a convenient direction. We shall illustrate this qualitative aspect by attempting to explain why the CH is not “true,” and why this is nontrivial.
As we have said, we want to construct a subsetM ofR having cardinality intermediate between the cardinality ofN and the cardinality ofR. We do this as follows: For any i ∈ I, let the random variable ¯xi : [0,1]I → [0, I] be the ith projection. Choose a subsetJ ⊂I such thatω0<card J <card I(this is possible ifI is large), and set
M ={xj|j∈ J } ⊂R.
Then card N < card M < card R is true in the usual meaning of the word. However, we must show that the corresponding assertion is “true” in our Pickwickian sense. But then the role of the integers is assumed by the “locally integral” random variables (whose values are integral with probability one), and these random variables can have cardinality much greater than ω0. Thus, the required lower estimate for cardM becomes much more serious. Similarly, if we formalize our naive description ofM and then interpret it inR, then M takes on a new meaning, and leads to a much larger set than the “real”M. Thus, it is also unclear that the upper inequality for cardM still holds. It seems almost miraculous that everything eventually falls into place.
The plan for the rest of the chapter is as follows. In §2 and §3 we give a (shortened) exposition for the second-order language of real numbers of this abbreviated version of the theorem that the CH is not deducible. If the reader is interested only in the complete proof for L1Set, he may skip to§4, where we introduce the Boolean-valued “universe of random sets,” which takes the place of V. In§§5−7 we verify that the Zermelo–Fraenkel axioms are “true,” and in
§8 we verify that the CH is “false.” Finally, in §9 we discuss Cohen’s original method, which is more syntactic and involves somewhat different intuitive ideas.
2.2. The language L2Real. The alphabet consists of the variable symbols x, y, z, . . .; the symbols for degree-1 functionsf, g, h, . . .; the constants 0 and 1;
the degree-2 operations + and·; the degree-2 relations = and; and the same connectives, quantifiers, and parentheses as in languages of L1. Thetermsare x, y, z, . . .and 0 and 1; and alsof(t), t1·t2, andt1+t2iff is a function symbol andt, t1, and t2 are terms. The terms are names of real numbers.
Theatomic formulasaret1=t2andt1t2, wheret1andt2are terms. The set of formulas is defined inductively exactly as in languages of L1, with one addition: ∀f(Q) and∃f(Q) are formulas if Qis a formula andf is the symbol for a variable function. The notions of a free occurrence of a variable (xorf), of a closed formula, and so on carry over to L2Real in the obvious way. We shall use the same type of abbreviated notation here as in Chapter I. The standard interpretation of formulas that is implicit in the language should be obvious from the definitions and from the following examples.
2.3. The formulaZ(y): “y is an integer.” It is perhaps not completely obvious how to write this formula. We can write, “ycan be obtained from 0 by repeatedly adding or subtracting 1,” or else “any functionf that has period 1 and vanishes at 0 must also vanish at y,” i.e.,
Z(y) : ∀f
f(0) = 0∧ ∀x(f(x) =f(x+ 1))
⇒f(y) = 0
.
2.4. The formula CH: “Any subset of Reither has the same cardinality as R, or else is countable or finite.”
We first restate the formula in different words: “Given a set of zeros of any functionh, either there exists a functiongmapping it onto allR, or else there exists a functionf mapping the integers onto all of this set.” We then have
CH:∀h
∃g ∀y ∃x(h(x) = 0∧y=g(x))∨ ∃f ∀y(h(y)
= 0⇒ ∃x(Z(x)∧y=f(x))) . Notice that the formulaZ(x) occurs as part of the CH.
We further write the completeness axiom C:
2.5.The formula C:“Any subset of R(the set of values of a function f)that is bounded from above has a least upper bound z.” We write
C : ∀f
∃y∀x(f(x)y)⇒ ∃z∀y(∀x(f(x)y)⇔zy) .
All the other formulas we are interested in are simpler and do not require any special comment.
We now give a precise definition of the property of “truth” for closed formulas in L2Real; this property was described informally in §1. We empha- size that it is not an absolute property, but rather depends on the choice of the probability space Ω that is used to construct the “model” of the real numbers.
2.6. The algebra of truth values. As in§1, we set I= a set;
Ω = [0,1]I with Lebesgue measure;
B= the algebra of measurable sets in Ω modulo sets of measure zero;
0 = the class of the empty set inB;
1 = the class of Ω inB.
We have the following operations in B:
a, the “complement” of the element a∈B;
a∧b, the “intersection” of two elements a, b∈B;
a∨b, the “union” of two elementsa, b∈B.
These operations satisfy the usual identities and give a Boolean algebra structure onB. We writeab ifa∧b=a.
Moreover, the operations of intersection and union extend uniquely to infinite families of elements, and continue to satisfy the usual identities that hold in the algebra of all subsets of any given set. We shall omit the verification of all this. We note only that sets here are identified “modulo sets of measure zero,” and that identities of the type (A mod 0)∧(B mod 0) = (A∩B) mod 0 do not carry over to infinite families.
Finally,B satisfies the followingcountable chain condition: ifaα∧aβ = 0 for all distinct indices α and β then aα = 0 for at most countably many indicesα. This follows because Lebesgue measure is positive and additive. Tech- nically speaking,B is acomplete Boolean algebra with the countable chain con- dition. The precise origin of B and the fact that it has a measure play a less important role.
2.7. The interpretation set. We now introduce a large set M, each point ξ of which corresponds to the assignment of certain values to all the symbols in the alphabet of L2Real. If ξ is fixed, each formula becomes a concrete statement about measurable functions (random variables) on Ω and about functionals on them (compare with§2 of Chapter II).
More precisely, we set
R= the set of measurable real-valued functions on Ω;
R(1)= the set of all possible maps ¯f :R⇒Rthat satisfy the condition
∀x,¯ y¯∈R
the set(
ω∈Ω|x(ω) = ¯¯ y(ω))
(
ω∈Ω|f¯(¯x)(ω) = ¯f(¯y)(ω)) mod 0
. The definition of R(1) has the following intuitive meaning. If we ignore the
“mod 0,” the condition simply means that the value of the random variable
f¯(¯x) at each trial (each point in Ω) must be determined by the value of ¯x at this trial. Of course, this is a very natural requirement if we want functions ¯f to be adequate reflections of properties of ordinary real-valued functions in the sense of§1. The addition of “mod 0” weakens this requirement by saying “with conditional probability one.”
We now return to the setM. A pointξ∈M consists of a choice of xξ∈R, for each variable symbolx;
fξ ∈R(1), for each symbolf for a variable function.
Here is the interpretation of the expressions in the language that corresponds to a given choice of ξ:
(a) Terms. Let t be a term, and let ξ ∈ M. Then tξ ∈ R is the random variable that is defined inductively in the obvious way.
(b)The truth function on atomic formulas.LetP be the atomic formula t1t2ort1=t2. Its truth value at a pointξ∈M is the element of the algebra B that is defined as follows:
t1t2(ξ) =
*
ω∈Ω|tξ1(ω)tξ2(ω) +
mod 0, and similarly fort1=t2.
(c) The truth function P(ξ) in the general case. The general definition proceeds by induction. The rules when formulas are joined by connectives are the same as in Section 5.7 of Chapter II:
¬P=P, P∨Q=P ∨ Q, P∧Q=P ∧ Q, P ⇒Q=P∨ Q,
P ⇔Q= (P ∧ Q)∨(P∧ Q).
Here, for brevity, we have omitted the ξ. Finally,
∀xP(ξ) =%
ξ
P(ξ) (over allξ that differ from ξonly by a variation ofx);
∃xP(ξ) ='
ξ
P(ξ) (over the sameξ);
and similarly when we quantify over variable functions. Intuitively, the value of the truth function of an assertion about random variables is the set of trials mod 0 for which this assertion becomes true as a fact about real numbers.
2.8. Lemma. If P is a closed formula, then P(ξ) does not depend on the choice of ξ∈M and takes only the value 0or 1.
This is proved by a simple induction on the length of P. It is just as easy to prove a more general fact: if P is any formula and ξ and ξ do not differ
on variables that occur freely in P, then P(ξ) = P(ξ). Compare with Proposition 2.10 in Chapter II.
This value ofP(ξ) that is common for allξifP is closed can be denoted simply by P. We are now ready to formulate the basic definition of this section:
2.9. Definition.A formula P in L2Real is said to be “true” ifP(ξ) = 1 for allξ∈M.