Provident sets and rudimentary set forcing

. Using the theory of rudimentary recursion and provident sets expounded in [MB], we give a treatment of set forcing appropriate for working over models of a theory P ROVI which may plausibly claim to be the weakest set theory supporting a smooth theory of set forcing, and of which the minimal model is Jensen’s J ω . Much of the development is rudimentary or at worst given by rudimentary recursions with parameter the notion of forcing under consideration. Our development eschews the power set axiom. We show that the forcing relation for ˙ ∆ 0 wﬀs is propagated through our hierarchies by a rudimentary function

HF = V ω = J 1 = L ω ; if an axiom of infinity be added to each theory, giving the theories KPI, PROVI and GJI, the minimal transitive models are then respectively the Jensen fragments J ω CK 1 , J ω and J 2 .The provident sets are HF and the transitive models of PROVI.We show that every provident set A supports the definition of the forcing relation − P when P ∈ A; our main result is that a set-generic extension of a provident set is provident.
For most of this paper we avoid use of the power set axiom; the paper [M4] discusses the problems and possibilities of set forcing over models of Mac Lane or of Zermelo set theory, two theories which include the power set axiom.
We draw on the notation ( 1 ) and results of [MB], and in particular we make heavy use of the rudimentary function T which was introduced in [M3]: its properties are that if u is transitive, then T(u) is transitive, with u both a member and a subset of it; every member of T(u) is a subset of u; further, the union over all n of T n (u) is the rudimentary closure of u ∪ {u}.
Provident sets.Let p be a set.Call a function x → F (x) p-rud-rec (short for p-rudimentarily recursive) if there is a rud function H such that for every set x, F (x) = H(p, F x).
Examples: the rank function, , and transitive closure, tcl, are ∅-rud-rec, meaning no parameters required; the evaluation val G (•) of the names of a forcing language using a generic G is G-rud-rec.
Rud recursion without parameters is treated in [MB,Section 4]; parameters are discussed in Section 5.
The axioms of PROV are such that its transitive models are those transitive sets A such that for each p, each p-rud-rec F and each x ∈ A, F (x) is in A.
Let c be a transitive set; using T we define in §2 a hierarchy giving an initial segment of L(c) by a recursion on the ordinals.The novelty of the definition is that the whole of c is not included at the start, but its members are fed in according to their rank: if we put c ν = {x ∈ c | (x) < ν}, then the following c-rud recursion on the ordinals holds: The canonical progress towards c is the hierarchy P c ν defined by setting P c 0 = ∅, P c ν+1 = T(P c ν ) ∪ {c ν } ∪ c ν+1 , P c λ = {P c ν | ν < λ} at limit λ.
( 1 ) Examples: (a, b)2, sometimes more informally written (a, b), is the Kuratowski ordered pair defined as {{a}, {a, b}}.(a, b, c)3 is the Kuratowski ordered triple, defined as (a, (b, c)2)2.a, b, c denotes this function with domain {0, 1, 2}: {(a, 0)2, (b, 1)2, (c, 2)2}.0.0.Remark.As c ν = c ∩ P c ν , we might have defined P c ν by a single c-rudimentary recursion on ordinals: The axiomatisation of PROV may then be summarised as extensionality + the empty set exists + all rudimentary functions are defined everywhere + every set has a rank + every set has a transitive closure + for every transitive c and ordinal ν the set P c ν exists Gentle functions.Let us review some material from [MB].
Definition.A gentle function is one of the form G • F where G is rudimentary and F is rud-rec.
A p-gentle function is one of the form G • F where G is rudimentary and F is p-rud-rec.
If A is a set, then a function is (A)-gentle if it is p-gentle for some p ∈ A.
To emphasise the absence of a parameter we may write ∅-gentle or, more accurately, (∅)-gentle.In that paper, Proposition 4.5 gives Bowler's example of two rud-rec functions whose composite is not rud-rec, and Theorem 4.9 his result that the composite of two gentle functions is gentle.
A point to note is that the composition of two p-gentle functions is liable not to be p-gentle but will be q-gentle for some parameter q in the provident closure of {p}.So for provident A the composition of (A)-gentle functions is (A)-gentle.Proposition 4.12 of [MB] proves that the characteristic function of a predicate B is gentle iff the separator x → x ∩ B is gentle.That yields the following principle: 0.1.Gentle Separation.Let A be provident, a and p in A, and B a p-gentle predicate; then a ∩ B ∈ A.
Proof.There will be some attempt f ∈ A at χ B , that is, a fragment of χ B , with a included in its domain, and then x ∩ {y | f (y) = 1} will be a set by ∆ 0 separation.0.1 Set forcing over provident sets.Let A be a transitive model of PROVI; let P be a separative partial ordering which is a member of A, and let P be the provident closure of {P}.Many functions and relations involved in the development of forcing are if not actually P-rud-rec, (P)-gentle.
The first goal of the paper is to prove that for each the forcing relation p − P ϕ, restricted to those sentences of the language of forcing that are ∆0 and of length at most , is (P)-gentle.
The second goal is to analyse the construction of names for the values of functions applied to objects in a generic extension.We speak of this task as the construction of nominators for the functions concerned.
The first stage of that is to show that for each rudimentary function R, say of two variables, there is a (P)-gentle function R P of two variables such that for (A, P)-generic G and all x, y in A, val G (R P (x, y)) = R(val G (x), val G (y)).
In many cases, we can do better, and indeed certain rudimentary functions have basic nominators.
We shall then find P-rud-rec functions P and tcl P that similarly build names for the rank and the transitive closure of a given object from its forcing name.
Finally we must build names for the stages of a progress ν → P d ν for d a transitive set in the generic extension.Here we shall repeatedly use the principle that functions rudimentary recursive in the (P)-gentle ternary relation p − P a = b are (P)-gentle, which follows from the results of Section 5 of [MB].
The main theorem will then follow easily for provident sets of the form P e θ , and will immediately extend to all provident sets containing P, using the fact-a special case of [MB,Theorem 7.0]-that every provident set is the union of a directed family of sets each of the form P c θ .In the rest of this paper gentle will normally be used to mean (P)-gentle: where it is necessary to emphasize the absence of parameters we shall write ∅-gentle.
1. Heuristic.We begin with some reminders of the general character of forcing: the present discussion is heuristic, to give the reader a feel for the way the forcing relation will operate.In particular, the methods used in this section for naming old and new objects are, dangerously, simpler than the methods of the formal development to be given in subsequent sections.
Suppose we face the following challenge: given a transitive M , to find a transitive N ⊇ M with On ∩ N = On ∩ M but where N contains a subset of ω not in M .
If M and N are both provident, such an N will necessarily violate the axiom of constructibility, for (L) N = (L) M .Thus we are aiming to add a set a ⊆ ω to M .1.0.We begin by asking questions about a. Suppose we have some information p about a: what statements about a will be forced to be true?For example if p is the statement that 5 ∈ a, then Not all members of a are even is forced by p.
Our beginning intuition for forcing is the idea that we have pieces of partial information about the new object we are adding, and that we build up a picture of the new model from this partial information.
Our pieces of information are called conditions, and to start with we suppose that the collection of conditions is a set, P .Experience shows that we should make the following assumptions about P : (1.0.0)P is partially ordered by a relation ≤; if p ≤ q we think that p contains more information than q.(1.0.1)To get something interesting we allow the possibility of two conditions being incompatible: we say that p is compatible with q if there is some r stronger than both: r ≤ p & r ≤ q; and we say that p is incompatible with q, in symbols p ⊥ q, if no such r exists.(1.0.2) We assume that any condition can be strengthened in two incompatible ways: ∀p ∃q ≤ p ∃r ≤ p q ⊥ r.
(1.0.3)We suppose that P has a greatest element 1 P , where this condition is the one that gives us no information at all.Thus 1 P is compatible with every condition.(1.0.4)Finally we suppose that P = df (P , 1 P , ≤) 3 is separative, that is, ∀p ∀q (p ≤ q ⇒ ∃r ≤ p r ⊥ q).Such a P is called a notion of forcing.Let us look at two examples.
Cohen's original forcing.We take the set of finite maps from ω to 2 = {0, 1}.
We define the ordering by reverse inclusion: This forcing is intended to add a "generic" a : ω → 2. We take a symbol ȧ and use it as a name for the new a that we are trying to add.
The intended meaning of a condition p : n → ω is that ȧ n = p.So if n = 6 and p(3) = 1, then p will force the statement that ȧ( 3) = 1.
We suppose that we are in one universe, which we call the ground model, describing a larger universe, which we call the generic extension; the new objects are only partially known to us, so we use dotted letters as names for them, as ȧ.The objects in the ground model are fully known to us, and we name them with hatted letters: thus 3 is our name for 3 in the language of forcing.
Continuing our discussion of Cohen's original forcing, we show now that the new real ȧ is not the same as any old real b: precisely, we prove the following: Notice the topological flavour to this proposition: it is saying that the set of conditions forcing a certain statement is dense.Indeed we may topologize P so that is exactly what is happening.
Proof of 1.1.Given p, let n = dom(p); n is of course the least natural number not in the domain of p. Look at b(n), and let So in any model in which everything that is forced on a dense set is true, ȧ will be a new subset of ω.
This notion of density is central to the concept of forcing.One of the properties of the forcing relation, which we shall refer to as the density property, is that p − P ϕ ⇔ ∀q ≤ p ∃r ≤ q r − P φ.
As we progressively extend the definition of the forcing relation to ever wider classes of formulae, we shall check at each stage that the density property and other characteristic properties of forcing are preserved.
Another example.Let η be an infinite ordinal.This time take As before, order by reverse inclusion: This forcing adds a generic ḟ : ω 1-1 − − → η: a condition p with domain n is a description of ḟ n. So Proof.By a density argument.Given p, n not in dom(p) and an ordinal ξ < η, suppose that ξ is not in the image of p.We find q ≤ P p such that n ∈ dom(q) and q(n) = ξ.
Thus we have shown that ∀ξ ≤ η ∀p ∃q ≤ p q − P ξ the image of ḟ .
2. Forcing in provident sets.Let M be a provident set, and P = (P , 1 P , ≤) 3 a separative partial order in M , with a top point 1 P .We suppose that ω ∈ M .
We aim to define within M a relation −, more exactly − P , describing an extension M [G] of M , where G is an (M, P)-generic filter.Each object in M potentially names an element of M [G].− is a relation between elements of P and sentences in a language of set theory that we shall gradually build up.In fact the full relation can only be defined schematically within M .This language will start from two two-place relations = and and will broadly resemble the formal languages introduced in [M2].We shall use our devices of dots and type-writer face as before; but the constants will play a different role, and hence we shall use a different mark.To each set x in the universe corresponds a name x for an object in the generic extension.Thus the statement p − P x y expresses information about the evaluation of the objects x and y functioning as names of sets to be created in the forcing extension given by the notion of forcing P.
− 0 is our first approximation to the relation −.
2.2.Definition.In future we shall write 2 x for x.

Definition
2.6.Lemma.For all p ∈ P, a and b: This last statement shows that − 1 improves − 0 and starts to resemble a forcing relation.
We define the relation p − b = c by recursion: 2.8.Definition.
The above definition is P-rud recursive in a suitable sense, which we must now articulate, and therefore will succeed in provident sets of which P is a member, or, more generally, in P-provident sets.
2.9.Definition.Let χ = (p, b, c) be the characteristic function of the relation p − P b = c, so that it takes the value 1 if p − P b = c and 0 otherwise.Our plan is to show that the graph of χ = on transitive sets is definable by a P-rudimentary recursion.
Proof.We must first say that everything in the domain of f is an ordered triple, of which the first component is a member of P ; and whenever (p, b, c) ∈ Dom(f ) and β and γ are in b and c respectively, and q ∈ P then (q, c, b) and (q, β, γ) are in the domain too.But all that is ∆ 0 (P, f ).
Then we must say that f respects the recursive definition: but all that is also a ∆ 0 statement about P and f .2.10 2.11.The Propagation Lemma.Let F (u) = χ = (P × u × u).There is a rudimentary function H = such that for any transitive P , if P ⊆ P + ⊆ P(P ), In the following argument, and elsewhere, (•) 3 i are basic "un-tripling" functions such that for a poset P = (P , 1 P , ≤) 3 , (P) 3 0 = P , (P) 3 1 = 1 P , and (P) 3 2 = ≤.On this occasion, but not in future, the restricted nature of a quantifier such as ∀r ≤ p has been made manifest by re-writing it as ∀r ∈ (≤"{p}) .
Proof.Much as in [MB].Iterating H = from χ = ∅ gives χ = P e ω ⊆ P e η 0 +ω ; and then continue the induction by showing for each limit λ that [i] if χ = P e λ ⊆ P e η 0 +λ then χ = P e λ is a member of P e η 0 +λ+ω ; [ii] if χ = P e λ is a member of P e η 0 +λ+ω , then so is χ = P e λ+k for each k ∈ ω.The proof of [i] uses the Definability Lemma 2.10 and the presence of the parameter P; and the proof of [ii] uses the Propagation Lemma 2.11.The integers s = and g = are the separational and generational delays calculated by [MB,Theorem 6.12 and Proposition 6.32].2.13 Propagation of χ = .The progress (P c ν ) ν was defined in §6 of [MB] for c a transitive set.
We could continue to work with progresses of the above kind, but a problem would then arise at the end of the paper, in the proof that a setgeneric extension of a provident set is provident.It is better to change tack now and work with other progresses, which might be called construction from e as a set and χ = as a predicate, with the definition of χ = evolving during the construction.An easy extension of [MB,Proposition 4.3] will assure us that the three functions we are about to define by simultaneous recursion are P-rud-rec.
(2.16.1):The '6' reflects the delay in the creation of Kuratowski ordered pairs.
(2.16.2):At limit stages, that delay no longer exists.

Proposition.
There is a natural number s such that for each ordinal ν η, χ P e;= ν ∈ P e;= ν+s , where e and η are as in Definition 2.14.
Proof.There are rudimentary functions R and S such that . We may take s = 12 + c S .2.20 Familiar properties of forcing.We check as our definition of forcing develops that it has the expected density properties, and we establish familiar properties of equality and the substitution properties of = for : (1) From the symmetry of the definition.
(2) If q − 1 α a then ∃r ≤ q ∃β (r − 1 β b & r − α = β), so ∃s ≤ r ∃γ (s − 1 γ c & s − β = γ); the t we seek is s; s − α = β ∧ β = γ; so, assuming we have minimised the rank of a possible failure b, s − α = γ and s − 1 γ c, as required.Forcing the negation of a statement.We have defined p − Φ for Φ of the form x y or x = y, and now wish to define p − Φ in these cases.The definition is characteristic of forcing; and we will maintain it as we extend our definition of forcing to ever larger classes of formulae.2.28.Definition.p − Φ ⇔ ∀q ≤ p q − Φ.
We shall use this definition in our next proposition: notice that it has the immediate consequence that ∀p ∃q ≤ p q − Φ v q − Φ .Proof.if p − Φ ∨ Ψ then ∀q ≤ p ∃r ≤ q (r − Φ v r − Ψ); but the first alternative is impossible for r ≤ p if p − Φ, and so r − Ψ; by density p − Ψ. 2.29 2.30.Remark.We have given this proof now; it only applies, of course, to those formulae for which a forcing definition has been given.
Installing the ground model.Before we turn to the definition of the generic extension of a model, we identify objects that will serve as names for the elements of the ground model, and thus ensure that our generic structure is indeed an extension of our ground model.It is convenient to assume that 1 P is actually the ordinal 1.

Definition
This is a rudimentary recursion in the parameter 1 P , being of the form where G is the rudimentary function (i, f ) → {i} × Im(f ); thus with our convention that 1 P = 1, the recursion may be regarded as pure.
We prove the third line inductively: suppose p − x = ŷ for a ∈ x and r ≤ p, r − 1 â x, there will be a b and s ≤ r with s − â = b and s − 1 b ŷ; but then b = ĉ for some c ∈ y; so s − â = ĉ, so by induction a = c; and thus x ⊆ y; similarly y ⊆ x and so x = y.
The next line is the contrapositive, by definition of forcing for negation.If p − x ŷ then for some r ≤ p and some b, r − x = b and r − 1 b ŷ so for some c ∈ y, b is ĉ; so r − x = ĉ; by line 3, x = c, and thus x ∈ y.
Thus the fifth line is proved; and the sixth is its contrapositive.2.33 3. Extension of the definition of forcing to all ∆0 sentences.So far we have set up the beginnings of a definition of forcing, for atomic sentences and their negations.We wish to extend the definition of − to all ∆0 sentences of the forcing language, on these lines: The Forcing Theorem for ∆0 sentences will follow if we can arrange that such classes as P ∩ {p | p p − P ϕ} and P ∩ {p | p p − P ϕ} are sets, where ϕ is a ∆0 sentence of the language of forcing.
The annotated language L. We must describe our language of forcing in greater detail.The first step is to define in some recursive manner a language L ⊆ HF, which is a first-order language with no constants other than those that we shall call token constants, with the two binary predicate symbols = and , connectives and ∧, and the restricted quantifier x y , where in the rules of formation y is required to be a distinct variable from x.There are no unrestricted quantifiers.Other propositional connectives and the existential restricted quantifier x y may be introduced by definition.
We shall need the customary notions of free and bound occurrence of a variable in a formula, and we imagine that each formula of L is accompanied by an annotation saying which occurrences of variables are bound by which occurrences of quantifiers.As we build up formulae, we have to update the annotations, and we imagine all that going on inside HF.
We then define L u as the language resulting from L when all token constants are replaced by constants a for a ∈ u, and E u to be the set of sentences of L u , meaning those wffs with no free variables.If u is rud closed and non-empty, and the map a → a is basic, then L ⊆ L u ⊆ u.
We shall denote by Subst(ϕ, x/a) the result of substituting the constant a for the free occurrences of the formal variable x in the formula ϕ; we give the familiar recursive definition of this process below, but in fact we shall think of it as accomplished by passing from ϕ ∈ L u to the corresponding formula ϕ 0 in L, with the constants of ϕ replaced (in a 1-1 manner) by token constants, then looking up inside HF the free occurrences of x in ϕ 0 and replacing them by occurrences of an as yet unused token constant, and then reverting to a formula of L u by replacing each occurrence of the new token constant by an occurrence of a, and reversing the other replacements of constants of ϕ.In this way Subst will become a function that is rudimentary in the subset of HF which is recursive in the ordinary sense and which codes all the necessary information about L. As that subset is recursive, we shall simply say that Subst is rudimentary.
3.1.Definition.We define the tree-rank τ of a formula, the substitution of a constant for a free occurrence of a variable in a formula, and, when the formula is a sentence, the set Rub(ϕ) of sentences to which reference will be made when deciding whether p − ϕ.In the following, x and y are distinct formal variables.
• ψ atomic: • ψ = y a ϑ: 3.2.Remark.As we have defined it above, viewing formulae as trees, substitution is ∅-rud-rec.But we have seen that we may treat substitution as rudimentary, by viewing formulae as annotated sequences.
3.3.Remark.To go from x ∈ a ϕ(x) to {ϕ[α] | ∃s ∈ P (s, α) ∈ a} is to form the image of the substitution function, and is thus rudimentary.The annotations will tell us where are the free occurrences of x in ϕ.
3.4.Definition.Let χ − u, for u a transitive set and k ∈ ω, be the characteristic function of the forcing relation restricted to those ∆0 sentences ϕ of the forcing language with τ (ϕ) and all a with a occurring in ϕ being in u.
3.5.Proposition.For u transitive and e and η as in Definition 2.14: and thus rudimentary in χ = u; (3.5.2) for each there is a natural number s such that for each ordinal ν η, χ − P e;= ν ∈ P e;= ν+s ; (3.5.3) for each limit ordinal λ > η, χ − P e;= λ is total and is a set, being definable over P e;= λ .
Proof.For (3.5.0), note that χ 0 − u is rudimentary in χ = u and χ u, which is rudimentary in χ = u.
(3.5.1):The passage from χ − u to χ +1 − u is rudimentary in P and Subst, being given uniformly by these clauses: ), 0 otherwise.The notation y/a indicates that the free occurrences of the variable y have been replaced by occurrences of the constant a.Only certain substitutions have been indicated explicitly: we think of wffs as accompanied by annotations, as described above; so, for example, in the longest line of the above equations, where visually ϕ should be Subst(ϕ, x/α), the details of the substitutions would be in the annotations.
(3.5.3) now follows.3.5 Once we know that there is at least one rud-closed transitive v containing P with χ − v total and a set, we may advance the definition of χ − more generally, as we now show in Proposition 3.11.
Rudimentary generation of the sentences of the forcing language.Suppose that u is transitive, that u ⊆ u + ⊆ P(u), and that S is a subset of E u , so that S is a set of sentences, all of whose constants are in u.We define a ternary function H E which will yield a larger set of sentences, all of whose constants are in u + , thus: 3.6.Lemma.H E is rudimentary in the parameter HF; more exactly, it is rudimentary in the subset of HF that codes the annotation, described above, of formulae of the language L.

3.9.
Lemma.There is a rudimentary function H − such that for every P, u and S as above, Proof sketch.H − would be built from the rudimentary function H = supplied by Lemma 2.11 and other rudimentary functions suggested by the clauses of the Proposed Definition 3.0.3.9 3.10.Lemma.Suppose that (u n ) n ω is a strict continuous progress, and that u 0 is a rud-closed transitive set.Suppose that E 0 = E u 0 , that for each n, Proof.Note that if ϑ ∈ Rub(ϕ) then τ (ϑ) < τ (ϕ).Now prove by induction on k ∈ ω that for each n ∈ ω, if τ (ϕ) = k and all a with a occurring in ϕ are in u n+1 then ϕ ∈ E n+k+1 .3.10 3.11.Proposition.Suppose that (u n ) n ω is a strict continuous progress, with u 0 a rud-closed transitive set containing P and χ − E u 0 total and definable over u 0 .Then the same is true of u ω .
Proof.By Lemma 3.10, any ϕ ∈ E uω E 0 is in some E n+1 , and Rub(ϕ) ⊆ E n .By Lemma 3.9, we can build χ − E n+1 inside u ω .So the definition of p − ϕ over u ω will take the expected form "there is a set E closed under Rub with φ ∈ E and a function χ defined on E that satisfies the recursive definition of χ − ; moreover, χ(p, ϕ, a) = 1." 3.11 A further speed-up.In [MB,Section 9], a definition of truth for ∆0 formulae is given that begins by de-nesting restricted quantifiers.That method could be used here: one would first define p − P ϕ for quantifierfree sentences ϕ and then extend that definition to de-nested ∆0 sentences, adapting appropriately the steps of [MB,§9].Finally one would argue that every ∆0 sentence is logically equivalent to a de-nested one.
This device then makes it evident that for each ϕ ∈ L and constants for a, p − P ϕ[ a] is rudimentary in an appropriate fragment of χ = and thus a gentle predicate of p, φ, and a.
Forcing ∆0 statements 3.12.Proposition.For every We shall often use the following general principle in our development.
3.13.Proposition.Suppose we have a name z such that for some formal wff ϕ.Then ∀p ∀α Proof.We gradually weaken the hypothesis.Suppose that p − 1 α z.Then for some q ≥ p, (q, α) ∈ z, so q − 0 α z; so q and therefore also p forces ϕ [α].Now suppose that p − α ∈ z.This tells us that Axioms of Extensionality and Foundation.We may now prove that the Axiom of Extensionality is forced: that reduces to proving the following, which presents no difficulty.Proof.Given x, consider A = {a | ∃p ∈ P (p, a) ∈ x}.Then A is a ∆ 0 subclass of 2 x and therefore a set; assuming it is non-empty, let c be an element of A of minimal rank.Then Preservation of ∆0 statements about the ground model.Now that we have defined forcing for ∆0 statements and have seen how elements of the ground model are named in the language of forcing, we may verify that ∆0 statements about them, if true, are forced.
Proof.The cases that Φ are either atomic or the negation of atomic are covered by Proposition 2.33.We then proceed by induction on the length of Φ; propositional connectives are easily handled, as a 0-1 law applies in this context; and the last lemma covers restricted quantifiers.3.17 The above is a schema expressed in the metalanguage: the version when we quantify over ∆0 wffs in the language of discourse would read 3.18.Proposition.Let ϕ(x 1 , . . ., x n ) be a ∆0 statement such that 4. Generic extensions of a transitive set.Let M be a transitive set and P a notion of forcing in M .The aim in life of an (M, P )-generic filter G is to create a transitive set M [G] out of the Lindenbaum algebra of the language of forcing, with the property that what is true in the model is what is forced by some p ∈ G.That principle is known as the Forcing Theorem.
Formally, those definitions work for any transitive set M .
For forcing over models of ZF that definition of generic would suffice to prove the Forcing Theorem; but for models of certain weaker theories such as KP, it is inadequate: we shall find in a later section that if M is admissible, we must require G to meet all dense open subsets of P that are unions of a Σ 1 and a Π 1 class over M if we are to show that M [G] will be admissible.Ironically, for models of the still weaker theory PROVI, all is well again: we shall show in this section that if a filter G is generic as defined above, then the Forcing Theorem will hold for ∆0 formulae.We assume henceforth that M models PROVI.
We know from its being a filter that G will be consistent in the sense that for no sentence ϕ of the language L P can there be a p ∈ G with p − P ϕ and a q ∈ G with q − P ϕ.Let ∆(ϕ) = {p ∈ P | p − P ϕ v p − P ϕ}; this is a dense open subclass of P .If ϕ is ∆ 0 , ∆(ϕ) will be a set of M by gentle separation.If G meets ∆(ϕ), then some p ∈ G decides ϕ in the sense that it forces either ϕ or ϕ.We shall refer to this property as the completeness of G.

4.3.
Remark.This is a rudimentary recursion with parameter G: φ(b) = H(G, φ b) where H(g, x) = df x"(Dom(x))"g).In [M4] it is shown that certain transitive models of Zermelo set theory fail to support such recursions: thus it is necessary to assume that PROVI is true in the "background" set theory.

4.4.
Remark.An immediate consequence of the definition is that if p ∈ G and either p − 0 α a or p − 1 α a, then val G (α) ∈ val G (a). 4.5.Proposition.For all a and b the following hold: We divide the proof into four lemmata.Proof by induction on rank.Let x ∈ val G (a).Let (q 0 , α) ∈ a with q 0 ∈ G and val G (α) = x.Let q ∈ G be below both p and q 0 .Consider the class That is dense below q, and is a set by ∆0 separation, once one has replaced the predicate r − α = β by an evaluation by an appropriate fragment of χ = .
It is therefore met by G; so let r ∈ G be below q and β ∈ 2 b with r − α = β and r − 1 β b.
From the second property of r, val G (β) ∈ val G (b), and from the first, applying the induction hypothesis, val G (α) = val G (β); thus x ∈ val G (b); as x was arbitrary, val G (a) ⊆ val G (b).
A similar argument shows that val G (b) ⊆ val G (a). Proof by induction.We show first that ∃p ∈ G p − x a x b.If not, then by density, ∃p ∈ G p − x a (x b); indeed there will then exist p ∈ G and α ∈ 2 a such that p − 1 α a and p − (α b).Given such p and α, val G (α) ∈ val G (a); so there is a β ∈ 2 b and a q ∈ G with (q, β) ∈ b, and val G (α) = val G (β).
By the induction hypothesis, there will be an r ∈ G, which we may suppose to be below both q and p, such that r − α = β and r − 1 β b; so r − α b, contrary to our hypothesis on p.
A similar argument will show that ∃p ∈ G p − x b x a; and we may now invoke the fact that Extensionality is forced, to conclude that there is a p ∈ G such that p − a = b.Proof.The hypothesis implies that there are q ∈ G and c such that (q, c) ∈ b and val G (a) = val G (c).By the previous lemma, there is a p 0 in G such that p − a = c; then if p ∈ G is below both p 0 and q, then p − 1 c b and so p − a b.4.11 The proof of Proposition 4.5 is now complete.4.5 4.12.Definition.
Proof.By the providence of M and an easy application of Proposition 2.33.4.13 We have a canonical name for G: 4.17.The Forcing Theorem.Given A, P and G, for each ∆0 formula ϕ and a 1 , . . .a n in A: Proof.First, the case of atomic ϕ is covered by Proposition 4.5.For Boolean conjunctions: ∃s ∈ G (s ≤ p & s ≤ q) iff p and q are both in G.
For negations: there is a dense class to be met by G, and we must show that the class in question is a set.Note that P ∩ {t | ¬∃r ≤ t (r − ϕ[β])} is a member of the provident set that is the ground model: take an attempt at χ = that covers all; then this set is obtainable as a separator that is ∆ 0 in that attempt.Then there is a q in G and an η such that q − 1 η b and q − a = η.Then densely below q, there are r such that r − ϕ [η].So some such r is in G; so Conversely, suppose that A[G] |= x b ϕ(x), and suppose that b ∈ A with b = val G (b).Let X is a set and is downwards closed, i.e. open in the usual topology on P.
∆ is a dense open set, and so meets Thus p / ∈ X; and so there is no q below p with q ∈ X.So which says precisely that p − x b ϕ(x).4.17 5. Construction of nominators for rudimentary functions.Our aim in this section is to prove a theorem about the construction of names in a provident set for the values of a rudimentary function in a set-generic ( 2 ) For explanation of the notation, please see the footnote on page 141.extension of that set.We state it now, and shall later give more precise formulations of particular cases.5.0.Theorem.Let R be a rudimentary function of some number of arguments.Then there is a function R P , of the same number of arguments, with the property that if A is a provident set and P ∈ A a notion of forcing, then A is closed under R P and, further, if G is an (A, P)-generic, then (to take the case of a function of two variables) for all x and y in A, Definition.We shall call the function R P the nominator of the function R. Usually its definition is uniform in P and A. We shall use the phrase "Cohen term" to speak of the value of a nominator given some arguments.Some general lemmata about forcing 5.4.We note alternative ways of naming an object.For given y, put 5.5.Remark.A 0 (y) ⊆ y; A 0 (y) ⊆ A 1 (y) ⊆ P × 2 y, so A 0 (y) and A 1 (y) are sets, whereas A(y) is a proper class whose intersection with a set will be a set provided one has rud-rec separation.5.6.Remark.A 0 (A 1 (y)) = A 1 (y) = A 1 (A 0 (y)) = A 1 (A 1 (y)) and A 0 (A 0 (y)) = A 0 (y).5.7.Lemma.If q − 1 w A 1 (y), then q − 1 w y; if q − 1 w y, then q − 0 w A 1 (y).5.8.Lemma.Let P ∈ M and let G be (M, P)-generic.
Suppose that p ∈ G and z ∈ M are such that p − z A 1 (y) and p − z y.Then there are w ∈ M and q ∈ G with q ≤ p, q − 1 w A 1 (y), (so by the previous lemma, q − w y), but also q − w = z and therefore q − w y, a contradiction.5.8 The proof.To prove Theorem 5.0 we begin by working through the list, given in [M3, p. 166] and discussed in greater detail at the beginning of [MB,§2], of the nine functions R 0 , . . ., R 8 , the auxiliary function A 14 , and some others, and show for each function how, given names (in the ground model) for its arguments in the generic extension, we may build names for its values.Certain of the nominators are rudimentary, even basic, functions of their arguments, others will be P-gentle.We shall do the rudimentary ones first.We assume that P = (P , 1 P , ≤ P ) 3 .
Although we shall not always obtain a nominator for a rudimentary function as a rudimentary function of the names of its arguments, as we shall see with x y, we may check as we go that we always find a nominator that is rudimentary in the relation χ − for some .5.9.Remark.Composition of rudimentary nominators will of course be rudimentary.
A basic nominator for x ∪ y 5.19.Definition.x ∪ P y = df x ∪ y.

5.24.
Remark.Hence if p − P γ ∈ P x, then there are many β and t (dense below p) for which t − β = γ and t − 1 β P x.
Conversely, if q − P y x z = {y }, then there are y and r ≤ q with r − 1 y x and r − P z = {y }.So (p, y) 2 ∈ x for some p ≥ r, so that (p, {y} P ) 2 ∈ F P 1 (x), r − 1 {y} P F P 1 (x) and so r − P z F P 1 (x).5.32 A basic nominator for [x] 2 5.33.Remark.
[x] 2 is easier to get than [x] 2 , because the latter will require us to be certain that two names denote different things; we could obtain such a term by using the identity [x] 2 = ∪(x × x); the following is slightly simpler.5.34.Definition.
Proof.If t − P a F P 2 (x), there are s ≤ t and b such that s − a = b and s − 1 b F P 2 (x).So there are r ≥ s, α and β with b = {α, β} P , and conditions p and q with (r then there are t ≤ s, α, β such that t − 1 α x & t − 1 β x & t − P a = {α, β }, so that there are p and q with t ≤ p, t ≤ q, (p, α) ∈ x and (q, β) ∈ x; so (t, {α, β} P ) 2 ∈ F P 2 (x), so t − P a F P 2 (x).5.36 A basic nominator for u .We recall the definition: and that for u transitive, u is transitive.Then a basic nominator for it can be found by composition using the preceding ones.
Gentle nominators for the other rudimentary generators.We show that for the remaining rud generators we get nominators of the form G(P, x, y) ∩ A where G is a rudimentary function and A is a separator that is rudimentary in an appropriate segment of χ = ; such nominators will be gentle by the principle of gentle separation.
Remark.That that should be so is suggested by our theory of companions, at least for DB functions.Each of them has a 2-companion W that is generated by and × and is therefore such that W P is basic; so if R(x, y) ⊆ W ({x, y}), then we may expect R P (x, y) to be of the form W P ({x, y} P ∩{(p, α) | p − α Ṙ(x, y)}; and as z ∈ R(x, y) is ∆ 0 , α Ṙ(x, y) will be ∆0 .
A gentle nominator for R 1 : x, y → x y.Set Then x P y, being a subclass of P × 2 x, will be a set if P is, being the application of a separator that is ∆ 0 in some appropriate attempt at χ .
Let z = x P y.For each p and α with p − 0 α z, p − α y ∧ α x so by our general principle, the same is true for each p and α with p − α z.
Conversely, suppose that q − β x ∧ β y.We seek s ≤ r and ( t, α) ∈ z with s ≤ t and s − α = β.We know that so that for such an s, s − 1 α x and s − α y, so (s, α) ∈ z.Hence we may take s = t = s.
A gentle nominator for R 5 : 2 is basic, so we can easily find a companion (i.e. a bounding set), and then apply the separator induced by the relation p − α β.

Proposition. 1 P
We have defined (•, •, •) P 3 above; it is basic; so we can use it to predict the whereabouts of δ; l, m, n must then be given appropriate values.5.42.Proposition. 1 P − P R P 6 (x) = Ṙ6 (x).
Then H is rudimentary in G, and R P 8 (x, y) is the result of applying a ∆ 0 separator to P × H(x, y).Thus there is a rudimentary function E such that for all such u, R P 8 (u×u) = E χ = (P × u × u) .5.48 5.49.Proposition. 1 P − P R P 8 (x, y) = Ṙ8 (x, y).5.50.Remark.Suppose that Q( x) = R(S( x), T ( x)), where Q, R, S, T are rudimentary, and that we have already found functions R P , S P , T P as in the statement of the theorem.We may obtain Q P by composition: define Q P ( x) = R P (S P ( x), T P ( x)).
The proof of Theorem 5.0 as stated is now complete.5.0 We must now prove that each of these functions is of uniform finite delay.
Propagation of nominators for rudimentary functions 5.51.Proposition.Let R be a rudimentary function of some number of arguments, and let R P be the corresponding nominator.There is a natural number s R such that whenever e is a transitive set with P ∈ e, and ν is an ordinal not less than (P), R P P e;= ν ∈ P e;= ν+s R .
Proof.We have seen that the nominators for R 0 , R 2 and R 4 proved to be themselves rudimentary, and hence s R can be taken in these cases to be 1 plus the rudimentary constant for u → R u.The nominators corresponding to the other functions in the standard generating set are all rudimentary in appropriate fragments of χ = , and so the proof in those cases follows from the corresponding result for χ = .We give the argument for R 8 .
Let ν η.We know that χ = P e;= ν ∈ P e;= ν+12 , and that for some rudimentary function E, R P 8 P e;= ν = E(χ = P e;= ν ), so we may take s R 8 = 12 + c E .Once the theorem has been established for the nine generators, it remains only to observe that the property in question is preserved under composition.If, for example, Q( x) = R(S( x), T ( x)), then s Q can be taken to be c R + max{c S , c T } + c • , where c • is the constant of the rudimentary function that composes fragments of R P , S P and T P to a fragment of Q P .5.51 5.52.Remark.At this point we know that all the axioms of GJ 0 are forced by the trivial condition.
No new ordinals!There is a long-established principle that a generic extension will contain no ordinals not in the ground model.In [M4] an admittedly pathological example of forcing over an improvident but transitive model of Zermelo set theory is presented where this principle breaks down.So our task here is to present a proof, working in the theory PROVI, of the following: Plainly the statement of the proposition requires every ordinal to have a hat; but hatting is 1 P -rud-rec, so available in PROVI.Proposition 3.16 then yields 5.54.Proposition.For each ordinal η, 1 P − P η is an ordinal.
The second requirement is that there should be enough set theory to prove that the principle of trichotomy for two ordinals is forced.So let ζ and η be ordinals.Proof.Consider the four statements We know that [(1a or 1b) and (2a or 2b)] holds.Of the four possibilities, (1b and 2b) is impossible, as it would imply ζ ∩ η ∈ ζ ∩ η, contradicting Foundation; the three disjuncts of the proposition correspond to (1b and 2a), (1a and 2a), (1a and 2b).5.56 The final requirement is that rank should be definable in the ground model; but is ∅-rud-rec.5.57.Lemma.− P Ȯn is transitive.

5.58.
Lemma.There are no p ∈ P and x such that p − x Ȯn ∧ (x) x.
Proof.Suppose such an x exists; let it be a member of the transitive set u.By rewriting its definition in terms of the attempts χ − u, u and • (u), we see that the class is a set by ∆ 0 separation, and non-empty by the initial supposition.Call that set A.
Let x be a member of A with (x) minimal.Then x ∈ A and 2 x ∩ A = ∅.Let η = (x), and let p − x Ȯn ∧ η x.So ∃q ≤ p ∃r ≥ q (r, y) ∈ x & q − η = y.
Proof.We have just seen that Trichotomy for ordinals is provable in GJ 0 , and we know that all axioms of GJ 0 are forced.5.59 Now Lemma 5.58 implies that if p − P x Ȯn, and η = (x) then p − P η x.By Trichotomy, p − P x η ∨ x = η; which implies that there are q ≤ p and ζ η with q − P x = ζ as required.5.53 5.60.Remark.In Section 6 of [M2], a forcing contruction is done over a non-standard model N, and it was there blithely stated without proof that the generic extension would bring no new "ordinals".Fortunately the model N was power-admissible, and therefore certainly a model of PROVI, which is a sub-theory of KP, so that the present remarks justify that blithe statement; that is reassuring in view of the models presented in [M4].
We record two related arguments.
Proof by eps-recursion on x.If p − ζ x then there are q and r with q ≤ p, q ≤ r, (r, y) ∈ x and q − ζ = y.Hence for all η < ζ, q − η y and so by induction, η < (y), so ζ ≤ (y) < (x).5.62 6. Construction of rudimentarily recursive nominators for rank and transitive closure.Rank and transitive closure are pure rud-rec; we show here that P-rud-rec nominators exist for them.
where for some p ∈ G, (p, y) ∈ x.So the LHS is contained in the RHS.

6.1.
Lemma.There is a rud function S P (•) such that val G (S P (x)) = S(val G (x)).
6.3.Remark.P is rud-rec in the parameter P.
Remark.That all makes sense: if x is in A, the name P (x) is in A. Note that (val G (x)) is evaluated in the universe.At present we do not know that the evaluation can be carried out in Proof.
6.6.Remark.tcl P is rud-rec in the parameter P.
Proof.By similar reasoning: properties of ∪ P and P = val G (tcl P (x)) by the definition of tcl P .6.7 6.8.Proposition.Let A be provident, and P ∈ A. Let G be (A, P, ∆0 )generic.Then A P [G] is closed under rank and transitive closure.6.9.Remark.A similar result will hold whenever F is rud-rec, given by G, where G P is rudimentary; G P may be permitted to have as a parameter a name a for a parameter val G (a) in the extension.
We pause for breath.The next stage will be to show that the generic extension is closed under the formation of certain canonical progresses; but we digress to discuss the case of primitive recursively closed sets, which is now easy.
7. Construction of primitive recursive nominators for primitive recursive functions.Jensen and Karp give, following Gandy, this definition: there are some initial functions, which are all rudimentary; two versions of substitution: F ( x, y) = G( x, H( x), y) and F ( x, y) = G(H( x), y); and this recursion schema: 7.0.Lemma.Let A be transitive and primitive recursively closed, and let F be primitive recursive.Then Proof.As before.7.0 For notational simplicity there is only one x in the following, but it could easily be replaced by a finite sequence.7.1.Proposition.Let A be transitive and primitive recursively closed.Let P ∈ A, and let G be (A, P)-generic.Suppose that G(f, z, x) is primitive recursive, and that it has a nominator, G P , primitive recursive in the parameter P, so that for all f, z, x in A, Then F P is primitive recursive in the parameter P, and for all z, x in A, so that F P is a primitive recursive nominator for F .Proof.For fixed x by recursion on z: The above confirms an observation made some years ago by Jensen: 7.2.Corollary.A set-generic extension of a primitive recursively closed set is primitive recursively closed.
8. Construction of nominators for the stages of a progress.Let e be a transitive set in the ground model of which P is a member, and let θ be indecomposable, exceeding η = df (e).Then P e θ is provident.Let ḋ be the Cohen term ê ∪ { Ġ} P , so that val G ( ḋ) will be the transitive set d = e ∪ {G}.8.0.Remark.ḋ will be a member of P e (P)+k for some (small) k, given the definition of Ġ, our convention that 1 P = 1 and the fact that • is 1-rud-rec.
Our task is to build for each ν < θ a name N (ν) for the stage P d ν of the progress towards d.
A simplified progress.Now (G) (P ) < η, so that for ν η, d ν = e ν ∪ {G}.It might be that (G) < (P ); to avoid building names which make allowance for that uncertainty, we shall build names for the terms of a slightly different progress ( 8.2.Proposition.If θ is indecomposable, then Q d θ is provident and equals P d θ .Proof.By [MB,Theorem 6.34].8.2 Names using dynamic predicates.With that in mind, we now define names 8.4.Lemma.For ν η, N (ν) ∈ P e;= ν+ω .Proof by cases.For ν = η, by inspection; for successor ordinals, by knowledge of the birthday of T P ; for limit λ by knowledge of the delay of P .8.4 8.5.Proposition.Each N (ν) for ν < θ is in P e θ .
Proof.All those names are in P e;= θ , which was shown in Proposition 2.15 to equal P e θ .8.5 8.6.Proposition.Let G be (P e θ , P) generic and let ν < θ.Then Proof.By induction on ν. 8.6 Pre-nominators for rud-rec functions 8.7.We may use the above idea to construct what one might call a pre-nominator of a rud-rec function.
Suppose that in M [G], F is given by G and the parameter g = val G (γ).Let H = H G be the rudimentary function ( [MB,5.9])such that for transitive P and P + with P + ⊆ P(P ), F P + = H(g, F P, P + ).
Define for ν < η, E(ν) = F P e ν ; thereafter define E(ν + 1) = H P (δ, E(ν), N (ν)), and for limit λ, E(λ) = P {(1 P , E(ν)) | ν ν < λ}.Then an easy induction shows that for every ν, val G (E(ν)) = F Q d ν+1 .8.8.Remark.To show that the sequences (N (ν)) ν and (E(ν)) ν are (P)-gentle, one could repeat their definition and those of (P e;= n u) ν and T P by a single simultaneous rudimentary recursion, a method seen in Definition 2.14 above and the proof of [MB,Proposition 4.19].With a little extra work one would then get a (P)-gentle nominator for F . 9. Generic extensions of provident sets and of Jensen fragments.We are now in a position to prove the following theorem: 9.0.Theorem.Let θ be an indecomposable ordinal strictly greater than the rank of a transitive set e which contains the notion of forcing, P. Let G be (P e θ , P)-generic.
and hence is provident.
Proof.(P e θ ) P [G] contains P e∪{G} θ , as we have for each ν < θ built a name in P e θ that evaluates under G to Q e∪{G} ν , and we know by Proposition 8.2 that Q e∪{G} θ equals P e∪{G} θ .
For the converse direction, we know that P e∪{G} θ is provident, and has G as a member and hence can support the G-rudimentary recursion defining val G (•).Further P e∪{G} θ includes (P e ν ) ν , which is defined by an e-rudimentary recursion, and so includes (P e θ ) P [G].9.0 Remark.Thus, in this special case, a generic extension of a model of PROVI is a model of PROVI.We shall use this result to establish it more generally.
Remark.Theorem 9.0 remains true if the hypothesis on θ is weakened to requiring that θ > (P).
Proof that a generic extension of a provident set is provident 9.1.Theorem.Let A be provident, P ∈ A and G (A, P)-generic.Then A P [G] is provident.Lemma.If θ is indecomposable and D is a collection of transitive sets each of rank less than θ and such that the pair of any two is a member of a third, then d∈D P d θ is provident.Take D = {c ∪ {G} | c ∈ T } to complete the proof.9.1 Remark.Theorem 9.0 and Corollary 4.16 give the elegant characterization noted by Bowler, that if A is provident, P ∈ A and G is (A, P)-generic then A P [G] = Prov(A ∪ {G}).
Generic extensions of P-provident sets.Our methods will prove the following more general result: 9.2.Theorem.Let λ be a limit ordinal ω 2 .Write η for the largest indecomposable not greater than λ.Let T be a set of transitive sets, and put A = c∈T P c λ and B = c∈T P c η .Suppose that both A and B are closed under pairing.Then for any P ∈ B, forcing with P is definable over A, and if G is (A, P)-generic, then A[G] is P-provident; indeed A[G] will be q-provident for every q in B[G].
Proof.In these circumstances, B will be provident, so forcing with P over B is by now well-established; each of the relations and functions χ = , χ − , R P and F P will progress through each P c λ for c ∈ T , and the closure of A under pairing shows that these functions and relations will be total on A. 9.2 Genericity at every limit level 9.3.Proposition.Let e be a transitive set with P ∈ e.Let θ be indecomposable, greater than (e).Let λ be a limit ordinal not less than θ.Let κ λ + ω, and let G be (P Here is an application, which fleshes out an argument outlined in a letter from Sy Friedman.9.6.Proposition.Let θ < η ζ < ξ be ordinals, with η indecomposable.Suppose that P ∈ J η and that G is for some y ∈ J ζ .Therefore some condition p in G forces y = x; so x = {ν < θ | p − ν y}.The map ν → ν, restricted to the ν less than θ, is in J θ+1 , and the relation p − ν z is rudimentary in χ = ; an appropriate segment of that characteristic function is in J ζ , by propagation starting from η; and therefore x ∈ J ζ .9.6 A similar argument leads to an extension of a result in [MB]: 9.7.Proposition.If a set c is such that there is a progress (Q c ν ) ν given by a c-rudimentary recursion with Q c λ = L λ for every limit λ ω, then c is not set-generic over L.
Proof.Suppose to the contrary that G is (L, P)-generic where P ∈ L and c ∈ L P [G].Let η be indecomposable with P ∈ L η , c ∈ L P η [G], and η ω ω , so that η = ωη, and therefore L η = J η = T η .Let λ = η + ω. 9.8.Remark.(i) for some a ∈ T λ , and there will be a condition p in the generic filter G with p − P Â = a; (iii) for such p and any z ∈ L, p − P ẑ a ⇔ z ∈ A.
From our analysis in §2 of the forcing relation, we know that the characteristic function of the predicate p − P ẑ a is P-gentle; and therefore from the discussion in [MB,§6] of the rate of convergence of a gentle function in a progress-see in particular the proof of [MB,Theorem 6.38]-we know the following: 9.9.Proposition.There is a natural number s such that if (P m ) m<ω is a strict progress, with P ∈ P 0 , P 0 P-provident, p ∈ P, then for any n and any a ∈ P n , P n ∩ {z | z p − P ẑ a} ∈ P (n+1)s .

9.10.
Lemma.There is a primitive recursive function f : ω × ω → ω such that for each k, n < ω, and A ∈ L, if A ⊆ L η+k and A = val G (a) for some a ∈ T η+n then A ∈ T η+f (k,n) .
Proof.Case k = 0. We suppose that A ⊆ L η = T η and for some a ∈ T η+n , A = val G (a).
Then there is a condition p ∈ G with p − P Â = a.So, using Remark (iii) and applying the proposition to the strict progress (T η+m ) m<ω , we shall have A = T η+n ∩ {z | z p − P ẑ a} ∈ T η+(n+1)s .
Case k = +1.We suppose that A ⊆ L η+ +1 and that for some a ∈ T η+n , A = val G (a).
Then there is a condition p ∈ G with p − P Â = a.Let B ∈ A. Then B ⊆ L η+ and for some b ∈ T η+n , B = val G (b).
[Actually b will be in a and therefore in T η+n−3 , if n 3.So we are giving a little ground: unimportant in the present case, but perhaps a fact to be stored for the future.]So by the case k = , B ∈ T η+f ( ,n) .Thus So we may set f ( + 1, n) = (f ( , n) + 1)s.9.10 The lemma is proved, and Remark 9.8(ii), implies, under the current hypotheses on c, that L λ ⊆ T λ ; but that is false, as L η+1 ∈ L λ but L η+1 / ∈ T λ = J η+1 , by the proof of [MB,Proposition 6.68].Thus c cannot be set-generic over L. 9.7 10.Persistence of certain systems.We suppose throughout that we are forcing over a provident set (or model, meaning possibly ill-founded) M by some P ∈ M .Our first task is for each of the schemes of foundation, collection and separation, to show that if they hold in M then in M [G].We call that the phenomenon of persistence.Establishing the persistence of a given axiom divides into two steps: first, showing that the axiom is forced; and second, showing that what is forced is true in the generic extension.The first step generally uses the truth of the axiom concerned, possibly in an apparently stronger form, in the ground model.The second step succeeds if the generic meets all appropriately definable dense subclasses of P, which it will do given the truth in the ground model of sufficiently strong instances of the separation scheme.
Extension of the definition of forcing to all formulae.We may extend the definition of forcing, schematically, to all formulae, thus: Proof.Already proved for atomic wffs, an easy induction thereafter.10.2 10.3.Exercise.Show that if p − x (Φ → Ψ(x)) and Φ is a sentence of L P (so that, intuitively, x has no free occurrence in Φ), then p − Φ → x Ψ(x).This exercise, coupled with our remarks about Modus Ponens above, ensures that we may apply mathematical reasoning to statements in our forcing language.
Complexity of some classes of formulae.For a class Γ of wffs, such as Σ k or Π k , let Σ 1 Γ be the class of wffs of the form ∃x Ψ where Ψ ∈ Γ; let ∆ 0 Γ be the class of those wffs where a formula from Γ is preceded by a finite string of restricted quantifiers; and let − Γ be the class of wffs of the form p − P Φ where Φ is in Γ. 10.4.Some computations for ϕ ∆0 : (10.4.3) p − P x a y ϕ is ∀x ∈ 2 a ∀q ≤ p ∃r ≤ q ∃t q − 1 x a ⇒ r − P ϕ[t, x] , and thus ∆ 0 Σ PROVI 1 and Σ KPI 1 ; (10.4.4) p − P y x ϕ(x, y) is ∀y ∀q ≤ p ∃r ≤ q ∃x r − P ϕ[x, y], and thus Π PROVI
More generally for positive k, all − Π k predicates are Π PROVI k , as ∆ 0 quantifiers such as ∀q ≤ p ∃r ≤ q will usually be interpolated between successive unrestricted quantifiers ∀∃ of the Π k formula, and thus absorbed, while − Σ k predicates will be ∆ 0 Σ PROVI k .When appropriate forms of Collection hold in M , those initial restricted quantifiers can also be absorbed; for example ∆ 0 -collection implies that every ∆ 0 Σ 1 formula is equivalent to a Σ 1 formula, so that − P Σ 1 predicates are Σ KPI 1 .Thus (in KP) if P is a set, − P restricted to Σ 1 wffs will be Σ 1 ; and in ZF, forcing for Σ n wffs is Σ n .
Remark.Initial restricted quantifiers can also be absorbed in contexts such as V = L when the lemma of Sy Friedman discussed in §5 of [M2] holds.
Completeness and consistency.By the consistency of forcing we shall mean the principle that no condition forces both a formula and its negation; by the completeness of forcing we shall mean the principle that every condition can be refined to one that decides a given formula.
The consistency and completenes of forcing are maintained by our system of definitions as we progressively enlarge the definition of forcing to wider classes of formulae: they follow from the way forcing handles negation.
Problem.Let us take G-completeness to mean that each statement is decided by some p ∈ G.The class of conditions that decide is certainly dense; but is it a set?P ∩ {p | p − Φ v p − Φ} will be a set if we have enough separation.If Φ is ∆0 , it will be a set by gentle separation.For other Φ, it will be a set if we have both ( − Φ)-separation and (∆ 0 ¬ − Φ)-separation.10.5.Remark.There is a point to be made here, similar to the problem of defining truth for all formulae.We have defined the forcing relation χ − for all ∆0 sentences ϕ of our formal language, essentially by a single recursion, so that every set is a member of a transitive set u such that χ − E u is total and a set.A definition of truth for a single ∆0 sentence is achieved by recursion over the transitive closure of the sets named in that sentence; in the case of forcing we must start our recursion from the transitive closure of the union of {P} and some set containing each set a of which the name a occurs in the sentence.
But we are not able to make a single definition for all formulae with arbitrarily many unrestricted quantifiers, but must introduce a sequence of definitions schematically.This would become very apparent in the Booleanvalued presentation of forcing, where truth values are assigned in a complete Boolean algebra, and we must invoke the axiom of replacement for each quantifier to see that the supremum over a class is actually the supremum over a set.
Then C is also (∆ 0 − P Γ), so, assuming A = ∅, let c be C-minimal.Then for some p, p − P Φ[c], but p will also force that c is Φ-minimal.10.6 Persistence of Collection.It is convenient to take Collection in the form ∀x ∈ a ∃y Φ ⇒ ∃b ∀x ∈ a ∃y ∈ b Φ.
Remark.Collection is more natural than replacement in the context of forcing.
Persistence of Separation.Given a and Φ, let b = In the other direction, suppose that p − α a ∧ Φ[α]; then ∀q ≤ p ∃r ≤ q ∃γ (r it would follow from ∆ 1 -separation in the ground model, which fortunately is a theorem of KP, together with TFT(∆ 0 ), which is free.
A teasing question 10.20.Remark.We have seen that we can reduce the amount of separation required to hold in M in proving the forcing theorem if, instead, we require the generic G to meet certain dense definable classes.On the other hand, that device apparently cannot be used to show that separation holds in the extension where it did not hold in the ground model; which raises the following question: 11.Definition of generic filter and extension in the ill-founded case.In Section 6 of [M2], forcing over an ill-founded model is used to obtain an independence result for H. Friedman's theory of power admissibility; contain some general remarks on such forcing.We go over some of this ground again as [M2] was written before the development of the theory of rudimentary recursion and in any case is chiefly concerned with set theories with the power set axiom.11.0.Suppose we have a countable ill-founded model M = M, R of PROVI, and P = (P , ≤) in M which M believes to be a separative poset.Working in M we can, as above, define a forcing relation − P .
The definition of an (M, P)-generic F will be much as before, but let us simplify our discussion by requiring F to meet every M-definable subclass of P .
In the case when M is transitive, we are able to define val G (•) by a rudimentary recursion; but in our present context, the model M is ill-founded, and so prima facie we cannot carry out that recursive definition.11.3.Then Q = df (Q, ∈ F ) is a perfectly reasonable countable set with a two-place relation on it, and we can ask which of the sentences of the forcing language are true in that model when we interpret = by equality and by ∈ F .
The Forcing Theorem in the general case.We wish to prove, to take the case of formulae with two free variables, that 11.4.Remark.That notation hints at a conflict of language level.We have ∆0 wffs which are sets, and over the set of which we can quantify; we are using these wffs, when their formal free variables are interpreted by constants, in two contexts; in our current universe, for which we have a truth definition |= 0 and in the generic extension via the definition of forcing p − ϕ.
To resolve that conflict for ∆0 wffs we should formulate the theorem thus: When unrestricted quantifiers are then "added by hand", the Forcing Theorem will extend schematically with no further problem.The proof will follow that for the transitive case, which relied entirely on the fact that F meets all the necessary dense classes, and made no use of the well-foundedness of the model under consideration.
Once that has been done, we may strengthen the ties between Q and N, by showing that we may treat Q as an extension of N by considering the map x → [x]; we may also show that F is in Q, being [ Ḟ].Here x is the canonical forcing name for the member x of the ground model, defined recursively inside N, (using which we may define a predicate V of the forcing language for membership of the ground model) and Ḟ is the canonical forcing name for the generic being added.11.6.The proof given in §5 that the generic extension has no new ordinals will go through in this case.So loosely we may say that the extension Q is no more ill-founded than is the starting model M. Further, Q considers itself to be a generic extension of M via P and F, the corresponding statement about P and Ḟ being forced.The author is much indebted to the anonymous referee for his meticulous reading of the manuscript and his thoughtful comments, which will now be addressed, on the past and possible future of work of this kind.11.8.Historical note.Since Cohen's creation of forcing as a construction of extensions of models of full ZF, many people have examined the possibility of forcing over models of weaker systems of set theory, to say nothing of those who have transplanted Cohen's ideas to other areas of enquiry outside set theory.Forcing over admissible sets was studied briefly by Barwise in his 1967 Stanford thesis, at greater length by Jensen in an originally unpublished treatise [J3] on admissibility that contained a proof of his celebrated "sequence-of-admissibles" theorem, in Steel's 1978 paper [St1], in Sacks' study [Sa], and in numerous writings of Sy Friedman such as his papers [F1] and [F2], which latter expounds inter alia that result of Jensen.In this connection, the referee draws my attention to the paper of Carlson [Ca] and the expositions of Ershov (the paper [E1], with a correction following a critical review by Blass; and the book [E2] which discusses forcing over models of KPU) and of Zarach [Z2].Something of the interplay between analysis and set theory is to be seen in a paper of Zarach and an unpublished manuscript of Gandy. 11.10.The referee also draws attention to the use of class forcing over admissible sets, which, it is hoped, might form the subject of a further paper.An early paper on class forcing in the context of Morse-Kelley theory is [Ch].Of the papers of Zarach, [Z1] cites a preprint form of [Ch].It discusses forcing with classes in the context of ZF-.[Z2] cites [Z1] and [Ch]: it discusses set forcing over admissible sets and certain cases of class forcing.[Z4] cites [Z1], [Z2] and [Z3]; it does both set and class forcing over models of ZF-.[Z3] cites none of the above, but it might be re-read in the light of the theory of rudimentary recursion.
A possible line of attack is this: suppose that M is a transitive model of some class theory, so that M has members of all ranks λ, where λ is a limit ordinal.For example, in the case of Morse-Kelley, M might be V κ+1 where κ is a strongly inaccessible cardinal.Pass to the provident closure, N of M , as defined in [M4] and [MB].N will be of height λω.Now the class forcing one had in mind for M will be a member of N , and therefore we can treat the problem as one of set forcing over the provident set N .The attraction of this approach is that names for members of N can be explicitly laid out, since the ordinals in N are all of the form λn + ζ where ζ < λ.

2. 18 .
Remark.This is not a definition by recursion: indeed, it is visibly rudimentary in p − b = c.2.19.Definition.Let χ (p, a, b) be the characteristic function of the relation p − P a b.

2. 22 .
Proposition.For all p ∈ P, a, b and c: (2.22.0)p − b = b; (2.22.1) if p − b = a then p − a = b; (2.22.2) if p − a = b and p − b = c then p − a = c.Proof.(0) Let b be a counter-example of minimal rank.The definition of p − b = b involves various r, β ∈ 2 b, for which r − β = β by the minimality condition on b.
2.22 2.23.Lemma.q ≤ p & p − a b ⇒ q − a b.2.24.Lemma.p − a b ⇔ ∀q ≤ p ∃r ≤ q r − a b.2.25.Proposition.If p − 1 a b then p − a b.Proof.Let r ≤ p; take s = r and β = a; then s − β = a and s − 1 β b. 2.25 2.26.Proposition.If p − a b and p − a = c then p − c b. Proof.Let s ≤ p.We seek t ≤ s and β ∈ 2 b such that t − β = c and t − 1 β b.We know that there are t ≤ s and β ∈ 2 b such that t − β = a and t − 1 β b; since p − a = c and t ≤ p, t − β = c.2.26 2.27.Proposition.If p − a b and p − b = d, then p − a d.Proof.Let s ≤ p.We seek t ≤ s and δ ∈ 2 d such that t − δ = a and t − 1 δ d.Since p − a b, there are r ≤ s and β ∈ 2 b such that r − β = a & r − 1 β b.Since p − b = d and r ≤ p, there are t ≤ r and δ ∈ 2 d such that t − β = δ and t − 1 δ d; as t − β = δ and t − β = a, t − δ = a.2.27
Lemma.If p ∈ G and p − a b then val G (a) ∈ val G (b). Proof.{r ∈ P | ∃β ∈ 2 b r − β = a & r − 1 β b} is dense below p and is a set, and so there is an r ∈ G and a β ∈ 2 b such that r − β = a, which by the previous lemma implies that val G (β) = val G (a), and, by Remark 4.4, such that val G (β) ∈ val G (b); so val G (a) ∈ val G (b). 4.9 4.10.Lemma.If val G (a) = val G (b), then for some p ∈ G, p − a = b.
Now consider the problem of a restricted quantifier ( 2 ).Suppose p − x b ϕ(x), and let b = val G (b). Suppose A[G] |= ã b where a = val G (a).
5.1.Corollary.Let A be provident, P ∈ A and G (A, P)-generic.Then A[G] is rud-closed and so a model of GJ 0 .Proof of the Corollary.Suppose (to take a function of two variables) that R(x, y) is a rudimentary function and that x and y are in A[G].Then there are a and b in A so that x = val G (a) and y = val G (b). Applying the nominator of R, the corresponding Cohen term R P (a, b) exists in A: letz = val G (R P (a, b)).Then z ∈ A[G],and by the theorem R(x, y) = z.Since R(x, y) = z is a ∆0 statement, and therefore absolute for transitive sets containing x, y and z, we know that it is true in A[G] that R(x, y) = z.5.1

6. 0 .
Lemma.Let A be provident and closed under F and F ": for example if F is gentle.Then val G ({(p, F (y)) | p,y (p, y) ∈ x}) = {val G (F (y)) | y ∃p ∈ G (p, y) ∈ x}.Proof.Let Z = {(p, F (y)) | p,y (p, y) ∈ x}.Then Z is in A by the hypotheses concerning the closure of A under F , F " and related functions.Then val G

Proof.
Let θ = df On ∩ A and let T = {c | c ∈ A & c is transitive & P ∈ c}.Then A = {P c θ | c ∈ T }, since the union on the right contains each element of A and is contained in A. It follows that A P [G] = c∈T (P c θ ) P [G].By Theorem 9.0, as each P c θ is provident and contains P,A P [G] = (10.4.0) p − P ϕ is a P-gentle predicate of p and φ, and therefore ∆ PROVI 1 ; (10.4.1) p − P x ϕ(x) is ∀x p − P ϕ[x], and thus Π PROVI 1 ; (10.4.2) p − P x ϕ(x) is ∀q ≤ p ∃r ≤ q ∃x r − P ϕ[x], and thus ∆ 0 Σ PROVI 1 ; 10.21.Problem.Can a set-generic extension satisfy more separation than held in the ground model?Here is one case, suggested by Kai Hauser in conversation, where a negative answer holds: 10.22.Proposition.Suppose that M is admissible and of the form P c θ for some transitive set c. Let P ∈ M , and let G be (M, P)-generic.If Σ 1 separation holds in M [G], then it holds in M .Proof.Inside M [G], M is a Σ 1 (c) class, and so if a and b are in M , the class a ∩ {x | x ∃y ∈ M Ψ(x, y, b)} is a set, d say, of M [G].For some ν < θ, d ∈ P c ν [G].∀x ∈ d ∃ξ (∃y ∈ P c ξ Ψ(x, y, b)), so by Σ 1 -collection in M [G], there is a ζ < θ such that for z = P c ζ ∈ M , d = a ∩ {x | x ∃y ∈ z Ψ(x, y, b)} ∈ M by ∆ 0 -separation.10.22 Instead we treat as a definition what in the transitive case was Proposition 4.5: 11.1.Definition.Define for all a and b in M the following equivalence relation:a≡ F b ⇔ ∃p ∈ F p − a = b.Let Q = Q F be the set of equivalence classes.Write [a] F for the ≡ Fequivalence class of a ∈ M .Define a relation ∈ F on Q by [a] F ∈ F [b] F ⇔ ∃p ∈ F p − a b.11.2.Remark.That that relation is independent of the chosen representives a, b of their equivalence classes follows from Propositions 2.26 and 2.27 established within M.
Hence inside Q the recursive definition of val F : M → Q by val F (b) = {val F (a) | ∃p ∈ F (p, a) 2 ∈ b} succeeds, using the predicate V identifying the members of M .11.7.Proposition.[a] F = val F (a). Proof.Use recursion inside the ill-founded model Q of PROVI.11.7 The results for ATR0 follow in the same way by taking RWO to be the class of all wellorderings of N or ω since the Mostowski Collapse ensures that every wellordering has an ordinal-valued rank function.It follows that PROVI and ACA + 0 have the same proof-theoretic ordinal: the first fixed point of the epsilon function.Similarly, PROVI + Mostowski Collapse and ATR0 have the Feferman-Schütte ordinal Γ0.All of these results only assume set-foundation in PROVI ; I'm still unsure what happens with Π1-foundation.
Consider now the special case that e = T η , a fragment of the constructible hierarchy, and let ξ be the least indecomposable not less than η.The above argument yields 9.4.Lemma.Let P ∈ T η .Suppose that ν is a limit ordinal ξ, that G is (T ν+ω , P)-generic, and that T P ν [G] = P d ν , where d = T η ∪ {G}.Theorem.Let P ∈ J ξ , where ξ is indecomposable.Let G be (L, P)generic.Then for each ordinal ζ ≥ ξ, J ζ