1. Introduction
This paper is written as a continuation of our earlier paper [
1] under the same title, which thereby has to be viewed as Part I of this paper.
Problems related to the well-orderability of the real line
emerged in the early years of set theory. The axiom of choice
AC implies that every set can be well-ordered, yet
AC does not yield a concrete construction of any particular well-ordering of
. The famous discussion between Baire, Borel, Hadamard, and Lebesgue in [
2] presents related issues widely discussed by mathematicians early in the 20th century.
Then, studies in descriptive set theory demonstrate that no well-ordering of
belongs (as a set of pairs) to the first-level projective classes
,
, see e.g., Sierpinski [
3]. This was a consequence of Luzin’s theorem [
4] saying that sets in
are Lebesgue measurable. (We refer to Moschovakis’ monograph [
5] in matters of both modern and early notation systems and early history of descriptive set theory. Yet, we may recall that
consists of all continuous images of Borel sets in Polish spaces,
consists of all complements of
sets,
consists of continuous images of
sets, and
, for all
).
For the sake of brevity, we let be the hypothesis saying:
“There is a well-ordering of the real numbers which belongs to Γ as a set of pairs.”
Here, is a given class of subsets of Polish spaces. Typical examples include projective classes defined as above, and their effective subclasses resp. , defined the same way but beginning with effective Borel sets, i.e., those that admit a Borel construction from an effectively (that is, computably) defined sequence of rational cubes.
Here, we can limit ourselves to classes
and
. Indeed,
because if a well-ordering ≼ of the reals is say
then it is
as well since
is equivalent to
. Therefore, the result above can be summarized as
.
At the next projective level, Gödel [
6] proved that
is consistent with the axioms of the Zermelo–Fraenkel set theory
ZFC. This was established by a concrete definition of a
well-ordering
of the reals in the constructible universe
. Then, Addison [
7] distinguished a crucial property of
now known as
goodness. Namely, a
-
good well-ordering is defined to be any
well-ordering ≼ such that the class
is closed under ≼-bounded quantification. In other words, it is required that if
is a binary
relation on the reals, then the following relations
belong to
as well. The result by Gödel–Addison then claims that, in
,
is a
-good well-ordering of the reals. It follows that the existence of such a well-ordering is a consequence of the axiom of constructibility
, and hence it is consistent with
ZFC. The
-goodness of
is behind many key results on sets of the second projective level, see for instance ([
5], Section 5A).
As for the opposite direction, studies in the early years of modern set theory (see, e.g., Levy [
8] and Solovay [
9]) demonstrated that the non-existence statement, saying that there is no well-ordering of
definable by any set-theoretic formula with ordinal and real parameters (this includes
as a small part), is consistent as well.
Modern research in connection with projective well-orderings touches on such issues as connections with forcing axioms [
10,
11], connections with large cardinals [
12,
13], connections with cardinal characteristics of the continuum [
14,
15], connections with the structure and properties of projective sets [
16,
17,
18,
19], and others. The following theorem contributes to this research field. The theorem is the first principal result of this paper.
Theorem 1. Let . Then, there exists a generic extension of , in which:
- (i)
is true, and moreover, there is a -long -good well-ordering of the reals;
- (ii)
is false, that is, there are no well-orderings of the reals, of any kind, i.e., not necessarily good.
Therefore, it is consistent that “ holds, even by means of a -good well-ordering, and in the same time the stronger statement fails”.
As an immediate corollary of Theorem 1, we conclude that, for any , the hypothesis is strictly stronger than because there exists a model in which the latter holds whereas the former fails. Thus, the strict ascending condition of the classes is adequately reflected in the property of the existence of a well-ordering of the reals in a given class.
Theorem 1 significantly strengthens a theorem in our previous paper [
1], where we defined a generic extension of
in which there is a
-good well-ordering but there do not exist
-good well-orderings of the reals. Thus, Theorem 1 improves the result in [
1] by eliminating the goodness property in part (ii). This improvement required some crucial modifications in the proof of the theorem in part (ii) in this paper. Indeed in [
1] we were able to use some well-known consequences of the
-goodness, in particular, the basis theorem saying that all non-empty
sets of reals contain
elements. This consequence is not available in the context of claim (ii) of the theorem since the
-goodness is not assumed. To circumvent this difficulty, this paper introduces an entirely new technique of working with the auxiliary forcing relation
, developed in
Section 23,
Section 24,
Section 25,
Section 26,
Section 27,
Section 28,
Section 29 and
Section 30 of this paper.
The other direction of the paper belongs to the context of the second-order Peano arithmetic
and related set and class theories. Theory
governs the interrelations between the natural numbers and sets of natural numbers, and is widely assumed to lay down working foundations for essential parts of modern mathematics including whatever is (or can be) developed by means of the theory of projective sets, see e.g., Simpson [
20].
In particular, claims (i) and (ii) of Theorem 1 can be adequately presented by certain formulas of the language of based on suitable universal formulas for classes and . Therefore, for any given , the statement (i) + (ii) of Theorem 1 is essentially a formula, say , of the language of , whose consistency is established by the theorem. Thus, it becomes a natural problem to prove the consistency of as in Theorem 1 on the base of tools close to , rather than (much stronger) tools. The next theorem, our second main result, solves this problem on the basis of (minus stands for the absence of the Power Set axiom), which is a theory equiconsistent with and thereby a substantial approximation towards .
Theorem 2 (in + ‘ exists’). Let . Then, the conjunction of (i) and (ii) of Theorem 1 is consistent with .
Further reduction to pure will be the topic of our subsequent planned paper.
2. Outline of the Proof
Given
as in Theorem 1, a generic extension of
, the constructible universe, was defined in [
1], in which there exist
-good well-orderings of the reals, but no
-good well-orderings. Here, to prove our main results, Theorems 1 and 2, we make use of a modified model. This model involves a product forcing construction in
, earlier applied in [
18,
21] for models with various effects related to the property of separation in the projective hierarchy, and also in [
17] for a model in which the full basis theorem holds in the effective projective hierarchy (all non-empty
sets of reals contain
elements), in the absence of a
well-ordering of the reals, for generic models with counterexamples to the countable axiom of choice
and dependent choices
DC in [
22], to name a few examples.
Following the earlier papers [
1,
17,
18], we make use of a sequence of forcings
,
, defined in
such that the product forcing
adds a sequence of generic reals to
, uniformly
-definable in two arguments. Each forcing notion
in this construction is a set of perfect trees
similarly to the Jensen minimal forcing defined in [
23]. See more in ([
24], 28A) on Jensen’s forcing. Infinite finite-support products of Jensen’s forcing were first considered by Enayat [
25], as demonstrated in [
1], following this modification of Jensen–Enayat construction results in the existence of
-good well-orderings in
-generic extensions, thus witnessing (i) of Theorem 1.
Yet, a substantial modification of the Jensen–Enayat forcing construction is maintained in this paper, in order to get rid of using countable models of
(i.e.,
without the Power Set axiom). Different tools based on such models were used in earlier papers, e.g., in [
1,
18], in particular, for evaluating the complexity of various sets, leke e.g., the forcing notion itself. However, as one of our goals is to reduce the whole complexity of the construction of the models required, we have to remove models of
from our instrumentarium. Getting rid of models of
is thereby a
principal technical achievement of this paper.
We begin in
Section 3,
Section 4,
Section 5,
Section 6 and
Section 7 with a rather routine material related to arboreal forcings (those with perfect trees in
as forcing conditions) and their countable finite-support products called
multiforcings, as well as finite tuples of trees called
multitrees. The principal
refinement relation
between multiforcings
is introduced in
Section 7. Roughly speaking, its meaning consists in the requirement that every multitree in
(all multitrees related to
) has to be meager in every multitree in
.
The second part of the paper (
Section 8,
Section 9,
Section 10,
Section 11,
Section 12,
Section 13,
Section 14 and
Section 15) develops the background for the above-mentioned technical achievement. It is based on the notion of
sealing refinement (
being a dense subset of
), which means that, besides
, every multitree
is covered by a finite collection of
-extendable multitrees in
(Definition 11). The following
transitivity property takes place: if multiforcings
satisfy
then
, where
is the component-wise union of multiforcings. We consider different types of dense sets to be sealed, including those which govern a kind of Cantor-Bendixson derivative procedure in
Section 12 and
Section 13.
Corollary 12 summarizes the transitivity property as above for different versions of . Theorem 4 proves the existence of sealing refinements. Theorem 5 provides consequences for generic extensions.
The next part of the paper (
Section 16,
Section 17,
Section 18,
Section 19,
Section 20,
Section 21 and
Section 22) presents the key constructions involved in the proof of Theorem 1. We fix a natural number
as in Theorem 1, and consider the constructible universe
as the ground model. Theorem 6 in
Section 19 introduces a
-long
-increasing sequence
of countable multiforcings, whose properties include; first, sealing a sufficient amount of dense sets during the course of the construction; second, a sort of definable genericity in
; and third, a definability requirement—as in Definition 23. The subsequent key forcing notion
(which depends on
) is defined in
Section 20. Its properties include CCC by Theorem 7. Then, we consider
-generic extensions of
, called key models. The main results regarding key models are: Theorem 8, which characterizes generic reals; and Theorem 9, which provides a
-good well-ordering in the generic model considered, with (i) of Theorem 1 as a consequence. Along with Theorem 7, these are the main results of this part of the paper.
Claim (ii) of Theorem 1 involves one more important technical tool related to the above-defined key forcing notion
. It turns out that the
-forcing relation of
formulas is equivalent (up to level
of the projective hierarchy of formulas) to a certain auxiliary forcing relation
defined and studied in the following part of the paper (
Section 23,
Section 24,
Section 25,
Section 26,
Section 27,
Section 28,
Section 29 and
Section 30). Theorem 11 proves the equivalence. This auxiliary forcing is invariant with respect to permutations of indices
(Theorem 12), whereas the forcing
itself is absolutely not invariant in that sense. Such a
hidden invariance plays a crucial role in the construction. It was applied in [
1] in the proof that
-generic extensions satisfy a weaker version of (ii) only for
-good well-orderings. Here, we make use of the invariance to prove, using Theorem 13, that the full version of (ii) also holds in the
-generic extensions. Theorems 11–13 are the main results related to
, and the introduction and the whole treatment of the auxiliary forcing in a form compatible with the system of sealing relations without any reference to countable models of
is our second principal technical achievement.
The final part of the paper contains
Section 31 with a short proof of Theorem 2 and a brief discussion of its possible reduction to a theory weaker than
‘
exists’. We finish in
Section 32 with conclusions and problems.
Here, we present a rather routine material on arboreal forcing notions, i.e., those with perfect trees in
in the role of forcing conditions. Then, in
Section 6, we consider countable finite-support products of arboreal forcing notions, called
multiforcings, as well as finite tuples of trees called
multitrees. We introduce and study a principal
refinement relation between arboreal forcing notions in
Section 4 and between multiforcings in
Section 7.
3. Trees and Arboreal Forcing Notions
Recall that is the set of all tuples (i.e., finite sequences) of . If and , then is the extension of t by i taking the rightmost position. If , then
Generally, ⊂ denotes a strict inclusion (the equality “=” not allowed) in all cases in this paper, i.e., the same as ⫋. The non-strict inclusion is ⊆. The length of t is denoted by , and we put , the set all tuples of length n.
Trees in
are considered. Thus,
is a
tree if
holds for all tuples
in
Then, the
body
is a closed set in
A tree
is:
- -
Pruned, if T contains no ⊆-maximal tuples;
- -
Perfect, if it is pruned and has no isolated branches;
- -
We let contain all perfect trees ;
- -
If then we put ; clearly as well.
If then is a perfect set in
Definition 1. If , then define ( are incompatible) if ; this is equivalent to being finite. Then, means the negation of .
A set is an antichainif holds for all in A.
Definition 2 (arboreal forcing notions). A set is an arboreal forcing if implies . We define to be the set of all arboreal forcings P.Any is:
- -
Regular, if, for all trees , the intersection is clopen in or in ;
- -
Special, if for some finite or countable antichain —note that in this case the antichain A is unique and the forcing P has to be countable.
Note that every special arboreal forcing is regular.
Example 1. For any define .Then, and . Then, is the Cohen forcing, a regular and special arboreal forcing. The set itself is a non-regular arboreal forcing.
Definition 3 (perfect kernels)
. The
perfect kernel of a tree
is the set
This is the largest perfect tree .
Definition 4 (meet of perfect trees). If then let .
The intersection may not even be pruned, but is a perfect (or empty) tree, , and the difference is at most countable.
Lemma 1. Let P be a regular arboreal forcing. Then,
- (i)
If and then X is a finite union of sets of the form , , and then is a perfect tree equal to a finite union of trees in P, and .
- (ii)
Any trees are -compatible (i.e., some tree satisfies ) if and only if , equivalently, .
Proof. (i) Assume that . By the regularity assumption, let be clopen in say . Then, there are tuples such that , where . However, as . As for , proceed by induction.
(ii) is an easy corollary of (i). □
Lemma 2. If and , , then there exists a tree satisfying and .
Proof. Let , where . □
4. Refinements of Arboreal Forcings
In this section, we introduce the key notion of refinement of arboreal forcings.
We remind that if is any poset then a set is:
- -
Dense in in case ;
- -
Open dense in if in addition ;
- -
Pre-dense in if the set is dense in .
An arboreal forcing Q is a refinement of an arboreal forcing P,in symbol , if:
- (1)
Q is dense in , so that for any there is with ;
- (2)
For any we have , meaning thatthere exists a finite satisfying , or equivalently ;
- (3)
If and then the intersection is clopen in , and —it follows that and the set is meager in .
Thus, trees in the refinement Q define closed sets that are essentially smaller in the sense of category than the trees of the original arboreal forcing P do.
Lemma 3. Assume that are arbitrary regular arboreal forcings. Then:
- (i)
The union is regular, too, and Q is open dense in
- (ii)
If , , and , then is a finite union of trees ,
- (iii)
If , , and , then there are trees and satisfying and .
Proof. To prove the regularity of in (i), make use of (3). To prove (ii) apply (3) once again. Finally prove (iii). By Lemma 1(i), there are trees such that . It follows by (ii) that there is a tuple such that . We observe that as Q is an arboreal forcing. Put . □
Lemma 4. If are arboreal forcings then , , .
Proof. Prove . Properties (1), (2) are rather obvious. To check (3), let and . By (2), there is a finite with . If , then is clopen in and is clopen in . Thus, is clopen in . To see that assume otherwise. Then, , and hence there is a tree such that is not meager in . On the other hand, is clopen in by (3). It follows that there are tuples satisfying . However, and This contradicts (3).
The relations and are easy consequences. □
Lemma 5 (see Lemma 5.2 in [
18])
. Let be any -increasing sequence of special arboreal forcings. Then, is a regular arboreal forcing.In addition, if then .
Moreover, if then is pre-dense in P and is dense in P.
Proof. If and , , then ; hence, is clopen in by (3) above. This implies the regularity of P. The additional claims are elementary as well. □
5. Sealing Refinements: Arboreal Forcings
Assume that . Then, a dense set is not dense in any more. Generally speaking, it may not even be pre-dense in . Yet, it happens that there is a special type of dense sets called sealed dense that preserves pre-density under refinements, and a special type of refinements that turns dense sets into sealed dense sets.
The case of arboreal forcings considered here is a simplified introduction into the more important case of multiforcings in the next section.
Definition 5. Let be an arboreal forcing. A set is sealed dense, if 1) it is open, i.e., , and 2) if then .
Lemma 6. Assume that D is a sealed dense set in . Then, D is open dense.
Proof. To prove the density, assume that . Then, , where for all i. At least one of the intersections is not meager in . Then, there is a tuple such that . Then, ; hence, by the openness. □
Lemma 7. Assume that are arboreal forcings, and D is a sealed dense set in P. Then, is a sealed dense set in Q.
Proof. Let
. By (2), in
Section 4, the tree
U is covered by a finite set of trees in
P, hence, by a finite set
of trees
because
D is sealed dense in
P. Then, any intersection
is clopen in
by (3) in
Section 4; hence,
is equal to a finite union of sets
,
. Thus, overall,
itself is equal to a finite union of sets
, where
is such that
for some
. It remains to note that each such
V belongs to
. □
Thus, the sealed denseness is preserved by the refinement operation. The next lemma shows that dense sets give rise to a sealed dense set by a certain kind of refinement.
Definition 6. If are arboreal forcings and , then means that every tree is covered by a finite union of trees in D.
Lemma 8 (see Lemma 5.4 in [
18])
. Assume that are arboreal forcings, , and . Then, and . 6. Multiforcings and Multitrees
By a multiforcing, we understand any map such that . The set of all multiforcings is denoted by . We can represent an arbitrary in the form of an indexed set , with for each , where all components ,, are arboreal forcings. Say that is:
- -
Small, in case both and eachset , , are countable;
- -
special,in case each is special (then countable) as in Definition 2;
- -
Regular,in case all are regular as in Definition 2.
Similarly, a multitree is any function with a finite domain . Let be the set of all multitrees. Any multitree can be represented in the form , where for all . The set is ordered component-wise: ( is stronger than ) if and for all .
Let be a multiforcing. Any multitree is called a -multitree,if and for each the tree belongs to . Clearly, the collection of all -multitreescan be identified with the finite support product of the arboreal forcings involved.
Definition 7. If then define ,the finite Cartesian product of the perfect sets , . If , then let , this is a cylinder in based on .
Definition 8 (extension). If and is an arbitrary multiforcing then we define .
Corollary 1 (of Lemma 1(i)). Let π be a regular multiforcing, , . Then, the intersection is equal to a finite (perhaps empty) union of sets , where and .
Definition 9. Multitrees are incompatible, in symbol , if , or equivalently, , holds for some index , and compatible otherwise. As usual, a set of pairwise incompatible multitrees is called an antichain.
Given a multiforcing π, multitrees are -compatible, if there existsa multitree such that and , and otherwise are -incompatible, in symbol . Sets of pairwise -incompatible multitrees are -antichains.
If multitrees are incompatible, then they are -incompatible for any . The next corollary shows that the inverse is true for regular multiforcings.
Corollary 2 (of Lemma 1(ii)). Let π be a regular multiforcing and . Then, are -compatible if are compatible as in Definition 9.
It follows that being an antichain is equivalent to being a -antichain.
Corollary 3 (of Lemma 2). Let π be a regular multiforcing and , . If for at least one , then there exists a multitree , , satisfying .
Let
be multiforcings. Define a multiforcing
(
the component-wise union), so that
and
If is a sequence of multiforcings, then the component-wise union is accordingly defined so that and for all . We observe that does not preserve regularity.
Definition 10 (component-wise meet of multitrees)
. Let π be a regular multiforcing. Say that a finite set of multitrees is compatible as a whole
if for any index , we have . (Here, and below, it is understood that whenever ). In such a case, let us define a multitreeso that and for all . Corollary 4 (of Lemma 1(i)). Suppose that π is a regular multiforcing, and a finite set of multitrees is compatible as a whole. Then, is a multitree, and is a finite union of sets of the form , where , .
Remark 1 (forcing). Let be an arboreal forcing. We may treat P as a forcing notion, so that if then T is a stronger condition. Clearly, P adjoins a real in .
If is a multiforcing then the set , ordered as above, is accordingly viewed as a forcing notion which adjoins a generic sequence , where every is a -generic real. Reals of the form will be called principal generic reals in the extension by a -generic set G.
7. Refinements of Multiforcings
Here, we extend the notion of refinement to multiforcings in component-wise way.
Let be arbitrary multiforcings. Then, is said to be a refinement of ,symbolically , if and we have in for all .
Corollary 5 (of Lemma 4). If are multiforcings then , , and .
Corollary 6 (of Lemma 3). Let be regular multiforcings. Then, so is , and is an open dense set in . Moreover, if , , and , then there are multitrees , satisfying and .
Corollary 7 (of Lemma 1(i)). Let be regular multiforcings, , , . Then, the intersection is a finite (perhaps empty) union of sets of the form , where and .
Remark 2. It follows from the above that the relations are strict partial orders on sets resp. . In addition, if are multiforcings and , then the relations and are equivalent, where .
The first goal of this Part is to introduce a notion of sealing refinements for multiforcings, similar to the sealing refinements for arboreal forcings as in
Section 5. This is a considerably more difficult case because obtaining adequate, working definitions both of the sealed density and the sealing refinements are somewhat less obvious. In particular, the notion of
sealing refinement (
being a dense subset of
), stipulates, that, besides
, every multitree
is covered by a finite collection of
-extendable multitrees in
(Definition 11). We consider different types of dense sets to be sealed, including those that govern a kind of Cantor-Bendixon derivative procedure in
Section 12 and
Section 13.
Corollary 12 summarizes the transitivity property as above for different versions of . Theorem 4 proves the existence of sealing refinements. Theorem 5 provides consequences for generic extensions. These are main results of Part II.
8. Sealing Refinements
Suppose that is a multitree and a set of multitrees. Define ,if there exists a finite subset such that 1) for all , and 2) . (Regarding we refer to Definition 7).
Definition 11. Let π be a multiforcing and .
- (1)
is sealed dense in if isopen in and holds for every .
- (2)
A multiforcing seals over π,symbolically ,if and, for every with , the relation holds.
- (3)
A multiforcing seals over π in the old sense, symbolically , if and the next condition is true:
if , , , , then there exists satisfying , also , and finally , where
The old definition (3) of sealing refinements was given in [
18] on the basis of earlier studies [
22,
26]. We use here a more flexible definition by (2).
Lemma 9. If are arbitrary multiforcings, , and then .
Proof. Apply (3) with (the empty multitree). □
We will use the notation as in Definition 8 in the following lemmas.
Lemma 10. Suppose that are regular multiforcings, and any set. Then:
- (i)
If is sealed dense in then is open dense, and moreover, if , , then there is a -extendable multitree with and
- (ii)
If is sealed dense in , then implies ;
- (iii)
If then is sealed dense in whereas itself is pre-dense in and in
- (iv)
If , and , , then there exists a -extendable multitree with and
- (v)
If is sealed dense in , and , then the set is sealed dense in and open dense in both and
Proof. (i) As the openness of is given, prove the ‘moreover’ claim. Let , . Then, , where the multitrees satisfy and are -extendable. Then, is compatible with at least one , and hence -compatible by Corollary 2, so that there is a multitree with , , and still . It remains to be shown that is -extendable.
By the choice of , there exists a multitree with and . Define a multitree so that , , and . Then, clearly, , and hence by the openness of .
(ii) Let , . By , there exists a finite collection of sets such that and . However, each , , is covered by a finite union of sets , where is -extendable.
(iii) The openness of in is obvious. To prove the sealed density, let . Since is a product, we can assume that . As , holds, where the multitrees satisfy and are -extendable; hence, there are such that and .
For each , it follows by Corollary 7 that there exists a finite set of multitrees satisfying still and . Let . Then, , so it remains to show that each is -extendable in .
Let and , so that and . As , there is a multitree with and . Then, ; therefore, witnesses that is -extendable. This completes the proof that is sealed dense in .
To prove the pre-density of in , let . As , there is , . There exists , , by the above. Then, . Thus, witnesses that are compatible; therefore, are -compatible by Corollary 2.
To prove (iv) make use of (iii) and apply (i) for .
Finally, (v) easily follows from (i) (to infer the open density in ), (ii), (iii), and Corollary 6 (to infer the open density in ). □
Thus, in the case of multiforcings, the sealed density is preserved by the refinement operation, and just a dense set converts to a sealed dense set by the refinement .
Lemma 11. Let be regular multiforcings and . Then:
- (i)
If are sealed dense sets in then is sealed dense as well;
- (ii)
If are open dense sets in , , and , then we have .
Proof. (i) Both are open dense in , and hence, so is . Now, let , . As is sealed dense, we have , where the multitrees satisfy and are -extendable. In other words, for any there is a multitree such that and . Let . As is sealed dense, we have , where the multitrees satisfy and are -extendable. Thus, for any there is a multitree such that and . Finally, each set is -extendable (to ), and .
(ii) Let , . The sets and are sealed dense in by Lemma 10(iii), hence so is by (i). It follows that we can w. l. o. g. assume that is already -extendable, so that there is multitrees , , and such that and , .
Then, , and on the other hand we have by Corollary 4, where , . Furthermore, by the open density assumption in (ii).
For each i, if , then, by , there are multitrees such that for all , and . We observe that for all by construction; hence, .
Now, let . Then, , , , and each is -extendable (to ), as required. □
Lemma 12. Let be regular multiforcings, . Then:
- (i)
- (ii)
- (iii)
where ;
- (iv)
and .
Proof. (i) Corollary 5 implies . Let , . As , there is a finite collection satisfying for all , and . As , we obtain for any and hence .
(ii) The relation is an easy corollary.
(iii) Let , . As , there exists a finite satisfying for each , and . Then, we have for every since . In other words, there exists a finite with . The multitrees in U can be refined using Corollary 7, so that we obtain a finite collection satisfying
(1) Still for any ;
(2) ;
(3) If then for some — and hence easily .
It follows that , as required.
(iv) The relation is an easy corollary of (3).
Finally, is established by the set W in the proof of (iii). □
9. Two Examples
Here, we consider two important types of dense sets that can be made sealed dense.
Example 2. If π is a multiforcing and (not necessarily ), then
- (∗)
the set is open dense in
by Corollary 3 in case , whereas if then .
If is another multiforcing then we write instead of , and say that seals over . Note that in this case.
In addition, if is sealed dense in then we say that is sealed by .Still in this case.
Corollary 8. Assume that are regular multiforcings, and . Then:
- (i)
If is sealed by π then
- (ii)
If then is sealed by , while is open dense in
Proof. We first recall (∗) in Example 2 and observe that . Then, to prove (i), (ii) apply Lemma 10. □
If is a multiforcing and are -incompatible multitrees in (not necessarily in ), then it is well possible that become -compatible for another multiforcing , even with . To inhibit such a case, the following condition is introduced.
Example 3. Let π be a multiforcing. If (not necessarily ), then let The set is open dense in by Corollary 3, provided and are -incompatible, but if are -compatible, then is not dense in .
We define to mean that , and are -incompatible, and . In this case, we say that seals over .
If , and are -incompatible, and is sealed dense in , then we say that is sealed by .
The following corollary reinterprets some key results above in terms of .
Corollary 9. Let and be regular multiforcings and . Then:
- (i)
If is sealed by π then
- (ii)
If then is sealed by , while is open dense in
- (iii)
If is sealed by π, , and , , , then is sealed by π as well.
Proof. The proof is similar to Corollary 8. We make use of Lemma 10 and Lemma 12(i),(iv), in view of the fact that . As for the extra item (iii), we obviously have provided and . □
10. Real Names and Direct Forcing
In this section, a notational system for names of reals in is introduced. It is appropriate for dealing with forcing notions .
Definition 12. We let a real name be any such that the sets satisfy the following condition: given , any , are incompatible, i.e., (Definition 9). Let ; .
A real name is small if every set is finite or countable — then both the set , and itself, are countable as well.
Given a multiforcing π, a real name is:
- -
-complete, whenever every collection (the -cone of ) is pre-dense (and then clearly open dense) in .
- -
sealed -complete, whenever each set is sealed dense in .
It is not assumed here that , or equivalently, , .
Suppose that is a real name. Say that a multitree :
Directly forces , where and —in case there is a multitree such that ;
Directly forces , where —in case directly forces for all , where ;
Directly forces , where —in case there is a tuple such that directly forces .
Lemma 13. Let π be a multiforcing, , , a -complete real name, .
There exists and a multitree , which directly forces .
There exists and a multitree , , which directly forces .
The definition of direct forcing is associated with the following notion of genericity.
Definition 13. Suppose that π is a multiforcing. A set is -generic if:
- (1)
For any in , implies .
- (2)
If then there is with , .
- (3)
G intersects every set of the form , .
Lemma 14 (obvious). Suppose that π is a multiforcing and is a -complete real name. Let be π-generic over . If some directly forces , or , or , then resp. , , .
Example 4. If , then let be a real name such that each set consists of a single multitree , satisfying (a singleton), and , where . Then, is a small real name, -complete for any multiforcing π. If a set is -generic over , then the real is identical to the real (see Remark 1). In other words, is a canonical name for .
11. Sealing Real Names and Avoiding Refinements
Here, we develop the idea of Definition 11 in the context of dense sets generated by real names.
Definition 14. Let be multiforcings and be a real name. We define that seals over π,in symbol ,in case seals each setover π, i.e., , in the sense of Definition 11. Corollary 10. Suppose that are regular multiforcings and is a real name. Then:
- (i)
If then the name is -complete, -complete, and sealed -complete;
- (ii)
If and is sealed -complete, then is sealed -complete.
Proof. To prove (i), (ii) apply Lemma 10 and observe that . □
If is a multiforcing then the forcing notion adjoins a family of principal generic reals , , where every is -generic over the ground set universe. Obviously many more reals are added. The next definition provides a sufficient condition for a -complete real name to generate not a real of the form .
Definition 15. Suppose that π is a multiforcing and . A real name is called non-principal at ξ over π,if the next set is open dense in : It will be demonstrated by Theorem 5(i) below that the non-principality at implies that is not a name of the real . Moreover, the avoidance condition in the following definition will be demonstrated to imply that is a name of a non-generic real.
Definition 16. Let π be a multiforcing and be a set of trees (e.g., for some ). A real name is said to avoid Y over π,if for each tree , the setis sealed dense (then open dense) in in the sense of Definition 11. Let be multiforcings, , be a set of trees. We write ,if for each tree , seals the set over π — that is formally .
The relation will be applied mainly in case for some .
Theorem 11.1 in [
18] demonstrates that if
is a small regular multiforcing,
, and a real name
is non-principal at
over
(in the sense of Definition 15) then there is a special multiforcing
with
(as in Definition 16). This fact will be used in the proof of Theorem 4 below.
Lemma 15. Let be regular multiforcings, be a set of trees, be a real name.
- (i)
If then avoids Y over ;
- (ii)
If avoids Y over π then avoids Y over as well.
Proof. (i) Let . The set is sealed dense in by Lemma 10(iii). However, clearly, ; thus, the set is sealed dense in as well.
(ii) Let . The set is sealed dense in by the avoidance assumption. Thus, is a sealed dense set in by Lemma 10(iii). However, clearly, . It follows that the set is sealed dense in as well. □
12. Inductive Analysis of Well-Foundedness
Here, we accomplish some work related to the combinatorial description of forcing of well-founded trees. This will be applied in Part IV as a tool to define an auxiliary forcing relation for formulas in and via the well-foundedness of certain trees.
A set
is called a
tree-name, if whenever
belong to
and
then
. Following
Section 10, say that a multitree
directly forces if
for some
such that
.
Definition 17. Assume that π is a multiforcing and is a tree-name.
If and then define Let the derivative contain all pairs such that is dense in (then clearly open dense too), so that . This is equivalent to saying that is dense in below p.
Note that is a tree-name. Define a descending sequence of tree-names , by transfinite induction, so that , for , and for limit . Then, eventually for some , and we let for this index . Thus, is a tree-name as well, and .
Lemma 16. Let be a tree-names, be regular multiforcings, and . Then, for all ν, and accordingly .
Proof. It suffices to prove that just ; all further inductive steps are similar. Recall that is open dense in by Corollary 6. It follows that one and the same set is dense in if it is dense in . □
Definition 18. A set is π-generic over τ if G is π-generic as in Definition 13, and G intersects every set of the form , dense in , where and . Put .
Thus, is a tree is because is a tree-name.
For any tree let be the pruned derivative, that consists of all that are not terminal nodes in T, and let be the pruned kernel, the largest subtree with no terminal nodes, that consists of all that belong to infinite branches .
Lemma 17. Assume that π is a multiforcing, is a tree-name, and a set is π-generic over τ. Then:
- (i)
is the pruned derivative of the tree
- (ii)
is the pruned kernel of
- (iii)
G remains π-generic over and over .
Proof. (i) The contrary assumption results in the two following cases.
Case 1: some is maximal in . In particular, we have multitrees and such that and . By definition, the set has to be dense in . Therefore, as G is generic over , and , some belongs to G as well. By definition, directly forces for some . Then, there is a multitree satisfying and . However, , contrary to the choice of s.
Case 2: a tuple belongs to but s does not belong to . Then, we have , , . It follows that , and hence , contrary to the choice of s.
Claim (ii) is a corollary of (i). To check (iii) note that for all . □
Corollary 11. Under the assumptions of the lemma, let .
- (i)
If directly forces then has an infinite chain containing
- (ii)
If no condition , directly forces , then is well-founded over s.
Proof. By Lemma 17, is the the pruned kernel of . Thus, includes an infinite chain containing some if . This easily implies both items. □
13. Absoluteness of the Derivative
The key result of this section will be to show that, under certain restrictions, the pruned derivative operation introduced in
Section 12 is absolute with respect to refinements of the multiforcings involved. We need, however, to introduce another property of the form of
sealing of dense sets, as in Definition 11.
Definition 19. Let π be a multiforcing, and is a tree-name, as in Section 12. Say that τ is sealed in π, if the following conditions hold: - (a)
If then ,
- (b)
If , , and (see Section 12) is a set dense (then open dense as well) in , then is sealed dense in .
Let be another multiforcing with , so that is a refinement of π. Say that seals τ over π,symbolically ,if the next two conditions hold:
- (c)
Just as (a) above;
- (d)
If and , and the set is dense (then open dense) in , then .
The following claims show the effect of in terms of Lemma 16.
Lemma 18 (obvious). Let be a tree-name, be regular multiforcings. Then for all ν, and accordingly .
Theorem 3. Let be a tree-name, be regular multiforcings, holds for all , and . Then, the following holds:
- (i)
If , then and
- (ii)
If , then the set is dense in ;
- (iii)
If τ is sealed in π, then ;
- (iv)
If , then for all ν, and accordingly
- (v)
If , then τ is sealed in ;
- (vi)
If , , and (see Example 3)holds for all such that are -incompatible— then the following are equivalent:
No multitree directly forces ;
No multitree directly forces ;
No multitree directly forces .
Proof. (i) Let belong to . There is a multitree with . Then, directly forces , some j, and hence, so does ; thus, .
(ii) Let , meaning that , , and also , where for some . Let . Note that both belong to ; therefore, we have and , and then by Lemma 11(ii). It follows by Lemma 10(iv) that there is a -extendable multitree , , with . In other words, there are multitrees with and , and with . As , the multitree cannot be incompatible with and with . Therefore, . This implies , and hence, . This ends the proof.
(iii) Suppose that , , and the set is dense in . Then, holds by Lemma 10(ii), as required.
(iv) In view of Lemmas 16 and 18, it suffices to prove .
Step 1:
. Let
. Then,
, and by definition, the set
is dense in
. We conclude that
(see
Section 12) is open dense in the whole
, and hence,
holds. Then,
is open dense in
by Lemma 10(iii),(v). Accordingly,
is open dense in
. Then,
, a bigger set by (i), is open dense in
, too. Thus,
.
Step 2: . Assume that , so that and the set is open dense in . Then, the set is dense in as well by (ii). We claim that the set is dense in .
Indeed, let . Then, there is , . By (ii), there exists a pair of multitrees and such that . Therefore, witnesses that the multitrees in are compatible, and hence, compatible right in by Corollary 2. Thus, we have established that the set is at least pre-dense, and then obviously dense in , as required.
(v) Prove (b) of Definition 19 for . In view of (iv) and Lemma 18, it suffices to only consider the case , i.e., given , and assuming that the set is dense in , we have to prove that is sealed dense in .
By definition, the set is dense (then open dense) in . It follows by (ii) that the set is also dense in . We conclude (see Step 2 above) that the set itself is dense in . Then, the set itself is dense in . Therefore, we have because is assumed.
It follows by Lemma 10(iii) that is sealed dense in . However, easily by (ii). This ends the proof that is sealed dense in .
(vi) Recall that is dense in by Corollary 6. Therefore, (iv) implies that . Moreover, (iv) implies as well that simply because . It remains to be proven that conversely .
Let (3) fail, that is, we assume that , , and directly forces . Then, by (iv) directly forces ; hence, holds for some with . Thus, are -compatible (by ). It follows that are -compatible as well. Indeed, otherwise we have by the assumptions of (vi). This implies that is sealed by by Corollary 9. However, this contradicts the -compatibility of . Finally, the -compatibility of means . □
14. Combining Refinement Types
Definition 20. Let be multiforcings and M be any set. W define to mean that the following conditions hold:
- (1)
If , , , D is pre-dense in , then we have ;
- (2)
If , , is open dense in , then we have ;
- (3)
If and , then ;
- (4)
If , , and are -incompatible, then ;
- (5)
If and is a -complete real name then we have ;
- (6)
If is a -complete real name, non-principal at over then ;
- (7)
If , is a tree-name, then
Corollary 12. Let M be a countable set, be regular multiforcings, and . Then, and .
Proof. Our basic reference is Lemma 12(i)(iv), which has to be applied for those sets involved in the definition of above (Definition 20). □
This follows a refinement existence result.
Theorem 4. If π is a small regular multiforcing and M a countable set, then there exists a special multiforcing satisfying and .
Proof (sketch)
. The proof is based on some rather difficult results in [
18] which we make use of here without proofs.
First of all, we can assume that , since all elements in are irrelevant. Let be the (countable) set containing , every , and every element of M. Let contain all sets , -definable over HC, with sets in allowed as parameters.
Definition 7.1 in [
18] introduces the notion of
-
generic refinements. Lemma 7.2 and Theorem 7.3 in [
18] prove the existence of a special multiforcing
, which satisfies
and is an
-
generic refinement of a given small regular multiforcing
. If
is such, then Theorem 8.1 in [
18] proves the relation
, and hence,
, for all open dense sets
,
. This implies (1)–(5) and (7) of Definition 20 because all dense sets involved there belong to
by construction.
Finally, (6) of Definition 20 is separately established by Theorem 11.1 in [
18]. We may note that (6) of Definition 20 differs from other items of this definition in that the list of the dense sets involved depends on the new multitree
(the one claimed to exist). Therefore, it needs a special theorem in [
18], namely Theorem 11.1. □
15. Consequences for Generic Extensions
Lemma 19 shows that real names provide a suitable representation of reals in -generic extensions. Then, corollaries for non-principal names will be the subject of Theorem 5.
Lemma 19. Assume that π is a regular multiforcing in the ground set universe , and is a -generic set over .
- (i)
If is a real in then there exists a -complete real name , , satisfying .
- (ii)
Let be a CCC forcing in , and , be a -complete real name. Then, there exists a small-complete real name , , such that every condition in forces the equality over .
As usual, the CCC property means here that every -antichain (i.e., antichain in , see Definition 9 and Corollary 2) is at most countable.
Proof. Claim (i) is a partial case of a general forcing theorem. To prove claim (ii), consider open dense set , choose maximal antichains in those sets, note that each is countable by CCC, and finally, define , where . □
Theorem 5. Suppose that π is a regular multiforcing, and . Then, the following holds.
- (i)
If is a CCC forcing, is a set generic over the ground set universe , is a real in , and , then there exists a small -complete real name , non-principal at ξ over π, satisfying .
- (ii)
If is a -complete real name that avoids over π, is a regular multiforcing, , and is generic over , then .
Proof. (i) By a general forcing theorem, there exists a
-complete real name
such that
and
forces that
. We can assume that
is small, by Lemma 19 (as
is CCC). Let us prove that
is non-principal at
over
, meaning that the set
is open dense in
. As the openness is clear, it remains to prove the density.
Let . Then, by the choice of , -forces . Thus, we may assume that, for some n, the inequality is -forced by . By Lemma 13, there is a tuple and a multitree , , such that directly forces the sentence . It remains to be checked that . Indeed, assume otherwise: . Then, the tree belongs to . Define a multitree by and for all . Then, and we have . However, directly forces both and to be equal to the same number , which contradicts the choice of n.
(ii) Suppose towards the contrary that and . Lemma 15 (ii) implies that avoids over as well. Lemma 10 implies that the set is open dense in . Therefore, the set itself is pre-dense in . We conclude that by the genericity, so that some multitree directly forces . It follows that , which is a contradiction. □
Here, we present the key forcing constructions of the proof of Theorem 1.
We consider the constructible universe as the ground model.
Fix a natural number as in Theorem 1.
Theorem 6 in
Section 19 introduces a
-long
-increasing sequence
of special multiforcings, whose properties include: first, sealing many dense sets during the course of the construction; second, a sort of definable genericity in
; and third, a definability requirement—as in Definition 23. The subsequent key forcing notion
(which depends on
) is defined in
Section 20. Its properties include CCC by Theorem 7. Then, we consider
-generic extensions of
, called key models. The main results about key models are Theorem 8, which characterizes generic reals, and Theorem 9, which provides a
-good well-ordering, with (i) of Theorem 1 as a consequence. Along with Theorem 7, they are the main results of this Part.
We begin with routine stuff on -increasing sequences of special multiforcings.
16. Increasing Sequences of Multiforcings
Based on Remark 2, we consider
-increasing sequences of multiforcings. Let
Thus, a multiforcing
(the set of all multiforcings) belongs to
if
is finite or countable and each
,
, is a special forcing. (See
Section 3 and
Section 6).
If , then let be the set of all -increasing sequences of multiforcings , of length , which are domain-continuous, so that if is a limit ordinal then .
Let (-increasing sequences of countable length).
The set is ordered by the relations ⊆, ⊂ of extension of sequences. Thus, means that a sequence properly extends .
If , then let (the component-wise union), , (a subset of ).
Lemma 20. Assume that and . Let for all α. Then:
- (i)
The multiforcings , , , are regular, and we have:, , and ;
- (ii)
The set is pre-dense in and is dense in .
Proof. To prove (i),(ii) apply Lemma 5. □
The following is a related form of -type definitions.
Definition 21. Let , , , and M be any set. Define , if (i.e., extends π) and , where is the component-wise union and is the first term in absent in .
Lemma 21. Assume that M is a countable set. Then:
- (i)
If and , then there exists a sequence such that
- (ii)
If , , , , and a set is open dense in , then , so that is pre-dense in .
Proof. (i) Let
. By Theorem 4, there is a special multiforcing
satisfying
and
, where
and
Each is open dense in because is pre-dense by Lemma 20(ii). Then, using Theorem 4 for in iteration, we define by transfinite induction special multiforcings , , such that and the sequence is just -increasing. Now, let , that is, for but for . Then, and by construction.
(ii) We have by Corollary 12, hence in particular . It follows by Lemma 10(iii) that is a pre-dense set in . □
17. Definability Lemma
Recall that is the set of all hereditarily countable sets. Thus, if the transitive closure is at most countable. Note that under .
We use the standard notation
,
,
(slanted lightface
)for classes of
parameter-free definability in
(no parameters allowed), and
,
,
for
full definability in
(parameters from
allowed). We will make use of the following known result, see e.g., Lemma 25.25 in Jech [
24]: if
and
then
and similar equivalences for the classes
,
,
,
instead of
,
.
Lemma 22 (in
)
. The following ternary relation belongs to the class : Proof. Note first of all that
, so that the claim makes sense. The proof goes on by routine verification that all sets and relations involved are definable by
formulas, i.e., those with only bounded quantifiers over suitable countable sets such as
or
despite the fact that their prima facie definitions may include quantifiers over uncountable sets such as
Consider for instance the relation
that participates in several definitions, e.g., in the definition of
regular arboreal forcing (Definition 2), in the definition of refinements in
Section 4,
etc. We observe that, because of the compactness of
if
then for
to be clopen in
it is necessary and sufficient that there exists a finite set
such that
and this condition is obviously
. Thus, this implies that the refinement relations ⊏ and
between arboreal forcings (
Section 4 and
Section 5) are definable by
formulas.
To check that
as a ternary relation (Definition 11) is definable by
formula, it suffices to prove the
definability of the relation
(see the beginning of
Section 8), where it is assumed that
,
is finite, and
for all
. Then, the relation
is equivalent to the following:
if for all then there is such that for all .
However, this condition is as required. □
18. Auxiliary Diamond Sequence
We argue in . Let us recall the technique of diamond sequences in .
Lemma 23 ( in ). There is a sequence of sets , such that
- (∗)
if then the set is stationary in ,so that it has a non-empty intersection with each club (i.e., a closed unbounded set) .
Proof. The existence of a sequence satisfying (∗) is the
diamond principle , see ([
24], Theorem 13.21). The
-definability (see is achieved by taking the
-least possible
at each step
, where
is the Gödel’s well-ordering of
, see ([
24], p. 558). □
Definition 22 (in ). We fix a sequence given by Lemma 23.
We let th element of in the sense of ; thus .
If and then let .
If then let . Then, is still a sequence.
Let .
Let . Then, is still a sequence.
An ordinal is a crucial ordinal for a sequence if the relation holds. This is equivalent to .
We obtain the following lemma as an easy corollary.
Lemma 24 (in ). (i) If then the set is stationary.
(ii) If for all n then the set is stationary.
Proof. To prove (i), let . The set is then stationary. However, easily . To prove (ii) put and apply (i). □
19. The Key Sequence
The next theorem (Theorem 6) is a crucial step towards the construction of the forcing notion that will prove Theorem 1. The construction employs some ideas related to definable generic transfinite constructions, and it will go on by a transfinite inductive definition of a sequence in from countable subsequences. The result can be viewed as a maximal branch in , generic with respect to all sets of a given complexity.
Definition 23 (in ). From now ona number as in Theorem 1 is fixed.
A sequence blocks a set W if either belongs to W (a positive block) or no sequence extends (a negative block).
Any sequence , satisfying (in ) the following four conditions (A)–(D) for this , will be called a key sequence:
- (A)
The set is equal to
- (B)
Every is a crucial ordinal for in the sense of Definition 22.
- (C)
If in fact and is a boldface set (a definition with parameters), then there exists an ordinal such that the subsequence blocksW — so that either , or there is no sequence extending
- (D)
The sequencev belongs to the definability class
Theorem 6 (in ). There exists a key sequence .
Proof. We argue under , with fixed. In case , let be a universal formula. In other words, the collection of all boldface sets is equal to the family of all sets of the form ,.
Claim 1. If then is a set.
Proof (Claim). We skip a routine verification that
is
. Further, if
and
then for
to block
it is necessary and sufficient that
This is a disjunction of and , hence, , and we are finished. □
Coming back to the proof of the theorem, a sequence is defined by induction on . To begin with, we put (the empty sequence).
Step . Assume that is already defined. Put , , and let be the -th element of in the sense of the Gödel well-ordering of .By Lemma 21(i), there is a sequence with , and then a sequence with . If then we trivially extend the last term of the construction by (see Example 1).
Finally if then there is a sequence satisfying and blocking the set , while if then simply put .
Thus, overall we have:
- (∗)
, , , and blocks in case .
Finally we let be the -least one of all sequences satisfying (∗).
Note the role of the blanket assumption in this construction (step ); otherwise, the -least choice of could not be executed.
Limit step. If is a limit ordinal then we obviously define .
We have by construction; hence, .
Let us check (D) of Definition 23. Note first of all that the relation is by Lemma 22. Easily “to block ” is a relation by Claim 1 above. On the other hand, it is known that choosing the -least element in each non-empty section of a set under results in a set (transversal) of the same class . Therefore, the assignment is as well. With these estimations, a routine calculation shows that the relation (∗) still is a relation (in ). This helps to easily accomplish the verification of (D), which we leave to the reader.
To check (A) of Definition 23, note that by construction.
To check (C) of Definition 23 (), note that any boldface set is equal to for some , so is as required.
Finally, (B) holds by by construction. □
Definition 24 (in ). From now on we fix a key sequence , given by Theorem 6 for the number fixed by Definition 23. It satisfies (A)–(D) of Definition 23. We call this fixed the key sequence.
Lemma 25. Assume that . Let be a set dense in . Then, there exists an ordinal satisfying .
Proof. By (C) of Definition 23, blocks W for some ordinal . The negative block is rejected because W is dense. Therefore, . □
20. The Key Forcing Notion
Based on Definition 24, we introduce some derived notions.
Definition 25 (in ). Using the key sequence ,we define the regular multiforcing , and the forcing notion .
We put , , .
We also put .
If , then, following (A) of Definition 23, we let be the smallest ordinal α with . Thus, an arboreal forcing notion is defined whenever the inequality holds. Moreover, is a -increasing sequence of special forcings , thus .
We will call the key multiforcing below, and accordingly the set will be our key forcing notion.The following lemmas present principal properties of in the ground universe , and of according -generic models in the next section.
Lemma 26 (in ). The sequences (of ordinals) and (of arboreal forcings) belong to the definability class .
Proof. The following double equivalence
holds by construction. Yet, “
” is a
formula by (D) of Definition 23. Therefore, the sequence
is
as well. The other sequence is treated similarly. □
Lemma 27 (in ). (i) is pre-dense in whenever and .
(ii) is a regular multiforcing and , thus (finite support).
(iii) is a club (closed unbounded) in , where .
Proof. (i), (ii) Use Lemma 5. To check that recall (A) of Definition 23.
(iii) is clear. □
The next lemma claims that satisfies CCC.
Theorem 7 (in ). The forcing notion satisfies CCC. Therefore -generic extensions of preserve cardinals.
Proof. Let be a maximal antichain. Then, is open dense in . In terms of Definition 22, the set of all limit ordinals , such that
- (∗)
, is a maximal antichain in , is open dense in , and the equality holds,
is a club. Therefore, by Lemma 24(ii), there is an ordinal such that and , and hence .
Note that , or equivalently , by (B) of Definition 23.
However, is open dense in by (*). Therefore, Lemma 21 implies that remains pre-dense in the whole set , and hence, itself by remains a maximal antichain in . We conclude that is countable. □
Corollary 13 (in ). Let a set be pre-dense in . There is an ordinal such that is already pre-dense in .
Proof. We can w. l. o. g. assume that D is even dense. Let be a maximal antichain in D. Then, A is a maximal antichain in since D is dense. Then, for some ordinal by Theorem 7. However, A is pre-dense in . □
21. The Key Model
We aim to prove Theorem 1 using -generic extensions of , which we call key models.We will mostly argue in and in -preserving generic extensions, in particular, in -generic extensions of (cardinal-preserving by Theorem 7). Therefore, we will always have . This allows us to view things so that (rather than ).
Definition 26. Let a set be generic over the constructible set universe . If , then, following the remark in the end of Section 6, - -
We put ;
- -
We define as the unique real which belongs to ;
- -
We finally let .
To conclude, the forcing notion adjoins an array of reals to , where every real is -generic over , and we have .
Theorem 8. Assume that a set is -generic over , , and Then, the following statements are equivalent:
- (1)
- (2)
the real x is -generic over
- (3)
we have .
Proof. The implication (1)(2) is routine (see Remark 1). To check (2)(3) note that, by Lemma 27, all sets are pre-dense in . Finally, prove .
Suppose that
but
, i.e., (1) fails. As
is CCC by Theorem 7, Theorem 5(i) implies the existence of a small
-complete real name
, such that
,
, and
is non-principal at
over
, meaning that
is a set open dense in
. By the smallness,
is a
-complete real name for some ordinal
.
In terms of Definition 22, the set of all limit ordinals , such that
- (∗)
and the set satisfies and is open dense in ,
is a club. Therefore, by Lemma 249ii), there is an ordinal such that and . Then, is non-principal at over . On the other hand, , by (B) of Definition 23. It follows that by Lemma 21(ii). Then, we have as well by Definition 20(6), since and because of the non-principality of .
Now, Theorem 5(ii) with and implies . (We observe that ).
In particular, x does not belong to . Thus, (3) fails, as required. □
Corollary 14. Let a set be -generic over . Then, it holds in that belongs to the definability class , and hence, to class by (1) of Section 20. Proof. By Theorem 8, it is true in
that
if and only if
which can be re-written as
Note that the equalities
and
belong to the class
by Corollary 26. This implies that the whole relation is
, since the quantifier
is bounded. □
22. Well-Orderings in the Key Model
According to the following theorem, the key model satisfies (i) of Theorem 1. The reals are treated as points of
the
Cantor space. The proof see Theorem 2 in [
1].
Theorem 9. Assume that a set is -generic over . Then, in , there is a -good well-ordering of of length , and hence (i) of Theorem 1 holds.
Our final step is to prove the result complementary to Theorem 9, that is, the key model also fulfills (ii) of Theorem 1. This will need some more effort. We will argue under the following assumption.
Assumption 1. We assume that from now on.
This leaves aside the case
in (ii) of Theorem 1. Therefore, this case requires a separate consideration to justify the assumption. Assume that
. We assert that (ii) of Theorem 1
holds in
(which is the key model), where
G is an arbitrary set
-generic over
. Suppose towards the contrary that (ii) of Theorem 1
fails, so that there is a
well-ordering of the reals. (We even do not assume that the well-ordering is good). Then, Theorem 25.39 in [
24] implies that
in
for some real
in
Yet this cannot be the case for the key models
we consider.
Indeed, arguing in , suppose to the contrary that and . Theorem 7 then implies that there exists an ordinal satisfying . However, the real does not belong to by the product forcing theory. We conclude that , which contradicts the choice of x.
Claim (ii) of Theorem 1 involves one more important technical tool related to the above-defined key forcing notion
. It turns out that the
-forcing relation of
formulas is equivalent (up to level
of the projective hierarchy of formulas) to a certain auxiliary forcing relation
defined and studied in
Section 23,
Section 24,
Section 25,
Section 26,
Section 27,
Section 28,
Section 29 and
Section 30 below. Theorem 11 proves the equivalence. This auxiliary forcing is invariant with respect to permutations of indices
(Theorem 12), whereas the forcing
itself is, generally speaking, not invariant in that sense. Such a
hidden invariance plays a crucial role in the construction. Here, we make use of the invariance to prove, using Theorem 13, that the full version of (ii) holds in
-generic extensions of
. Theorems 11–13 are the main results of this Part.
23. Auxiliary Forcing Relation
We argue in . We make use of the second-order arithmetic language. It involves variables (type 0) assumed to run over , and variables (type 1) over The atomic formulas are only those of the form . Consider the extension of this language, which allows us to substitute natural numbers for variables of type 0, and small real names (Definition 12) for variables of type 1.
We define natural classes , of -formulas, as usual.
Given a formula in (resp., ), let bethe result of canonical transformation of to the (resp., ) form.
Now, we introduce a relation between multitrees , small multiforcings , and closed -formulas in , , which will approximate the true -forcing relation. The definition proceeds by induction on the -structure of .
- 1°.
Let be a small regular multiforcing, (not necessarily ), and be a closed formula. We assume that has the following canonical form:
- (f1)
where is a recursive relation and every is a small real name.
Consider a tree-name which consists of allpairs such that there exist tuples where , and multitrees , , (see Definition 12), satisfying:
- (I)
for all ;
- (II)
is equal to the multitree
—see
Section 6 on
, therefore
satisfies
for all
, and hence
directly forces
for all
.
We define ifthe following conditions (a)–(d) and (e1) hold:
- (a)
Every is sealed by (see Example 2);
- (b)
If and then is sealed by (see Example 3);
- (c)
Every name in is sealed -complete (see Definition 12);
- (d)
is sealed in (see Definition 19);
- (e1)
directly forces , i.e., there is a multitree with .
- 2°.
Let be a small regular multiforcing, (not necessarily ), and be a closed formula. We assume that has the following canonical form
- (f2)
where is a recursive relation and every is a small real name.
We define a tree-name as above. Then, define ifconditions (a)–(d), as above, hold and the following condition (e2) holds too instead of (e1) above:
- (e2)
if , , then does not directly force .
- 3°.
If is a formula, , then we define if and only if there exists a small real name such that .
- 4°.
If is a closed formula, , then define if and only if there is no special multiforcing and such that , , and .
Remark 3. If holds then it is not necessary that , and it is not necessary that every name in φ satisfies
Definition 27. Given a class K of the form , (), we let contain all triples satisfying .
Then, is a subset of . Recall that if then .
Lemma 28 (in ). and belong to .
Given any , belongs to and belongs to .
Proof. Relations such as “being a small regular multiforcing”, “being a formula in , ”, , etc., are definable in by bounded formulas, hence . Such also are the operations , and , (provided is small, as in 1°, 2°). This wraps up the estimation for the cases of and .
The inductive step by 3° is quite simple.
Now, for the step by 4°, assume that , and is already established. Then, if is a small regular multiforcing, , is a closed formula, and, by 4°, there is no triple such that is a special multiforcing, , , , and is . This clearly implies the estimation of as required. □
24. Elementary Properties of the Auxiliary Forcing
We still argue in .
Lemma 29. Assume that are small regular multiforcings, , , φ is an -formula. Then implies .
Proof. If
is a formula in
as in 1° of
Section 23, and
, which is witnessed by (a)–(d) and (e1), then
also holds.
Indeed, condition (a) transfers from to by Corollary 8(i), (ii).
Condition (b) transfers to by Corollary 9(i)(ii) (with same ) and (iii) (to ).
Condition (c) transfers to by Corollary 10 (ii).
Condition (d) transfers to by Theorem 3(iii)(v).
Finally, (e1) transfers to because by Theorem 3 (iv).
The case is rather similar, yet the transfer of (e2) from to deserves attention. Note that all premises of Theorem 3(vi) hold for , and . That is, holds by Theorem 3(iii) and (d), whereas holds for all , -incompatible with , by Corollary 9(i) and (b).
Now, condition (e2) for and means that (1) of Theorem 3(iv) fails for , and . Therefore, (3) fails as well, that is, no , , directly forces . However, , and therefore, we have (e2) for and (instead of ), as required.
The induction step , as in 3°, is pretty elementary.
Now, for the induction step , as in 4°, assume that and is a closed formula in satisfying . Suppose that fails. Then, by 4° there exist: a special multiforcing and a multitree such that , , and . However, then we have and . We conclude that fails by 4°. □
Lemma 30 (in ). Let π be a small regular multiforcing, φ a formula in , , , and if then . Then implies .
Recall that is the canonical transformation of to the prenex form.
Proof. If then the result follows from definition 4°. Therefore, let , so that, by the contrary assumption and Lemma 29, there exists such that both and . However, then (e1) immediately contradicts (e2) with . □
Lemma 31 (in ). Let π be a small regular multiforcing, , φ a formula in , , and each name in φ is -complete. Then, there is a special multiforcing and a multitree such that , , and either or .
Proof. Suppose that , so that has the canonical form (f1) with a recursive R. Then, is a tree-name. As each name in is -complete, Theorem 4 gives a special multiforcing satisfying for each multitree in , for each name in , and .
Case 1: some directly forces . Then, by 1°.
Case 2: not case 1. Then, we have
by 2° of
Section 23.
If
, then the result follows from definition 4° of
Section 23. □
25. Forcing the First Level Formulas
The following theorem shows that the auxiliary forcing relation is properly connected with the ordinary forcing at least for formulas in .
Theorem 10 (in ). Let and . Assume that and , φ is a formula in , and . Then, -forces the sentence over the universe in the ordinary sense.
Proof. Case 1: is a formula in
, of the canonical form (f1), that is,
where
is a recursive relation and every
is a small real name, and
, so that properties (a)–(d) and (e1) of
Section 23 hold for
,
, and
. In particular, (∗)
directly forces
by (e1).
Now, consider any set , generic over the given universe and containing ; the goal is to prove that holds in . The following lemma simplifies the task. □
Lemma 32. The set is -generic over τ in the sense of Definition 18, and is -generic over each name occurring in φ, in the sense of Definition 13.
In addition, for any such name , as well as .
Proof (Lemma).
First of all, we have to check (2) of Definition 13 for
. Thus, let
belong to
. However,
are sealed by
by (a) of
Section 23, thus the sets
and
are sealed dense in
. Then,
is sealed dense in
as well by Lemma 11(i). Therefore, if
then
is a sealed dense, and therefore open dense, set in
by Lemma 10(iii). We conclude, by the genericity of
G, that there is a multitree
that belongs to
for some
. Then, there is a multitree
satisfying
. We have
since
. On the other hand,
as well; therefore,
cannot be incompatible with
. It follows that
and
.
This ends the proof of (2) of Definition 13 for .
Now, check the special condition of Definition 18. Let
,
, and let the set
be dense in
; prove that
. Note that
is sealed in
by (d) of
Section 23. It follows that
is sealed dense in
(see Definition 19(b)). Therefore, by Lemma 10(iii), if
, then the set
is dense in
. It follows, by the genericity of
G, that there is a multitree
, for some
. Then, we have a multitree
satisfying
. Then,
since
, and we are finished.
The genericity over the occurring names is verified similarly, starting from (c).
Finally, prove the additional part of the lemma. To check
, assume that
, meaning that there is
with
, and thus,
. If follows by by (a) of
Section 23 that
is sealed by
, in other words, the set
is sealed dense in
. Then, arguing as above we prove that
for some
, and hence there exists
. However,
is impossible since
either, so we have
. Thus,
witnesses that
.
The proof that for any name in is similar. □
We return to the proof of Theorem 10.
We know that directly forces . The set is -generic over by Lemma 32. Therefore, by Corollary 11, is ill-founded, i.e., has an infinite chain, and hence, is ill-founded because still by Lemma 32.
Our goal is to prove that the sentence
(see Formula (
2) above) holds in
, where
,
. We put
Thus, is a tree, and is true if is ill-founded. However, is ill-founded, see just above. Thus, it remains to be shown that .
Let
,
. We have
for some
. By definition, there are tuples
satisfying (I) and (II) of
Section 23; in particular,
directly forces
by (II) for all
, and hence,
for all
i. Thus,
by (I).
Conversely let , that is, holds for all , where for all i. As , there is a family of conditions , , . Then, the multitree belongs to G as well as G is generic, and by definition we have . It follows that .
Thus, , as required.
Case 2: is a formula in
. We write
instead of
. Thus, let
be a
formula of the form (f2), i.e., essentially the negation of
above, and let
, so that properties (a)–(d) and (e2) of
Section 23 hold for
,
, and
. In particular, if
,
, then
does not directly force
, by (e2). Given any
, generic over
and containing
, the goal is to prove that
holds in
.
As and is -generic over by Lemma 32, Corollary 11(ii) implies that is well-founded. Then, the tree , defined as above, is well-founded either, since it is equal to . Therefore, holds, as required.
26. Forcing inside the Key Sequence
It is implied by Theorem 11 below that the forcing relation , considered with the terms of the key sequence , really approximates the true -forcing relation at level and below. Recall that is assumed (see Assumption 1).
We argue in . Recall that the key sequence , satisfying (A), (B), (C), (D) of Definition 23 was defined by Theorem 6, is the key multiforcing, and is our forcing notion.
Definition 28. We write instead of , for the sake of brevity. Let mean: for some .
Lemma 33 (in ). Assume that , , and . Then, the following holds:
- (i)
If , , and , then
- (ii)
If , , then there exists , such that ;
- (iii)
If and then ;
- (iv)
It follows that, first, if , , and then , and second, and are incompatible.
Proof. Lemma 29 implies (i). To prove (ii), choose an ordinal with , satisfying , and apply (i). To check (iii), we observe that are incompatible in , as otherwise (i) leads to contradiction. On the other hand, multitrees incompatible in are by Corollary 2. □
Theorem 11. Assume that and ψ is a closed formula in , with all names small and -complete, and . Then -forces over in the usual sense, if and only if .
Proof. Let be the ordinary -forcing relation over .
Part 1:
is a
formula, of the canonical form (f2) in
Section 23, with a recursive
R. Define the tree-name
by (I), (II) in
Section 23.
If then for some , and then by Theorem 10.
To prove the converse suppose that . By the choice of , there is an ordinal such that all names in are -complete, where . Thanks to Lemma 24(ii), the set C of all ordinals such that
- (∗)
contains all names in , all multitrees in , and as well,
is stationary. Therefore, there is an ordinal , .
Property (B) in Definition 23 implies . In particular, it follows by (∗), that for all , for all , -incompatible with , for all names in , and .
We conclude that properties (a)–(d) of
Section 23 hold for
in the role of
(and for the given
). Indeed, to check (a), recall that
holds for all
by the above; hence, every
is sealed by
by Corollary 8(ii). To check (b), (c), (d) for
argue similarly but refer to resp. Corollary 9(ii), Corollary 10(i), Theorem 3(v).
We claim that (e2) of
Section 23 also holds, i.e., if
,
, then
does not directly force
. Indeed, suppose to the contrary that
,
is a counterexample, so that
directly forces
. Then,
, where
is
, by the definition in 1° of
Section 23. We conclude that
by Theorem 10, that is,
. However, this contradicts the assumption that
.
Part 2: the step (). Let be a formula. Suppose that . Then, by definition, we have for a small real name . The inductive hypothesis implies ; hence, . To prove the converse, let . As is CCC, there exists a small real name (in ) satisfying . Then, by the inductive hypothesis; therefore, .
Part 3: the step (). Let be a closed formula, and . Lemma 33(iv) implies that no multitree , , satisfies . We conclude that by the inductive hypothesis.
Conversely, let . There is an ordinal such that every name in is -complete, where . Consider the set U of all sequences , of successor length , such that and there is a multitree , , such that . Then, U is a set (defined with , as parameters) by Lemma 28; hence, U belongs to . (Recall that by Assumption 1. Therefore, using Theorem 6 (and (C) of Definition 23) there is an ordinal such that the restricted sequence blocks U.
Case 1: , so that , , and there is a multitree satisfying and . Then, holds by the inductive hypothesis, which contradicts the choice of . We conclude that Case 1 is impossible.
Case 2: no sequence in U extends . We can assume that and is a successor, . (If not, replace by ). We asert that . Indeed, otherwise, 4° implies that there exists a special multiforcing and a multitree , satisfying , , and . However, then the extended sequence belongs to U. However, , which contradicts the Case 2 assumption. Thus, , as required. □
27. Permutation Invariance
The theory of forcing admits various invariance theorems. Theorem 12 is related to the invariance of the auxiliary forcing under permutations.
We argue in . Let PERM be the set of all bijections , satisfying and such that the non-identity domain is at most countable. Bijections in PERM will be called permutations.
We extend the action of any as follows:
If is a multitree then is a multitree, , and for each ;
If is a multiforcing then accordingly is a multiforcing, and for each ;
If is a real name, then we define , so that is a real name as well;
If , then , this is still a sequence in ;
If is a -formula (all names indicated), then is
Many notions and relations defined above are rather obviously
-invariant. Thus,
if
,
if
,
et cetera. As the next lemma shows, the invariance also holds with respect to the relation
. An obvious proof by induction on
n is left to the reader. (See Theorem 24.1 in [
18]).
Theorem 12. Assume that , π is a small regular multiforcing, , , and a closed formula φ belongs to . Then, if .
28. Embedding Multiforcings in the Key Sequence
The following lemma proves that any special multiforcing admits a suitable embedding into the key sequence , due to the generic properties of the latter.
In , if , then we define shift permutations so that
and for all ,
and for ,
where as usual, and . In other words, , are order-preserving shifts between and resp. , .
Lemma 34 (in ). Let be a special multiforcing, . There exist a special multiforcing , and ordinals , such that , , , and the multiforcings , satisfy , i.e., and .
Under the conditions of the lemma, it follows by the definition of
and
above that
and
. This lemma will have two applications below. One of them (the proof of Lemma 35) will really need the effect of both
and
. The other one (
Section 30) will involve only
.
Proof. We argue in . First of all, fix any special multiforcing satisfying and . Let U be the set of all sequences such that:
- (†)
there are ordinals , , and the shifted multiforcings , satisfy .
Easily, U is a set (with as the only parameter of the definition in ); therefore, a set, because by Assumption 1. It follows by Definition 23(C) that there exists an ordinal such that blocks U. We can w. l. o. g. assume that is a successor; otherwise, just substitute for . We put .
Case 1: no sequence in U extends . To demonstrate that this is inconsistent, let be the least ordinal satisfying . Let be any special multiforcing satisfying and still .
We define , . Thus, are special multiforcings with disjoint domains , , . ’ It follows that the simple union is still a multiforcing, and by the way since . It follows that the extended sequence belongs to and , , and , so that .
We conclude that , but this contradicts the Case 1 assumption.
Case 2: . Let this be witnessed by an ordinal and as in (†). Then, , satisfy , and hence, this accomplishes the proof of the lemma. □
29. The Non-Well-Orderability Claim, Part I
Here, we begin
the proof of Theorem 1 in part (ii). It will be completed in the end of
Section 30. We are going to establish the following even somewhat stronger result.
Theorem 13. Assume that a set is -generic over . Then, in , there is no well-orderings of the reals, and moreover, no relation well-orders the set .
Our plan is to infer a contradiction from the next assumption contrary to Theorem 13.
Assumption 2. Assume that is a parameter-free formula, , , and p -forces, over , that “the relation , defined by if , strictly well-orders the set in ”.
We begin with the next lemma. See Example 4 regarding real names of the form .
Lemma 35 (in ). Under Assumption 2, suppose that is a small regular multiforcing, , and . Then, there exists a special multiforcing π, a multitree , and an ordinal , such that , , and .
Proof. We argue in . We recall that . By Lemma 34, there exist a special multiforcing , and ordinals , such that , , , and the shifted multiforcings , satisfy .
Let
,
. Pick a multitree
with
. Then
hence,
. We observe that
and
.
Note that the sets , , are subsets of the pairwise disjoint intervals resp. , , . Then, the simple union is still a multitree in stronger than each of .
Then, by the choice of , there exists a condition , , which either -forces or -forces . Let -force say over . We can assume that for some , satisfying and . Using Theorem 11, we have
- (1)
, for some .
Here, can be chosen large enough for by Lemma 29. Then, there exists a multitree satisfying and . Lemma 29 implies:
- (2)
,
and then acting by on (2), we obtain by Theorem 12:
- (3)
—by Theorem 12,
where and because .
It remains to be observed that ; hence, , and further because by construction. Therefore, since . Acting by , we obtain . We similarly obtain because by construction. This ends the proof of the lemma, with . □
The goal of the next lemma is to strengthen Lemma 35 to the effect that a whole dense set of conditions with the same property will be obtained.
Lemma 36 (in ). Under Assumption 2, there is a sequence satisfying the following for all
- (i)
If , where , then there is a condition , , and an ordinal , such that
- (ii)
The set is dense (then in fact open dense by Lemma 29) in .
Proof. (i) Using Lemma 34, we define
by induction so that for each
k there is a certain pair of
and
, satisfying:
Moreover, the enumeration by and can be arranged so that for each ordinal and condition there exists k such that and . However, is as required. Claim (ii) is just a reformulation of (i). □
Corollary 15 (in ). Under Assumption 2, let satisfy (ii) of Lemma 36: if then is open dense in , where . There is a special multiforcing , such that for each k, and each set , , is sealed dense in .
Proof. Consider the (countable) collection of all sets , . By Lemma 21(i), there exists an extension of , by the rightmost term , satisfying for all . Then, each set is sealed dense in by Lemma 10(iii). □
30. The Non-Well-Orderability Claim, Part II
Still arguing under the conditions of Assumption 2, we proceed with the following construction.
- (I)
Pick
by Lemma 36, so that
is an open dense set in
for any
, where
.
- (II)
Then, pick a special multiforcing by Corollary 15, so that for each k, and if then , and hence, the set is sealed dense in .
- (III)
By Lemma 34, there exist a special multiforcing
, and ordinals
, such that
,
, and the multiforcing
satisfies
. Accordingly, we let
,
,
,
, so that
for all
k, and
. If
then put
Lemma 37. If and is a special multiforcing, , then the set is sealed dense and open dense in .
Proof. Let us make use of the action of on (II) above. We assert that
- (∗)
.
Indeed, suppose that and show that , where accordingly . By definition there is an ordinal , such that . Then, by Theorem 12, where . This completes the proof of (∗) from right to left. The inverse implication is similar.
Now, it follows from (∗) and that each set , where is sealed dense in . However, we have . It follows that the set is sealed dense and open dense in simply because , whereas is sealed dense and open dense in by Lemma 10(ii),(iii). □
We now obtain a related the pre-density result in the context of the the key sequence . Recall that means that holds for some . (See Definition 28).
Lemma 38. If , then the following set is open dense in : Proof. The openness holds by Lemma 29. To show the density, pick any . The goal is to find an ordinal , and a multitree such that and . As is -increasing, there is an ordinal and a stronger multitree , . It follows from (III) that ; hence, , which implies for each ; therefore, for the ordinal chosen just above. However, the set is open dense in by Lemma 37 (with ). It follows that there is a multitree , .
Then, , where , and hence, for some and k. We conclude that then by Lemma 29; thus, , as required. □
Let be the -forcing relation over . It essentially coincides with by Proposition 11. Therefore, the lemma implies the following corollary.
Corollary 16. If then the following set is open dense in : Proof of Theorem 13. It follows from Corollary 16 that if G if -generic over , then the set contains no -minimal element, which contradicts Assumption 2. The contradiction negates Assumption 2 and thereby proves Theorem 13. □
Combining Theorems 13 and 9, we complete the proof of Theorem 1.
This final Part contains
Section 31 with a short proof of Theorem 2 and a brief discussion of its possible reduction to a theory weaker than
‘
exists’. We finish in
Section 32 with conclusions and problems.
31. Proof of Theorem 2 and Comments
First of all, we recall that is a subtheory of obtained as follows:
- (a)
We exclude the Power Set axiom PS;
- (b)
The well-orderability axiom WA, which claims that every set can be well-ordered, is substituted for the usual set-theoretic Axiom of Choice AC of ;
- (c)
The Separation schema is preserved, but the Replacement schema (which happens to be not sufficiently strong in the absence of PS) is substituted with the Collection schema: .
A comprehensive account of main features of
is given in e.g., [
27,
28,
29].
Proof of Theorem 2. Arguing in , let us drop to the subuniverse of all constructible sets in the universe of discourse. Then, satisfies too, and if exists then exists in . Thus, instead of + ‘ exists’, we argue in the theory + () + ‘ exists’, whose universe is .
Now, the existence of the power set leads to the existence of sets such as and , and basically, the existence of all sets involved in the construction of the key forcing notion (including itself). After this remark, all arguments in the proof of Theorem 1 in Parts I, II, III, and IV above naturally go through, giving the proof of Theorem 2 by means of a -generic extension of . □
It is really interesting to further reduce the assumptions of Theorem 2 down to
(see [
20,
30,
31] and elsewhere on second-order Peano arithmetic
) or
without the extra assumption of the existence of
, or the associated class theory
, which is formalized in a two-sorted language with separate variables and quantifiers for sets and classes, so that lower-case letters are used for set variables, whereas upper-case letters are used for class variables. The minus
still reflects the absence of the Power Set axiom.The axiomatization of
(see e.g., [
29]) includes axioms for sets (exactly those of
) and those for classes. In particular, (1) extensionality for classes; (2) the class replacement axiom asserting that every class function restricted to a set is a set; and (3) a predicative comprehension schema asserting that every collection of sets, definable by a formula with no quantified class variables, is a class.
Theories
,
, and
have been known to be equiconsistent for a while, see e.g., [
20,
30,
31] for
vs.
, and [
32,
33,
34] for
vs.
.
Such objects as
and HC are legitimate classes in
, and such are all
-sets that play any role in the proof of Theorem 1 above, with one notable exception. The exceptional case being the
-sequence used in Lemma 24. The
construction of such a sequence (as e.g., in [
24]) can be maintained as a proper class in
as well as in
. However, unfortunately, the proof of the
-property of the resulting sequence does not go through in
because the
proof involves ordinals beyond
, and hence, does not directly translate to the
level. This will be the subject of our forthcoming paper aimed at solving this technical obstacle by means of recently discovered methods as, e.g., in [
35,
36].
32. Conclusions and Problems
In this study, the method of finite-support products of Jensen’s forcing was applied to the problem of obtaining a model of
in which, for a given
, there is a
-good well-ordering of the reals, but no well-orderings of the reals exist in the class
at the preceding level of the hierarchy. This is achieved by Theorem 1, our
first main result. We also demonstrate that this theorem can be obtained on the basis of the consistency of
(i.e.,
sans the Power Set axiom) plus the claim that
exists, which is a much weaker assumption than the consistency of
usually assumed in such independence results obtained by forcing method. This is achieved by Theorem 2, our
second main result. Two
principal technical achievements, related to getting rid of countable models of
as a technical tool and according treatment of the auxiliary forcing, were mentioned in
Section 2. These are new results in such a generality (with
arbitrary), and valuable improvements upon our earlier results in [
1]. They may lead to further progress in studies of the projective hierarchy.
From our study, it is concluded that the technique of
definable generic inductive construction of forcing notions in
that carry
hidden automorphisms, developed for Jensen-type product forcing in our earlier papers [
17,
18,
21], succeeds to solve other important descriptive set theoretic problems of the same kind, using Theorems 1 and 2.
These results (Theorems 1 and 2) continue the series of recent research such as a model [
37] in which there is
real singleton
that codes a cofinal map
, while every
set
is constructible, and hence, cannot code a cofinal map
, and another model [
38], in which there is a non-ROD-uniformizable
set with countable cross-sections, while all
sets with countable cross-sections are
-uniformizable—in addition to the research already mentioned in
Section 2 above.
This study may also be a contribution to the search for solutions of several similar and still open problems related to the projective hierarchy, such as separation of the countable
at different levels of the projective hierarchy, a similar problem for the principle
of dependent choices, and a critically significant problem posed by S. D. Friedman in ([
39], p. 209) and ([
40], p. 602): assuming the consistency of an inaccessible cardinal, find a model for a given
n in which all
sets of reals are Lebesgue measurable and have the Baire and perfect set properties, while there is a
well-ordering of the reals.
From the result of Theorem 1, the following more concrete problems arise.
Problem 1. Prove that it is true in the key model of Section 20 that there is no boldface well-ordering of the reals. The boldface specification means that the real parameters are allowed in the definitions of pointsets, whereas they are not allowed in the lightface case. This is a principal difference.
Problem 2. Prove a version of Theorem 1 with the additional requirement that the negation of the continuum hypothesis holds in the generic extension considered.
The model for Theorem 1 introduced in
Section 20 definitely satisfies the continuum hypothesis
. The problem of obtaining models of
ZFC in which
and there is a projective well-ordering of the real line, has been known since the beginning of modern set theory. See, e.g., problem 3214 in an early survey [
41] by Mathias. Harrington [
42] solved this problem using a generic model in which
and there is a
well-ordering of the continuum, using a combination of methods based on such coding forcing notions as the almost-disjoint forcing [
43] and a forcing by Jensen and Johnsbråten [
44]. Such a different forcing notion as the product/iterated Sacks forcing [
45,
46] may also be of interest here.