A = {Ak : k ∈ ω} be afamily of nowhere dense sets such that Mn =

⋃A. Then Mn ⊆⋃

k∈ω Ak and for eachk ∈ ω, Ak ∈ NR. Now define a function f on ω such that for each k ∈ ω, f(k) = Ak.Clearly f ∈ En. By CAC(R), let E = {fn : n ∈ ω} be a choice set for E. Then{fn(k) : n, k ∈ ω} is a countable cover of

⋃M consisting of closed nowhere dense

sets. Clearly⋃M =

⋃n,k∈ω(fn(k) ∩ (

⋃M)) and fn(k) ∩ (

⋃M) is a nowhere dense

set. Thus,⋃M is a meager set as required. Part (ii) is straightforward because each

countable set is meager. ✷

Th e o r em 2. CAC(R) implies that ℵ1 is a regular cardinal (Form 34).P r o o f . Assume on the contrary that there exists an increasing sequence (an)n∈ω

of ordinals in ℵ1 such that limn∈ω an = ℵ1. Without loss of generality, we may assumethat for each n ≥ 1, an \ an−1 is an infinite set. Let f : R −→ ℵ1 be a surjection (see [4,p. 148]). For each n ≥ 1, set Sn = {g ∈ R

ω : f [ran(g)] = an \ an−1}. By CAC(R), wehave that Sn �= ∅ for every n ≥ 1. Indeed, let A = {f−1(x) : x ∈ an \ an−1}. Since Ais a countable set of non-empty subsets of R, CAC(R) implies that there is a choice setc = {rx : x ∈ an \ an−1} for A. As |c| = ω, there exists a bijection g : ω −→ c. Clearlyg ∈ Sn. Identify R

ω with R and let {gn : n ≥ 1} be a choice set for {Sn : n ≥ 1}(which exists by CAC(R)). Clearly G =

⋃n≥1 ran(gn) =

⋃n≥1{gn(i) : i ∈ ω} is a

countable set. Let G = {rn : n ∈ ω} be an enumeration of G. It is evident that


for every x ∈ ⋃n≥1(an \ an−1) there is an n ∈ ω such that f(rn) = x. This fact

immediately yields an injection h :⋃

n≥1(an \ an−1) −→ G and consequently ℵ1 is atmost countable, a contradiction. ✷

In view of Theorem 2 and the discussion in the introduction we see that thestatement “ℵ1 is a regular cardinal” lies in strength between CAC(R) and Form 38.

Another implication whose status was unknown is “Form 170 ⇒ CAC(R)” (seeTable 1 in [3]). By the fact that Form 170 does not imply “ℵ1 is a regular cardinal”(see [3]) and by Theorem 2, we deduce that

(∗) Form 170 does not imply CAC(R).

Co r o l l a r y 1. The statement “ℵ1 is a regular cardinal” holds in every Fraenkel-Mostowski model.

P r o o f . Since CAC(R) holds in all Fraenkel-Mostowski models ([3]) the conclusionfollows from Theorem 2. ✷

L emma 2.

(i) CAC(R) if and only if PCAC(R).(ii) CACω(R) if and only if PCACω(R).

P r o o f . For (i) it suffices to show that PCAC(R) implies CAC(R) as the otherdirection is evident. Let A = {An : n ∈ ω} be a countable family of non-empty setsand let B = {∏i≤n Ai : n ∈ ω}. Apply the partial choice form to B and we obtainthe corresponding choice form. The proof for (ii) is similar. ✷

Th e o r em 3. (i) DCR implies CAC(R). (ii) Form 203 implies DCR.P r o o f .(i) Fix a family A = {Ai : i ∈ ω} of non-empty subsets of R. Without loss

of generality we may assume that the members of A are pairwise disjoint. LetB0 = R \ (⋃A), and for each n ≥ 1 let Bn = An−1. Then

⋃n∈ω Bn = R. De-

fine a relation R on R by requiring xR y if and only if x ∈ Bn for some n ∈ ω impliesy ∈ Bm for some m > n. It is straightforward to verify that R satisfies the hypothesesof DCR. Thus, there exists a sequence (xn)n∈ω such that for each n ∈ ω, xn Rxn+1.Without loss of generality we may assume that x0 /∈ B0. Then {xn : n ∈ ω} is achoice set for the family {Ain : n ∈ ω}, where in is the unique i such that xn ∈ Ai.The conclusion now follows from Lemma 2.

(ii) Let R be a relation on R such that for every x ∈ R there exists y ∈ R suchthat xRy. Since |(0, 1)| = |R|, we may consider R as a relation on (0, 1) such thatdom(R) = (0, 1). For each n ∈ ω, set

An = {f ∈ (0, 1]ω : (∀i ≤ n) (f(i) ∈ (0, 1) ∧ f(0)Rf(1)R · · · Rf(n))∧ (∀i > n) (f(i) = 1)}.

For each n ∈ ω and g ∈ An, set A(g,n) = {f ∈ An+1 : (g|n+1 = f |n+1)}. Clearlythe family A = {A(g,n) : n ∈ ω, g ∈ An} ∪ {(0, 1]ω \

⋃{A(g,n) : n ∈ ω, g ∈ An}}is a partition of (0, 1]ω. Since we can identify (0, 1]ω with R, let, by Form 203,{f(g,n) : n ∈ ω, g ∈ An} be a choice set for {A(g,n) : n ∈ ω, g ∈ An}. By inductionwe define a sequence (xn)n∈ω of real numbers and a sequence (gn)n∈ω of real valuedfunctions such that xn Rxn+1 for all n ∈ ω. Let g0 ∈ A0 and define gn+1 = f(gn,n).


For each n ∈ ω define xn = gn(n). Then the sequence (xn)n∈ω satisfies the conclusionof DCR. ✷

In view of Theorem 3 and (∗) above we see that Form 170 does not imply DCR.It is shown in [3] that the statement “ℵ1 is a regular cardinal” implies none

of the choice forms AC(R), CAC(R), CMC, DMC, DCR and Form 203. It is un-known whether the implications Form 170 ⇒ CACω(R), Form 170 ⇒ Form 38 andForm 170 ⇒ CUC(R) are valid (see Table 1 in [3]). However, we show next that astronger version of Form 170 implies both CACω(R) and Form 38 whereas Form 170does not imply CUC(R).

T h e o r em 4.(i) CACω(R) implies Form 38.(ii) The statement “ℵ1 can be embedded in every uncountable subset of R” implies

CACω(R).(iii) Form 170 + CUC(R) implies Form 34.P r o o f .(i) Suppose, by contradiction, that the set {An : n ∈ ω} is a countable collection

of countable subsets of R that covers R. For each n ∈ ω, put Sn = {f ∈ (An)ω :f is a bijection}. As |Rω| = |R|, let σ be a bijection from R

ω onto R. For each n ∈ ω,let mn be the least m such that σ(Sn) ∩ Am �= ∅. By CACω(R), let {xn : n ∈ ω} bea choice set for the family {σ(Sn) ∩ Amn : n ∈ ω}. Using {fn = σ−1(xn) : n ∈ ω},we can construct a bijection g : ω −→ ⋃

n∈ω An. Thus {An : n ∈ ω} is not a coverof R. So Form 38 holds.

(ii) Let A = {Ai : i ∈ ω} be a family consisting of countable subsets of R. If⋃A is countable, then A has a choice set. Assume⋃A is an uncountable subset

of R. By our hypothesis we can view ℵ1 as a subset of⋃A. Since ℵ1 is uncountable,

there exists an infinite subfamily A∗ = {Aij : j ∈ ω} of A such that ℵ1 ∩ Aij �= ∅ forevery j ∈ ω. Then c = {cij : j ∈ ω} with cij = min(ℵ1 ∩ Aij) is a choice set for A∗.Since an infinite subfamily of A has a choice set, by Lemma 2, A has a choice set.

(iii) Let (αn)n∈ω be an increasing sequence of ordinals in ℵ1. Then⋃

n∈ω αn ⊂ ℵ1.By Form 170 we may view ℵ1 and thus

⋃n∈ω αn as subsets of R. Since each αn is

a countable set, we deduce by CUC(R) that⋃

n∈ω αn is a countable set and thus⋃n∈ω αn �= ℵ1. So ℵ1 is regular. ✷

It is known that Form 170 does not imply Form 34. In the Cohen model M36(Figura’s model) in [3], Form 170 is true whereas Form 34 is false. By Theorem 4,we have that CUC(R) fails inM36 also.

It is still an open problem of whether Form 170 implies Form 38, but we showbelow that Form 170 does not imply Form 36.

T h e o r em 5. Form 36 iff CUC(R).P r o o f . By mimicking the proof of Form 36 ⇒ Form 38 in Note 110 in [3] we get

that Form 36 ⇒CUC(R). For the converse, fix A ⊆ Rn such that A ∩ B is countable

for every bounded B. Since |Rn| = |R|, A can be considered as a subset of R. Let Bbe the set of all open intervals with rational endpoints. By our hypothesis and thefact that B is countable we have that A = {A ∩ B : B ∈ B} is a countable family ofcountable sets. Thus, CUC(R) implies that A =

⋃A is countable. ✷


It is shown in [3] that Form 170 does not imply Form 6, therefore, the followingcorollary follows immediately from Theorem 5.

C o r o l l a r y 2. Form 170 does not imply Form 36.L emma 3. The conjunction of the statements “ℵ1 is a regular cardinal” and

“ℵ1 can be embedded in every uncountable subset of R” implies CUC(R).P r o o f . Let A = {Ai : i ∈ ω} be a family of countable subsets of R. Since ℵ1 is

a regular cardinal, it cannot be written as a countable union of countable sets. Thusℵ1 cannot be embedded in

⋃A. Thus ⋃A is countable. ✷

Th e o r em 6. Form 51 implies AC(WO,LO).P r o o f . Let A = {Ai : i ∈ µ} be a family, where µ is an ordinal and

⋃A islinearly ordered. For each i ∈ µ define Bi =

∏j≤iAj. Let ≤i be a lexicographic order

on Bi. Let B =⋃

i∈µ Bi. For each x, y ∈ B we write x � y if and only if one of thefollowing conditions holds: (a) x = y, (b) x ∈ Bi, y ∈ Bj and i < j, (c) x, y ∈ Bi andx ≤i y. Since (B,�) is a linearly ordered set there is a well-ordered subset C of Bsuch that C is cofinal in B. Since C is cofinal in B there exists a strictly increasingsequence (ij)j<κ of ordinals cofinal in µ such that C ∩ Bij �= ∅ for every j < κ. Foreach j < κ, let cj = min(C ∩ Bij ). Then {cj : j < κ} is a choice set for {Bij : j < κ}.For each i ∈ µ define ai = cj(i), where j is least such that ij ≥ i. Then {ai : i ∈ µ}is a choice set for A. ✷

Co r o l l a r y 3. AC(WO,LO) implies DCR.P r o o f . It is known that the axiom of choice for well ordered families of non-

empty sets implies the principle of dependent choices (see [4, Theorem 8.2]). Theproof of this fact adapts to show that AC(WO,LO) implies DCR. The only partof the proof of this Theorem 8.2 we need to justify is the place where AC|α| isused. We claim that the sets Sξ are linearly ordered. Indeed, letting b = sup(A),A = {u ∈ On : X � u for some X ⊆ R} we see that every well ordered subset Xof R is order isomorphic with some ordinal of b. Thus, every element of Sξ can beviewed as an element of R

b. As Rb with the lexicographic ordering is linearly ordered,

it follows that Sξ is linearly ordered. Thus, the family G = {Sξ : ξ ∈ α} is a wellordered family of linearly ordered sets. Therefore it has a choice set. ✷

Other implications whose status was unknown (see Table 1 in [3]) are

Form 51 ⇒ Form 170, Form 203 ⇒ Form 170, Form 203 ⇒ Form 368,Form 203 ⇒ Form 369, Form 212 ⇒ Form 369.

In Lemma 4 (i) below we show that AC(WO,R) implies Form 170, so we conclude byTheorem 6, that Form 51 implies Form 170. In Lemma 4(ii)-(v), we prove the latterfour implications and in (vi) we state an old result of Tarski [7].

L emma 4.

(i) AC(WO,R) implies Form 170.(ii) Form 203 implies Form 170.(iii) Form 203 implies Form 368.(iv) Form 203 implies Form 369.(v) Form 212 implies Form 369.(vi) Form 368 implies Form 170.


P r o o f .(i) As in Theorem 2, there is a function f from R onto ℵ1. Consequently,

AC(WO,R) implies that {f−1(i) : i ∈ ℵ1} has a choice function. Therefore, ℵ1 ≤ 2ℵ0.(ii) Each well ordering of ω is a subset of ω × ω and |ω × ω| = |ω|. Conse-

quently, each well ordering of ω corresponds to a real number. Therefore, the setW = {R : R well orders ω} is in one-to-one correspondence with a subset of R. De-fine an equivalence relation ≡ on W so that if R1, R2 are in W , then R1 ≡ R2 if andonly if R1 and R2 have the same order type. Let W≡ be the set of equivalences classesof W . It follows from Form 203 that W≡ has a choice set and, since each countableordinal is included in the set of order types, it follows that ℵ1 ≤ 2ℵ0.

(iii) Let D be the set of all denumerable subsets of R. For each X ∈ D, letDX = {f ∈ Xω : f is a bijection}. For all X, Y ∈ D, if X �= Y , then DX ∩ DY = ∅.Since |Rω| = |R|, let F : R

ω −→ R be a bijection. Let A = {F [DX] : X ∈ D} and letB = A ∪ {R \⋃A}. Thus, B is a partition of R so Form 203 implies that A has achoice set C = {rX : X ∈ D}. It is clear that |R| ≤ |D| = |C| ≤ |R|. So, |D| = 2ℵ0 .

(iv) Suppose Form 369 is false. Since |R×R| = |R|, we may assume that R×R canbe partioned into two sets, A and B, each of which has cardinality different from 2ℵ0.Let S = {(R × {x}) ∩ A : x ∈ R}. Then (S, (R × R) \S) is a partition of R × R.If (R × {x}) ∩ A �= ∅ for all x ∈ R, it follows from Form 203 that 2ℵ0 ≤ A, whichcontradicts our assumption on A. Thus, there is an x ∈ R such that (R×{x})∩A = ∅.Then R× {x} ⊆ B, and this contradicts our assumption on B.

(v) Using essentially the same proof as that in part (iv), it is easy to see that Form212 may be used instead of Form 203 to get the same contradiction. ✷

For our last result, we prove some additional independence results. All the modelsmentioned are in [3]. G. H. Moore [6, p. 324] asked whether Form 38 implies Form36. We answer that question in the negative. In Theorem 5 we proved that Form 36is equivalent to Form 6 and in Lemma 5(v) we prove that Form 34 is true in M6.However, Form 34 implies Form 38, so Form 38 is true in M6 and Form 6 is false.(See [3, p. 152].)

L emma 5.

(i) Form 170 is true in M1.(ii) Form 369 is false in M1.(iii) AC(WO,LO) is false in N53.(iv) Form 369 is true in M12(ℵ).(v) Form 34 and Form 170 are true inM6.

P r o o f .(i) LetM1 be the basic Cohen model. Since GCH holds in the ground model, the

reals in M have cardinality ℵ1. Since cardinalities are preserved in the constructionofM1, in M1, ℵ1 ≤ |R|. Thus, Form 170 is true in M1.

(ii) G. P. Monro (Fund. Math. 80 (1973), 101 – 104) has shown that the assertion“every cardinal that cannot be well ordered is decomposable” (i. e. is the sum of twosmaller positive cardinals) (Form 277 in [3]) is true in M1. Since R cannot be wellordered in M1, it follows that Form 369 is false.


(iii) InN53, the set Q = {Qn : n ∈ ω} is well ordered, ⋃Q can be linearly ordered,

but, because supports are finite, Q has no choice function. Therefore, AC(WO,LO)is false.

(iv) In M12(ℵ), every uncountable set of reals has a perfect subset. Since it canbe shown in ZF0, (see [0 AB] in [3]) that every perfect subset of R has cardinality 2ℵ0,it follows that Form 369 is true.

(v) GCH holds in the ground model for M6. Consequently, Form 34 and Form170 hold in the ground model. Sageev (Ann. Math. Logic 8 (1975), pp. 130ff.) hasshown that in M6 any set of mutually incompatible conditions are countable. Itfollows from this that alephs are preserved. Therefore, Form 34 and Form 170 holdin M6. ✷

3 Summary

The following implications are clear, where “x” denotes “Form x”, from the Intro-duction and [3]:

130 ⇒ AC(R); AC(R) ⇒ 369; AC(R) ⇒ 212;AC(R) ⇒ AC(WO,R); 212 ⇒ DCR; AC(R) ⇒ 203;DMC ⇒ DCR; DMC ⇒ CMC; CMC ⇒ CAC(R);AC(WO,LO) ⇒ AC(WO,R); AC(WO,R) ⇒ CAC(R); 6 ⇒ CACω(R).

In the paper we proved the following implications:

CAC(R) ⇒ 35 (Theorem 1(i)); 203 ⇒ DCR (Theorem 3(ii));51 ⇒ AC(WO,LO) (Theorem 6); AC(WO,LO) ⇒ DCR (Corollary 3);DCR ⇒ CAC(R) (Theorem 3(i)); CAC(R) ⇒ 34 (Theorem 2);AC(WO,R) ⇒ 170 (Lemma 4(i)); 203 ⇒ 170 (Lemma 4(ii));203 ⇒ 368 (Lemma 4(iii)); 203 ⇒ 369 (Lemma 4 (iv));212 ⇒ 369 (Lemma 4(v)); 35 ⇒ 38 (Theorem 1(ii));CACω(R) ⇒ 38 (Theorem 4(i)); CAC(R) ⇒ 6 (Lemma 1);6 ≡ 36 (Theorem 5).

In addition we have the following independence results (all the models given canbe found in [3]):

1. 6, 34, 35, 170 � 369, CAC(R), AC(WO,R) (Lemma 5(i), (ii), modelM1)2. AC(WO,R), AC(WO,LO) � 79, 130, 203 (model M2)3. 369 � AC(R), AC(WO,R), 130, 170, 203, 368 (model M5(ℵ))4. 130, DCR, AC(WO,R) � AC(WO,LO) (Lemma 5(iii), N53)5. 34, 170 � 5 (Lemma 5(v), modelM6)6. 369 � CAC(R), DCR, 34, 212 (Lemma 5(iv), model M12(ℵ))7. DCR � DMC and CAC(R) � CMC (model N1)8. 6, 38 � 34 (model M12(ℵ))


9. 170 � 6, 34, AC(WO,R) (Corollary 2, Theorem 5, modelM36)10. CAC(R) � 170, AC(WO,R) (model M38)11. DCR � AC(WO,LO), 203 (model M38).

For the rest of the independence and positive results appearing in the matrix belowthe reader is referred to [3]. In the matrix the numerals have the following meaning:“0” means that it is unknown whether the row form implies the column form, “1”means that that the row form implies the column form; “3” means that the row formdoes not imply the column form in ZF, and “5” means that the row form does notimply the column form in ZF0.

5 6 34 35 38 51 79 92 94 106 126 130 170 203 211 212 337 368 369

5 1 0 3 0 1 3 3 3 3 3 3 3 3 3 3 3 3 3 36 1 1 3 0 1 3 3 3 3 3 3 3 3 3 3 3 3 3 334 3 3 1 0 1 3 3 3 3 3 3 3 3 3 3 3 3 3 335 0 0 0 1 1 3 3 3 3 3 3 3 3 3 3 3 3 3 338 3 3 3 0 1 3 3 3 3 3 3 3 3 3 3 3 3 3 351 1 1 1 1 1 1 0 1 1 5 5 0 1 0 1 0 1 0 079 1 1 1 1 1 3 1 1 1 3 3 0 1 1 1 1 3 1 192 1 1 1 1 1 3 3 1 1 3 3 3 1 3 0 0 3 0 094 1 1 1 1 1 3 3 3 1 3 3 3 3 3 0 0 3 3 0106 1 1 1 1 1 3 3 3 1 1 1 3 3 3 1 0 3 3 0126 1 1 1 1 1 3 3 3 1 3 1 3 3 3 0 0 3 3 0130 1 1 1 1 1 3 1 1 1 3 3 1 1 1 1 1 3 1 1170 3 3 3 0 0 3 3 3 3 3 3 3 1 3 3 3 3 0 3203 1 1 1 1 1 3 0 0 1 3 3 0 1 1 1 0 3 1 1211 1 1 1 1 1 3 3 3 1 3 3 3 3 3 1 0 3 3 0212 1 1 1 1 1 3 0 0 1 3 3 0 0 0 1 1 3 0 1337 1 1 1 1 1 0 3 1 1 5 5 3 1 3 1 0 1 0 0368 0 0 0 0 0 3 0 0 0 3 3 0 1 0 0 0 3 1 0369 0 0 3 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 1

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(Received: September 2, 1999; Revised: July 20, 2000)

Non-constructive Properties of the Real Numbers

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