(where we haveh(¯¯ xj) = 0= 0 by 3.7(b)). Hence, for everyj there exists an integern(j) such that 0=x¯j= ¯f(n(j)). SinceJ is uncountable, there exist ann0 and an uncountable subsetJ0⊂ J such thatn(j0) =n0for allj0∈ J0. Then thex¯j = ¯f(n0) forj∈ J0 form an uncountable set of pairwise disjoint nonzero elements ofB. This contradicts the countable chain condition onB.
are defined (intuitively, they should be thought of as the “probability that X is an element of Y” and the “probability that X coincides with Y,”
respectively).
By assumption, this data satisfies the following conditions:
(a) Ifβ1β2α1, thenVβB
1 VβB
2.
(b) Ifβ < αand X∈Vβ+1B \VβB, thenD(X) =VβB. (1)α (c1)X ∈Y='
Z∈D(Y)(X =Z ∧ Z ∈Y)
(the condition (1)α expresses the requirement that the formula x ∈ y ⇔
∃z(x=z∧z∈y), which is easily deduced from the Zermelo–Fraenkel axioms, must be “true”).
(c2)X=Y= %
Z∈D(X)
Z∈X∨ Z ∈Y
∧ %
Z∈D(Y)
Z∈Y∨ Z ∈X
(2)α
(this condition expresses the “truth” of the formula x = y ⇔ (∀z (z ∈ x→ z ∈y)∧ ∀z (z ∈y ⇒z ∈x)). We note that it is not completely clear at this point why, for example, in (1)αwe took the union only overZinD(Y); it would seem natural to take allZ. Later we shall see that the formula remains true if we take the Boolean union over allZ.
This completes the description of the data forVαB. We now give explicitly the recursive construction ofVα+1B and the corresponding data.
Definition of Vα+1B and D. We setVα+1B =VαB∪Vα+1B∗ , where Vα+1B∗ consists of all possible functions Z with domain of definition VαB and range of values⊂B that satisfy the following “extensionality condition”:
X =Y ∧Z(X) =X =Y ∧Z(Y), for allX, Y ∈VαB. (3) A little later we shall defineX∈Z=Z(X) forX ∈VαB andZ∈Vα+1B \VαB. Thus, as before, (3) can be thought of as reflecting the formula
(x=y∧x∈z)⇔(x=y∧y∈z).
Compare also with the comment in 2.7 concerning the definition ofR(1). We shall call the elements of Vα+1B \VαB new elements (of rankα+ 1), and we shall call the elements ofVαB oldelements. We setD(Z) =VαB ifZ is a new element.
Definition of the Boolean truth functions.These functions have already been defined for pairs of old elements. We further set
X ∈Y=Y(X), ifX is old andY is new; (4)
X =Y= %
Z∈D(X)
Z∈X∨ Z∈Y
∧ %
Z∈D(Y)
Z∈Y∨ Z∈X
. (5)
Because of (2)α, (5) automatically holds ifX andY are both old elements; in the other cases, (5) uniquely determines X =Y if we use (4) and the fact that Z runs only through old elements in (5). Finally, we set
X ∈Y= '
Z∈D(Y)
X=Z ∧ Z∈Y (6)
if X is a new element and Y is either new or old. The right side is uniquely determined using (4) and (5), sinceD(Y)⊂VαB.
Formulas (4) and (6) show the following. As a first approximation we might say that a random setY of rankα“consists” of setsZof lower rank that occur inY with probabilityY(Z); these probabilities can be chosen rather arbitrarily, subject only to the extensionality condition (3).
However, we then find (in formula (6) for newX and old Y) that we must automatically “include” more and more elements X in Y with probabilities already assigned by formula (6). It is conditions (3) and (6) that prevent our sets from being completely random.
Definition of VαB and other data for limiting ordinals α.We simply set VαB =∪β<αVβB, and then all the other data has already been determined.
4.3. Verification that the definitions are correct.Properties 4.2 (a) and (b) are obviously preserved in going fromαtoα+ 1; we must verify (1)a+1and (2)a+1. Now the only identity here that is not completely obvious is obtained by taking X old andY new in (1)a+1:
Y(X) = '
Z∈Vαβ
X =Z ∧Y(Z).
This is verified as follows. We obtain by writing the right-hand side in the form ,
ZX = Z ∧Y(X) using (3). We obtain by considering the term with Z =X and taking into account that X =X = 1 for all X (as follows immediately from (5)).
This completes the construction of the Boolean-valued universe.
4.4. Examples and remarks. We examine some special cases of these constructions in order to clarify their structure.
(a) Obviously V1B = {∅}, since there exists a unique “empty” function whose domain of definition is the subsetV0B=∅. We computeV2B=V1B∪V2B∗.
We let {∅}b ∈V2B∗ denote the function of the one-element setV1B that takes the value b∈B. All these functions are extensional, so that
V2B={∅,{∅}b, for allb∈B}. It follows from (4) that
∅∈ {∅}b=b.
It is clear from (5) that
∅={∅}b=b.
Intuitively, these formulas mean that {∅}b consists of one element∅“overb”
and is empty away fromb. Again applying (5), we obtain
{∅}a={∅}b= (a∨b)∧(a∨b) = (a∧b)∨(a∧b).
Thus, {∅}a and{∅}b coincide when either they are both empty or they both consist of one element∅: this agrees with intuition. Now applying (6), we obtain
{∅}a∈ {∅}b={∅}a=∅ ∧ ∅∈ {∅}b=a∧b
(i.e., the only possible inclusion, which has the form∅∈ {∅}, holds when{∅}a
is empty and{∅}b is nonempty).
Finally, let X ∈ V3B∗ be an extensional function on the subset V2B with values inB. Then, by (6),
X ∈ {∅}b=X =∅ ∧ ∅∈ {∅}b=X =∅ ∧b, and by (5),
X =∅= %
a∈B
{∅}a∈X
∧ ∅∈X
= '
a∈B
{∅}a∈X ∨ ∅∈X
.
Thus, intuitively, X =∅ means the complement of the support ofX in B, andX∈ {∅}bis the set where bothX is empty and{∅}bis nonempty, which again agrees with the usual formula ∅∈ {∅}. This shows how new objectsX can be random elements of old objects with nonzero probabilities.
(b) We consider the case B = {0,1}. The corresponding probability space consists of one point, so our random sets become completely determined. What happens is this: the universe VB maps naturally onto the von Neumann uni- verseV in such a way that if ˜X denotes the image ofX∈VB, then allX and Y satisfy the conditions
X ∈Y= 1⇔X˜ ∈Y ,˜ X =Y= 1⇔X˜ = ˜Y .
To construct this map we first set ˜∅= ∅. We now suppose that the map Vα{0,1} → Vα has already been constructed with the required properties, and we extend the map to α+ 1. To do this, for any new element X ∈Vα+1{0,1} we first find the subset ofVα{0,1} on whichX takes the value 1, and we then take the image of this subset in Vα, which is an element ˜X of P(Vα) = Vα+1; by definition, our map takesXto this ˜X. We leave the verification of the properties of this map to the reader.
(c) Boolean truth functions for the formulas inL1Set.
We define these truth functions in an analogous manner to§2. We introduce the interpretation class M: each pointξ∈M assigns to every variable symbol x in L1Set some object xξ =X of the universe VB. We further assume that every point ξmaps the symbol∅in L1Set to the empty set.
IfP is the atomic formulax∈y or x=y in L1Set, then P(ξ) is defined to be xξ ∈ yξ ∈ B or xξ = yξ ∈ B, respectively. The value of P(ξ) for all other P is defined inductively using exactly the same formulas as in Section 2.7. We need only note that although the expressions,
ξaξ and&
ξaξ must be taken over families indexed by the class M when we compute with quantifiers, all the different elements of such a family form a subset of B, so that such an expression makes sense. We shall call a formulaP “true” (in the modelVB) ifP(ξ) = 1 for allξ, and we shall callP “false” ifP(ξ) = 0 for allξ.
As in§3 of Chapter II, it can be verified that all the tautologies and logical quantifier axioms are “true” and that the rules of deduction preserve “truth.”
Hence, it remains for us to show that the Zermelo–Fraenkel axioms are “true”
(for anyB) and that the continuum hypothesis is “false” (for suitableB).