Summing first overY, we rewrite (5) in the form 0=aGis a function ∧ '
β<ω1
β, Z(α)ˆ B ∈G.
Hence, for everyα < ω2 there is aβ(α)< ω1 such that
0=aα=Gis a function ∧ β(α), Z(α)B ∈G.
Then there exist aβ0< ω1and a subsetJ ⊂ω2 of cardinalityω2such that 0=aα=Gis a function ∧ βˆ0, Z(α)B∈G, for allα∈ J. As in 8.11, we obtain a contradiction to the countable chain condition if we show that aα∧aβ= 0 forα=β. But this follows from
aα∧aβZ(α) =Z(β)= 0
by Lemma 8.5.
of the CH. One possible approach to constructing a model (in the usual rather than Boolean sense) of L1Set in whichP is true is as follows.
We take our original countable transitive modelM of set theory (i.e., of the special axioms of L1Set), which was shown to exist in§7 of Chapter II. LetXM
and YM be the “representatives” ofX andY in M. (This means that if, say, X is defined by the formula ∃!x P(x), then XM = xξ, where ξ is a point of the interpretation class for which |P(x)|M(ξ) = 1; see §7 of Chapter II.) We assume thatXM is infinite andYM is nonempty. Then “from an external point of view”XM is countable andYM is at most countable, so there automatically exists a function F that mapsXM onto all ofYM. A natural idea would be to add (the graph of)F toM, i.e., to consider the least countable modelN of the axioms that containsM and F. Then N has a map fromXM ontoYM, but it is very likely thatXN =XM andYN =YM. What we need in N is a map from XN ontoYN.
As we have shown when discussing Skolem’s paradox in Chapter II, at least for certain pairs (such asX =ω0, Y =P(ω0)), we cannot obtain a map from X ontoY in this way. In those cases in which we can construct such a map, we must chooseFvery carefully. Cohen’s idea was thatF, rather than being chosen so as to satisfy some conditions, should be chosen so as to avoid reflecting any specific properties of M, i.e., F should be “generic.” We shall formulate this more precisely.
It turns out to be important to start not by choosing F directly, but by choosing the set
G={restrictions ofFto finite subsets ofXM}.
Clearly,F is uniquely determined fromG:F =∪g∈Gg (recall that a function is the same as its graph). Hence F is contained in any model that containsG.
But now we must give an axiomatic characterization of the suitableGwithout usingF explicitly. Here are the properties that G must satisfy:
9.3.
(a)G⊆C, whereCis the set of maps from finite subsets ofXM toYM. It is important thatC∈M, because the formula in L1Set that defines Cis (M, V)- absolute. We need this remark in order to motivate the general definitions later.
(b)∅∈G; ifp∈Gand q∈C, whereq⊆p, thenq∈G; for anyp1, p2∈G there is p∈Gsuch thatp⊇p1∪p2.
Suppose we have chosen such a setGof maps from finite subsets ofXM to YM. Then ∪g∈Gg is also a map from some subset ofXM to YM. In order for this map to be defined on all of XM and to be surjective, it is necessary and sufficient for the following additional conditions to hold:
∀Z ∈XM, G∩ {p∈C|pis defined atZ} =∅,
∀Z∈YM, G∩ {q∈C|qtakes the value Z} =∅.
We call a subset D ⊆ C dense in C if for all p ∈ C there is a q ∈ D with p⊆q. The set of maps pdefined atZ and the set of mapsq taking the value
Z are dense, and, moreover, are elements of M by the same consideration of (M, V)-absoluteness. Hence the two requirements at the end of the last paragraph are included in the last condition, thatGbegeneric:
(c)G∩W =∅for all dense subsetsD⊆C that are elements ofM. Although it is not yet evident, it is precisely the condition thatGbe generic that ensures that the properties of the sets XM and YM will be preserved as much as possible after we addGto the model.
We now define the general concept of “forcing conditions.”
9.4. Forcing conditions. These are the elements in any partially ordered set (C, <) that has a maximal element 1. UsuallyCand<lie in the original model M.
A setGis calledgeneric overM (relative toC) if the following conditions hold:
(a) G⊆C;
(b) 1 ∈G; if p∈G andq ∈ C, where q p, then q∈ G; for anyp1, p2 ∈G there is ap∈Gsuch thatpp1 andpp2;
(c) G∩D=∅for all dense subsetsD ⊆C withD∈M (D is dense if for all p∈C, there is aq∈Dwithqp).
If the reader compares this definition with the special case in 9.3, he or she will notice that we have replaced ⊆byand∅by 1. This is in keeping with Cohen’s original point of view, according to whichpqif, whenpis considered as a “condition” imposed, say, on F, moreF’s satisfy pthan q. (Eachp fixes the restriction of F to some finite subset ofXM.)
9.5.The existence of generic sets. LetM andC be fixed. IfM ∩ P(C)is count- able, then for every p∈C there exists a generic setG containingp.
In fact, we index the elements ofM∩ P(X) asX1, X2, X3, . . .and then set p1=p, pn+1=
pn, ifpnqfor allq∈Xn;
anyq∈Xn such thatq < pn, otherwise.
Finally, we setG={q∈C|∃n(pn q)}.
Conditions (a) and (b) for Gto be generic are trivial to verify. Condition (c) follows because ifD∈M andDis dense, then there existnandqfor which D=Xn, q∈Xn, andqpn, so thatpn+1∈D∩G.
9.6.The connection with Boolean models. As mentioned before, we have consid- erable freedom in our choice of the setC of forcing conditions and the generic subset G⊆C. Exactly how one “forces” a given propositionP was explained briefly in 9.2. We now show how to construct an axiom model M[G] that containsM andG, onceC andGhave already been chosen.
The article by Shoenfield gives a direct construction, but we shall make use of an analogy with VB, as in Jech’s presentation. In this approach M[G] is constructed in three basic steps:
(a) Corresponding to the set C we construct a canonical complete Boolean algebraB.
(b) We construct a Boolean universeMBoverB that is “relativized” by means ofM.
(c) We construct a canonical maximal idealIG⊆B determined byGand the
“fiber” of the universe MB over the quotient algebraB/IG ∼={0,1}. It is this fiber that will be the modelM[G].
We now discuss these steps separately and in more detail.
9.7.Ordered sets and Boolean algebras. Every Boolean algebraBhas a canonical partial ordering: a b if a∧b = a. All elements of the structure of B are uniquely determined by this partial ordering. The induced ordering onB− {0} is separable. By definition, this means that ifa, b = 0 and a b, then there exists ca, c= 0, such that there is nod= 0 for which db anddc. (It suffices to takec=a∧b.) Suchbandc are calleddisjoint.
Now letC be a fixed partially ordered set. We consider the class of (non- strictly) order-preserving maps ofCinto differentcompleteBoolean algebrasB such that 0 is not contained in the image.
9.8. Proposition. In this class of maps there exists a unique universal map e:C→B with the following properties:
(a) e(c)is the maximal separable ordered quotient set ofC such thatc1, c2∈C are disjoint⇔e(c1), e(c2)∈B are disjoint;
(b) e(c)is dense inB− {0}.
Bcan be realized as the algebra of regular open sets in the space C with the topology defined by the basisUC ={x∈C|xc}, c∈C.
Now we can indicate howIG is constructed from the generic subsetG⊆C:
G1={b∈B|∃p∈G, e(p)b}, IG =B\G1.
It is not hard to prove that IG is a maximal ideal in B, i.e., the kernel of a Boolean homomorphism B → {0,1}. The set G1 is precisely the preimage of 1 under this homomorphism. Since G is generic in C, we have the following property of G1: for any subset A⊆ B such that ,
a∈Aa= 1 and a1∧a2 = 0 whenever a1=a2∈A, there exists a unique elementa∈A∩G1.
9.9.The universeMB. This universe is constructed fromM andBin exactly the same way as VB was constructed fromV andB, with one essential difference:
all constructions are relativized with respect to M.This means that instead of B, we take the algebra BM that “represents”B in M (see 9.2); only ordinals α∈M are used in the construction ofMαB, and so on. A rigorous presentation of these constructions would require much more formalization using the expressive means in L1Set than seems desirable in this section. In such a presentation both the general plan and the details of the work would remain essentially the same as before.
The basic result of these constructions is that to every closed formulaP in L1Set with constants inM corresponds a Boolean truth valueP ∈BM. Here the value 1 corresponds to the axioms, and deductions preserve “truth.”
The next step cuts down the size ofMB, again giving a transitive standard submodel.
9.10. ConstructionofM[G]. For brevity, we shall writeB instead ofBM, and so on. The construction essentially consists in going from “random” sets X, Y ∈MB to “determined” sets X, Y, where we say thatX ∈Y if the truth valueX ∈Y goes to 1 under the homomorphismB →B/IG={0,1}, i.e., if X ∈Y ∈G1 (see 9.8). More precisely, we inductively define
i(∅) =∅,
and let M[G] denote the image of the mapi:MB →V. This notation is jus- tified by the following result. Suppose thatC and<belong toM and that the subsetG⊆C is generic.
9.11.Proposition.M[G]is a model for the Zermelo–Fraenkel axioms that con- tainsM andG. IfM is countable, thenM[G] is the least such model.
M[G] containsM for the following reason. If we letX →Xˆ denote the map M →MB that is constructed as in 8.1, then it is easy to show that ˆX=X.
M[G] contains G because G = G , where G is the object in MB that collects all the ˆb, b∈B, with probability 1.
M[G] is an axiom model basically becauseMB is a Boolean axiom model.
However, here we use in an essential way the assumption that G is generic.
(Shoenfield verifies this result directly, without using MB.)
9.12.Example. We return to the assertion “cardP(ω0)(ω)2” in 9.2. By the above discussion, to prove that it is consistent with the axioms we choose a countable modelM and then set
C={maps of finite subsets ofP(ω0) toω2}, G⊆C= a generic subset ofC.
If we consider a map from a subset ofP(ω0) toω2as a function fromω0×ω2to {0,1}, and if, instead of “relative” constructions inM, we consider “absolute”
constructions in V, then the Boolean algebraB that we obtain from C turns out to be the same algebra that was constructed in 8.3 and 8.4. This explains the appearance ofB. The idealIG did not play any role in§8 because we were not trying to construct a standard model.
9.13. We conclude with a very general theorem of Easton, which shows how little we understand the behavior of the function 2k (ka cardinal).
Letαbe a limit ordinal. Itscofinalitycf (α) is the least ordinalβ such that αis the union ofβordinals less thanα. An infinite cardinalkis calledregularif cf(k) = k and is called singular if cf (k) < k. K¨onig (1905) proved that cf(2k)> k.
9.14.Theorem(Easton, 1965).Let F be any (nonstrictly) monotonic function on a subclass of the regular cardinals that takes values in the class of cardi- nals and that satisfies: cf (ℵF(k))>ℵk. Then the assertion “∀ regular k ∈ dom F,2ℵk =ℵF(k)” does not contradict the Zermelo–Fraenkel axioms.
If the domain ofF is a set, Easton’s theorem can be obtained using a model of the form M[G], where M is a model in which the generalized continuum hypothesis holds (G¨odel proved that such anM exists; see the next(chapter).
If the domain of F is a class (for example, the class of all regular cardinals), the concept of forcing must be generalized to the case that C is a class.
For singular cardinalsκ, the following result is known (Silver’s theorem).
Letκbe singular, cf(κ) uncountable. Denote by κ+ the successor cardinal to κ. If 2cf(λ)=λ+ for all infinite cardinalsλ < κ, then 2cf(κ)=κ+.