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Jensen’s P∗Theory and the Combinatorial

arXiv:math/9212201v1 [math.LO] 2 Dec 1992Jensen’s P∗Theory and the CombinatorialContent of V = LSy D. Friedman*M.I.T.An awkward feature of the fine structure theory of the Jα’s is that special param-eters are required to make good sense of the notion of “Σn Skolem hull” for n > 1.The source of the problem is that parameters are needed to uniform Σn relationswhen n > 1.The purpose of this article is to indicate how a reformulation of Jensen’s Σ∗theory (developed for the study of core models) can be used to provide a moresatisfactory treatment of uniformization, hulls and Skolem functions for the Jα’s.Then we use this approach to fine structure to formulate a principle intended tocapture the combinatorial content of the axiom V = L.Section One Fine Structure RevisitedWe begin with a simplified definition of the J-hierarchy. Inductively we defineeJα, α ∈ORD (and then Jα = eJωα) : eJn = Vn for n ≤ω.

Suppose eJλ is defined fora limit λ and let W λn (e, x) be a canonical universal Σn( eJλ) predicate (also definedinductively).For e ∈eJλ let Xλ1 (e) = {x|W λ1 (e, x)} and for n ≥1, Xλn+1(e) ={Xλn(¯e)|W λn+1(e, ¯e)}. Then eJλ+n = {Xλn(e)|e ∈eJλ}.

For all limit λ, eJλ = S{ eJδ|δ <λ}. It is straightforward to verify that the eJλ, λ limit behave like, and in fact equal,the usual Jα’s.Let M denote some Jα, α > 0.

(More generally, our theory applies to “acceptableJ-models”.) We make the following definitions, inductively.1) A Σ∗1 formula is just a Σ1 formula.

A predicate is Σ∗1 (Σ∗1, respectively) ifit is definable by a Σ∗1 formula with (without, respectively) parameters. ρM1= Σ∗1projectum of M = least ρ s.t.

there is a Σ∗1 subset of ωρ not in M. HM1= HMωρM1 =sets x in M s.t. M-card (transitive closure (x)) < ωρM1 .

For any x ∈M, M1(x) =First reduct of M relative to x = ⟨HM1 , A1(x)⟩where A1(x) ⊆HM1codes the Σ∗1*Research supported by NSF contract # 8903380.1

2theory of M with parameters from HM1 ∪{x} in the natural way: A1(x) = {⟨y, n⟩|the nth Σ∗1 formula is true at ⟨y, x⟩, y ∈HM1 }. A good Σ∗1 function is just a Σ1function and for any X ⊆M the Σ∗1 hull (X) is just the Σ1 hull of X.2) For n ≥1, a Σ∗n+1 formula is one of the form ϕ(x) ←→Mn(x) |= ψ, whereψ is Σ1.

A predicate is Σ∗n+1 (Σ∗n+1, respectively) if it is defined by a Σ∗n+1 formulawith (without, respectively) parameters. ρMn+1 = Σ∗n+1 projectum of M = least ρsuch that there is a Σ∗n+1 subset of ωρ not in M. HMn+1 = HMωρMn+1 = sets x in Ms.t.

M-card (transitive closure (x)) < ωρMn+1. For any x ∈M, Mn+1(x) = (n + 1)s.t.

reduct of M relative to x = ⟨HMn+1, An+1(x)⟩where An+1(x) ⊆HMn+1 codes theΣ∗n+1 theory of M with parameters from HMn+1∪{x} in the natural way: An+1(x) ={⟨y, m⟩| the mth Σ∗n+1 formula is true at ⟨y, x⟩, y ∈HMn+1}. A good Σ∗n+1 function fis a function whose graph is Σ∗n+1 with the additional property that for x ∈Dom(f),f(x) ∈Σ∗n hull (HMn ∪{x}).

The Σ∗n+1 hull (X) for X ⊆M is the closure of Xunder good Σ∗n+1 functions.Facts. (a) ϕ, ψΣ∗n formulas −→ϕ ∨ψ, ϕ ∧ψ are Σ∗n formulas.

(b) ϕΣ∗n or Q∗n (= negation of Σ∗n) −→ϕ is Σ∗n+1. (c) Y ⊆Σ∗n hull (X) −→Σ∗n hull (Y ) ⊆Σ∗n hull (X).

(d) f good Σ∗n function −→f good Σ∗n+1 function. (e) Σ∗n hull (X) ⊆Σ∗n+1 hull (X).

(f)There is a Σ∗n relation W(e, x) s.t. if S(x) is Σ∗n then for some e ∈ω,S(x) ←→W(e, x) for all x.

(g) The structure Mn(x) = ⟨HMn , An(x)⟩is amenable. (h) HMn = JAnωρMn where An = An(0).

(i) Suppose H ⊆M is closed under good Σ∗n functions and π : M −→M, Mtransitive, Range(π) = H. Then π preserves Σ∗n formulas: for Σ∗nϕ and x ∈M,M |= ϕ(x) ←→M |= ϕ(π(x)).Proof of (i).Note that H ∩Mn−1(π(x)) is Σ1-elementary in Mn−1(π(x)). Andπ−1[H ∩Mn−1(π(x))] = ⟨JAωρ, A(x)⟩for some ρ, A, A(x).

But (by induction on n)A = AMn−1 ∩JAωρ, A(x) = An−1(x)M ∩JAωρ and ρ = ρMn−1.⊣Theorem 1. By induction on n > 0 :1) If ϕ(x, y) is Σ∗n then ∃y ∈Σ∗n−1 hull (HMn−1 ∪{x})ϕ(x, y) is also Σ∗n.

32) If ϕ(x1 · · · xk) is Σ∗m, m ≥n and f1(x), · · · , fk(x) are good Σ∗n functions, thenϕ(f1(x) · · ·fk(x)) is Σ∗m.3) The domain of a good Σ∗n function is Σ∗n.4) Good Σ∗n functions are closed under composition.5) (Σ∗n Uniformization) If R(x, y) is Σ∗n then there is a good Σ∗n function f(x)s.t. x ∈Dom(f) ←→∃y ∈Σ∗n−1 hull (HMn−1 ∪{x})R(x, y) ←→R(x, f(x)).6)There is a good Σ∗n function hn(e, x) s.t.

for each x, Σ∗n hull ({x}) ={hn(e, x)|e ∈ω}.Proof. The base case n = 1 is easy (take Σ∗0 hull (X) = M for all X).

Now we proveit for n > 1, assuming the result for smaller n.1) Write ∃y ∈Σ∗n−1 hull (HMn−1 ∪{x})ϕ(x, y) as ∃¯y ∈HMn−1ϕ(x, hn−1(e, ⟨x, ¯y⟩))using 6) for n −1. Since hn−1 is good Σ∗n−1 we can apply 2) for n −1 to concludethat ϕ(x, hn−1(e, ⟨x, ¯y⟩)) is Σ∗n.

Since the quantifiers ∃e∃¯y ∈HMn−1 range over HMn−1they preserve Σ∗n-ness.2)ϕ(f1(X) · · ·fk(x)) ←→∃x1 · · · xk ∈Σ∗n−1 hull (HMn−1 ∪{x}) [xi = fi(x)for 1 ≤i ≤k ∧ϕ(x1 · · · xk)]. If m = n then this is Σ∗n by 1).

If m > n thenreason as follows: the result for m = n implies that An(⟨f1(x) · · ·fk(x)⟩) is ∆1 overMn+1(x). Thus Am−1(⟨f1(x) · · ·fk(x)⟩) is ∆1 over Mm−1(x).

So as ϕ is Σ∗m we getthat ϕ(f1(x) · · ·fk(x)) is also Σ1 over Mm−1(x), hence Σ∗m.3) If f(x) is good Σ∗n then dom(f) = {x|∃y ∈Σ∗n−1 hull of HMn−1∪{x}(y = f(x))}is Σ∗n by 1).4) If f, g are good Σ∗n then the graph of f ◦g is Σ∗n by 2). And f ◦g(x) ∈Σ∗n−1hull(HMn−1 ∪{x}) since the latter hull contains g(x), f is good Σ∗n and Fact c) holds.5) Using 6) for n −1, let R(x, ¯y) ←→R(x, hn−1(¯y)) ∧¯y ∈HMn−1.

Then R is Σ∗nby 2) for n −1 and using Σ1 uniformization on (n −1) s.t. reducts we can define agood Σ∗n function ¯f s.t.

R(x, ¯f(x)) ←→∃¯y ∈HMn−1R(x, ¯y). Let f(x) = hn−1( ¯f(x)).Then f is good Σ∗n by 4).6) Let W be universal Σ∗n as in Fact f).

By 5) there is a good Σ∗n g(e, x) s.t.∃y ∈Σ∗n−1 hull(HMn−1 ∪{x}) W(e, ⟨x, y⟩) ←→W(e, ⟨x, g(e, x)⟩) (and g(e, x) defined−→W(e, ⟨x, g(e, x)⟩)). Let hn(e, x) = g(e, x).

If y ∈Σ∗n hull ({x}) then for somee, W(e, ⟨x, y′⟩) ←→y′ = y so y = hn(e, x). Clearly hn(e, x) ∈Σ∗n hull ({x}) sincehn is good Σ∗n.⊣

4Section Two The Combinatorial Content of V = LIn this section we provide an axiomatic treatment of the Σ∗theory introducedin Section One. When establishing combinatorial principles in L[R], R a real, onemakes use of a standard Skolem system for R (defined below), of which the system ofcanonical Σ∗n Skolem functions for the JRα ’s constitutes the canonical example.

Ourprincipal goal is to provide combinatorial axioms for a system of functions whichguarantee that it is in fact a standard Skolem system for some real. These axiomscan then be used to formulate a single combinatorial principle which captures thefull power of Jensen’s fine structure theory.Some notation:For δ = λ + n, λ limit or 0 and n ∈ω, Seq(δ) denotes all finitesequences from λ together with all finite sequences from δ of length ≤n.

Let x ∗ydenote the concatenation of the sequences x, y. For λ limit or O, eJRλ denotes JRδwhere ω · δ = λ.A standard Skolem system for a real R is a system ⃗F = ⟨F δn|n > 0, δ ∈ORD, n >1 −→δ limit⟩where F δn is a partial function from ω × Seq(δ) to δ, obeying (A) –(E) below.

For any limit λ, x ∈Seq(λ), n ≥1 let Hλn(x) = {F λn (k, x)|k ∈ω} andif ¯λ = ordertype (Hλn(x)) let πn¯λλ(x) :¯λ −→λ be the increasing enumeration ofHλn(x). We say y ∈Hλn(x), for y ∈Seq(λ), if y∗∈Hλn(x) where y∗is a canonicalordinal code for y.

(A)(Monotonicity) δ1 ≤δ2 −→F δ11⊆F δ21 , x ∈Hλ1 (x) ⊆Hλ2 (x) ⊆· · · ⊆λfor limit λ, x ∈Seq(λ). (B)(Condensation)Let π = πn¯λλ(x).

Then for m ≤n, and ¯x ∈Seq(¯λ),π(F ¯λm(k, ¯x)) ≃F λm(k, π(¯x)). And ˜π(F¯λ+m1(k, ¯x)) ≃F λ+m1(k, ˜π(¯x)) for ¯x ∈Seq(¯λ +m), where ˜π is the extension of π to ¯λ + m obtained by sending ¯λ + i to λ + i.

(C) (Continuity) For limit λ, F λ1 = S{F δ1 |δ < λ}. There is a p ∈Seq(λ) suchthat for all x ∈Seq(λ) and y < λ, F λn+1(x) ≃y ifffor some z ∈Seq(λ), F ¯λn+1(¯x) ≃¯ywhere ¯λ = ordertype (Hλn(z)), πn¯λλ(z) sends ¯x, ¯y to x, y and p ∈Hλn(z).

(D)⟨F δn|δ < λ, n < ω⟩is uniformly ∆1( eJRλ ) for limit λ, in the parameter R.(E) For limit λ, Hλ1 (x) = λ ∩Σ1 Skolem hull of x in eJRλ for x ∈Seq(λ) and forsome fixed p ∈Seq(λ), SnHλn(x) = λ∩Skolem hull of x in eJRλ whenever p belongsto Hλn(x) some n, x ∈Seq(λ).Intuitively, F λn is a Σ∗n Skolem function for eJRλ and F λ+n1is the nth approximation

5to F λ+ω1.Proposition 2. For every real R there exists a standard Skolem system for R.Proof.

Let ψ 7−→ψ∗n be a recursive translation on formulas so that for limit λ,eJRλ+n |= ψ ←→eJRλ |= ψ∗n (where eJRα is defined just like eJα, but relativized toR). Fix a recursive enumeration ⟨ϕk(v)|k ∈ω⟩of ∆0 formulas with a predicate Rdenoting R and sole free variable v. Let

Xλ,Rn+1(e) = x, Xλ,Rn+1(f) = y) or(λ = 0 and x 0, n > 1 −→δ limit⟩as follows:(a) F n1 (k, x) ≃y iffLRn |= ∃w s.t. ⟨y, w⟩is

ϕk(⟨x, y, w⟩). (b) For λ limit, F λ1 = ∪{F δ1 |δ < λ}.

(c)For λ limit, n > 0, F λ+n1(k, x) ≃y ifffor some m ≤n, eJRλ+m |= (∃w s.t.⟨y, w⟩is

(d) For λ limit, n > 1, F λn is the canonical Σ∗n Skolem function for eJRλ (restrictedto ω × Seq(λ)) as in 6) of Theorem 1.The verification that ⃗F is a standard Skolem system for R is straightforward asCondensation is guaranteed by (c) above and (C), (E) are satisfied by letting p bethe full standard parameter for eJRλ .⊣An abstract Skolem system is a system ⃗F obeying properties (A), (B), (C) fromthe definition of standard Skolem system. We would like to prove that every ab-stract Skolem system is a standard Skolem system for some real.

However standardsystems share one further property which we must also impose:(Stability) For λ limit, x ∈Seq(λ) let π : ¯λ −→λ be the increasing enumerationof Hλ1 (x). Then π extends uniquely to a Σ1-elementary embedding of ⟨eJ ⃗F¯λ , ⃗F ↾¯λ⟩into ⟨eJ ⃗Fλ , ⃗F ↾λ⟩.

Also for λ limit there is p ∈Seq(λ) such that for all x ∈Seq(λ), ifπ : ¯λ −→λ is the increasing enumeration of Hλ(x) = SnHλn(x) and p ∈Hλ(x) thenπ extends uniquely to an elementary embedding of ⟨eJ ⃗F¯λ , ⃗F ↾¯λ⟩into ⟨eJ ⃗Fλ , ⃗F ↾λ⟩.Though stability is not combinatorial we shall see that any abstract Skolemsystem can be made stable without changing its “cofinality function”. This fact will

6enable us to formulate combinatorial principles which are universal for principleswhich depend only on cofinality.Theorem 3. The following are equivalent:(a)⃗F is a stable, abstract Skolem system.

(b)⃗F is a standard Skolem system in a CCC forcing extension of V.Note that (b) −→(a) follows easily, using the absoluteness of the concept of stability.We now develop the forcing required to prove (a) −→(b).Fix a stable, abstract Skolem system ⃗F and let M denote L[ ⃗F], Mλ = ⟨eJ ⃗Fλ , ⃗F ↾λ⟩for limit λ. The desired forcing P is a CCC forcing of size ω1 in M. It is designedso as to produce a generic real R which codes ⃗F ↾ω1 via a careful almost disjointcoding.We will demonstrate that R in fact codes all of ⃗F using condensationproperties of ⃗F.We begin our description of P. A limit ordinal λ is small if for some x ∈Seq(λ)and some n, Hλn(x) = λ.

Let n(λ) be the least n s.t. such an x exists and let pλbe the least p ∈Seq(λ) s.t.

Hλn(λ)(p) = λ. We now define a canonical bijection¯fλ : λ −→ω.

First let g : λ −→ω be defined by g(δ) = least k s.t. δ = F λn(λ)(k, pλ).Then ¯fλ(δ) = m if g(δ) is the mth element of Range(g) under < on ω.

Now letfλ : ω −→Mλ be g∗◦¯f −1λwhere g∗: λ −→Mλ is a canonical ∆∼1(Mλ) bijection.Now choose Aλ ⊆ω to code Mλ using fλ and let bλ+n(λ) be a function from ωto ω which is ∆n(λ)+1⟨Mω, Aλ⟩yet eventually dominates each function from ω toω which is ∆n(λ)⟨Mω, Aλ⟩. Also require that Range(bλ+n(λ)) ⊆∗Range(b¯λ+n) forall ¯λ < λ, n < ω where we have (inductively) defined b¯λ+n.

(⊆∗denotes inclusionexcept for a finite set. )We also define bλ+n for n = n(λ)+m, m > 0.

For this purpose define Fλ+n1(k, ¯x) ≃¯y to mean F λ+n1(k, x) ≃y where x(i) = λ+¯x(i) if ¯x(i) < n, ¯x(i) = n+x(i) otherwise(similarly for y). Let Aλ+m ⊆ω code.⟨Mλ, Fλ+n(λ)1, · · · , Fλ+n(λ)+m−11, F λn(λ), · · · , F λn(λ)+m−1⟩using fλ and let bλ+n(λ)+m be a function from ω to ω which is ∆n(λ)+m+1⟨Mω, Aλ+m⟩yet eventually dominates ∆n(λ)+m⟨Mω, Aλ+m⟩functions.

Also require that Range(bλ+n(λ)+m) ⊆∗Range(bλ+n(λ)+m−1). We use the bλ+n, n ≥n(λ) to facilitate the desired almost dis-joint coding.

7An index is a tuple of one of the forms ⟨λ + n, 1, k, ¯x, ¯y⟩, ⟨λ, n, k, ¯x, ¯y⟩where λis small, n ≥n(λ) and Fλ+n1(k, ¯x) ≃¯y, F λn (k, ¯x) ≃¯y, respectively. Let ⟨Ze|e ∈ω⟩be a recursive partition of ω −{0} into infinite pieces.

For each index x we definea “code” bx as follows:If x = ⟨λ + n, 1, k, ¯x, ¯y⟩, ⟨λ, n, k, ¯x, ¯y⟩then bx = bλ+n ↾Zewhere fλ(e) = ⟨n, 1, k, ¯x, ¯y⟩, ⟨0, n, k, ¯x, ¯y⟩, respectively. A restraint is a function ofthe form bx, x an index.

We sometimes view bx as a subset of ω by identifying itwith {⟨n, m⟩|bx(n) = m}, ⟨· , ·⟩a recursive pairing on ω.A condition in P is p = ⟨s, ¯s⟩where s : |s| −→2, |s| ∈ω, ¯s is a finite setof restraints and when i = ⟨m, k, x, y⟩< |s| then s(i) = 1 ←→F m1 (k, x) ≃y.Extension is defined by:(s, ¯s) ≤(t, ¯t) iffs ⊇t, ¯s ⊇¯t and s(i) = 1 −→t(i) = 1 ori /∈S ¯t. (Recall that we can think of bx ∈¯t as a subset of ω.

)This is a CCC forcing and a generic G is uniquely determined by the real R =S{s|(s, ¯s) ∈G for some ¯s}. Fix such a real R.Lemma 4.

⟨F δn|δ < λ, n < ω⟩is uniformly ∆1( eJRλ ) for limit λ, in the parameterR.Proof. By induction we define F λn , F λ+n1for λ limit or 0, n ∈ω.

If λ = 0 thenF n1 can be defined directly from R by the restriction we placed on s for conditions(s, ¯s). For λ limit, F λ1 is defined by induction and Continuity.

Also, induction andContinuity enable us to define F λn , F λ+n1provided n ≤n(λ) ̸= 1 or n(λ) is notdefined. Thus if λ is not small we’re done and otherwise we can define fλ, bλ+n, byinduction.

Let fλ(e) = ⟨n, 1, k, ¯x, ¯y⟩. Then Fλ+n1(k, ¯x) ≃¯y iff⟨λ + n, 1, k, ¯x, ¯y⟩is anindex iffR is almost disjoint from bλ+n ↾Ze.

The definition of F λn is similar, using⟨0, n, k, ¯x, ¯y⟩.⊣Our next goal is to establish a strong statement of the definability of the forcingrelation for P. For any infinite ordinal δ we let P(δ) denote those conditions in Pinvolving restraints with indices ⟨λ+n, 1, k, ¯x, ¯y⟩, ⟨λ, n, k, ¯x, ¯y⟩where λ+n < δ. Forp ∈P we let p ↾δ be obtained from p by discarding all restraints which are not ofthe above form.Lemma 5. (Persistence) let λ be small and for p ∈P(λ + ω) let p∗be obtainedby replacing each of its restraints of the form bx, x = ⟨λ + n, 1, k, ¯x, ¯y⟩, ⟨λ, n, k, ¯x, ¯y⟩by ⟨n, 1, k, ¯x, ¯y⟩, ⟨n, k, ¯x, ¯y⟩, respectively.

(Then p∗∈Mλ.) Suppose W ⊆P(λ +

8n(λ) + m) and W ∗= f −1λ [{p∗|p ∈W}] is Σn(λ)+m over ⟨Mω, Aλ+m⟩. Then D ={p ∈P(λ + n(λ) + m)|∃q ∈W(p ≤q) or ∀q ≤p(q /∈W)} is predense on P.Proof.

Given p ∈P we must find q ≤p such that q ↾λ + n(λ) + m belongs to D.Write p = (s, ¯s∪¯t) where p ↾λ+n(λ)+m = (s¯s), ¯s∩¯t = ∅. For each n let sn extends by assigning ⟨m0, m1⟩to 0 whenever ⟨m0, m1⟩/∈Dom(s) and m0 ≤m1 ≤n.

(We intend that n 7−→sn is recursive.) If (sn, ¯s) belongs to D for some n thenwe are done since (sn, ¯s ∪¯t) extends p. If not then we can define a Σn(λ)+m over⟨Mω, Aλ+m⟩function n 7−→tn so that for some ¯tn, (tn, ¯tn) ≤(sn, ¯s), (tn, ¯tn) ∈W,using the fact that Aλ+m codes ⟨Mλ, Fλ+n(λ)1, ·, Fλ+n(λ)+m−11⟩and hence “codes”P(λ + n(λ) + m).

Then f(m + 1) = length (tf(m)), f(0) = 0 defines a Σn(λ)+mover ⟨Mω, Aλ+m⟩function and every such function is eventually dominated by thefunction bλ+n(λ)+m. Thus there must be infinitely many ℓsuch that [f(ℓ), f(ℓ+ 1)]is disjoint from Range(bλ+n(λ)+m).

As Range(b) ⊆∗Range(bλ+n(λ)+m) for all b ∈¯tit follows that for some ℓ, [f(ℓ), f(ℓ+ 1)] is disjoint from ∪{Range(b)|b ∈¯t}. Butthen (tf(ℓ), ¯tf(ℓ) ∪¯t) = q ≤q and q ↾λ + n(λ) + m belongs to W ⊆D.⊣Corollary 6.

The forcing relation {(p, ϕ)|p ∈P(λ) and p ⊩ϕ in P(λ) where ϕ isa ranked sentence in Mλ} is Σ1 over Mλ, for limit λ.Proof. By induction on λ.

Note that if ¯λ < λ, ¯λ limit then for p ∈P(¯λ), ϕ rankedin M¯λ we have p ⊩ϕ in P(¯λ) iffp ⊩ϕ in P(λ). The reason is that by Lemma5, every P(λ)-generic is P(¯λ)-generic for ranked sentences, since by induction theP(¯λ) forcing relation for ranked sentences is Σ1 over Mλ.Thus we are done by induction if λ is a limit of limit ordinals.

Now suppose thatwe wish to establish the Corollary for λ + ω. We may assume that λ is small asotherwise P(λ + ω) is a set forcing in Mλ+ω.

Now any ranked sentence ϕ in Mλ isequivalent to a Σn(λ)+M statement about Mλ[R] for some m(R denoting the genericreal). But then by Lemma 5, p ⊩ϕ in P(λ + ω) iffp ⊩ϕ in P(λ + n(λ) + m) forp ∈P(λ + n(λ) + m).

As the latter is Σ1-definable over Mλ+ω, we are done.⊣Corollary 7. Suppose λ is small and W ⊆P(λ) is Σn(λ) over Mλ.

Let D = {p ∈P(λ)|∃q ∈W(p ≤q) or ∀q ≤p(q /∈W)}. Then D is predense on P.Proof.

Let m = 0 in Lemma 5.⊣Now we are prepared to finish the proof of the Characterization Theorem. Note

9that the only remaining condition to verify in showing that ⃗F is a Standard Skolemsystem is condition (E), where stability is used.Lemma 8. For λ limit, x ∈Seq(λ), Hλ1 (x) = λ ∩Σ1 Skolem hull of x in eJRλ .

Forλ limit there is p ∈Seq(λ) s.t. for all x ∈Seq(λ), Hλ(x) = SnHλn(x) = λ∩Skolemhull of x in eJRλ whenever p ∈Hλ(x).Proof.

We begin with the first statement. The inclusion Hλ1 (x) ⊆Σ1 Skolem hullof x in eJRλ follows from Lemma 4 and Continuity.

To prove the converse we makea definition: R is Σn −generic for P(λ) if for any Σn(Mλ) W ⊆P(λ) there existsp ∈G ∩P(λ), G denoting the generic determined by R, such that either p extendsa condition in W or p has no extension in W. By Corollary 7, if λ is small then Ris Σn(λ)-generic for P(λ).Suppose ϕ(x, y) is a Σ1 formula with parameter x. Let π :¯λ −→λ be theincreasing enumeration of Hλ1 (x) and let π(¯x) = x.

By Corollary 6 the forcingrelation for P(¯λ) is Σ1(M¯λ) is Σ1(M¯λ) for ranked sentences. Since R is Σ1-genericfor P(¯λ) there is p ∈G ∩P(¯λ) s.t.either p ⊩ϕ(¯x, ¯y) in P(¯λ) for some ¯y or p ⊩¬∃¯yϕ(¯x, ¯y) in P(¯λ).

Since ⃗F is stable we have that p ⊩¬∃yϕ(x, y) in P(λ) orp ⊩ϕ(x, y) where y = π(¯y). (Note that π extends to a Σ1-elementary embedding˜π : M¯λ −→Mλ such that ˜π(p) = p.) If λ is small then R is Σ1-generic for P(λ) andthus we have shown that λ ∩Σ1 Skolem hull of x in eJRλ is contained in Hλ1 (x).

Butthe above shows that if R is Σ1-generic for P(λ) for all small λ then R is Σ1-genericfor all λ. So we’re done.To prove the second statement, choose p to witness stability for ⃗F.

The directionHλ(x) ⊆Skolem hull of x in eJRλ follows again from Lemma 4. For the converse,handle each formula ψ(x, y) as in the Σ1 case, using stability and the assumptionthat p ∈Hλ(x).⊣This completes the proof of Theorem 3.Universal Combinatorial Principles.Inherent in any abstract Skolem system ⃗F is its cofinality function cof⃗F definedat limit ordinals λ as follows: cof⃗F (λ) = least ordertype of an unbounded subset ofλ of the form Hδn(γ ∪{p}) = S{Hδn(x∗p)|x ∈Seq(γ)} for some δ ≥λ, n ≥1, γ ≤λ,p ∈Seq(δ).

For any inner model M let cofM be the cofinality function of M. And

10cof = cofV .Lemma 9. Suppose ⃗F is an abstract Skolem system.

Then there exists a stableabstract Skolem system ⃗G such that cof⃗G = cofL[ ⃗F] .Proof. Let ⃗G be obtained from ⃗F just as in the proof of Proposition 2, with Rreplaced by ⃗F.

Then ⃗G is stable. Since ⃗G codes L[ ⃗F], cof⃗G(λ) ≤cofL[ ⃗F ](λ) all λ.But ⃗G is ⟨L[ ⃗F], ⃗F⟩-definable, so cof⃗G = cofL[ ⃗F] .⊣We now state our Universal Combinatorial Principle P.Principle P. There is an Abstract Skolem System ⃗F such that cof⃗F = cof .We show that P implies all “fine-structural principles” for L.Definition.

A fine-structural principle is a statement of the form ∃Aψ(A),where A denotes a class and ψ is first-order, such that:(a) For every real R and every Standard Skolem System ⃗F for R, L[R] |= ψ(A)for some A which is definable over ⟨L[ ⃗F], ǫ, ⃗F⟩. (b) If M, N are inner models of ZFC, A is amenable to both M, N, cofM = cofNand ⟨M, A⟩|= ψ(A) then ⟨N, A⟩|= ψ(A).Theorem 10.

P implies all fine-structural principles.Proof. Suppose M |= P with witness ⃗F and let ϕ be fine-structural.

Then cof⃗F =cofM = cofL[ ⃗F ], since L[ ⃗F] ⊆M. By Lemma 9 there is ⃗G amenable to M suchthat cof⃗G = cofM and ⃗G is stable.By the Characterization Theorem there isa (generic) real R such that ⃗G is a Standard Skolem System for R and henceL[R] |= ϕ with witness A definable over ⟨L[ ⃗G], ǫ, ⃗G⟩.

Then A is amenable to M andcofM = cof⃗G = cofL[R], so M |= ϕ.⊣□and Morass are fine-structural but ⋄is not. To obtain a universal principlewhich also implies ⋄we introduce a strengthening of P.Principle P∗.V = L[ ⃗F] where ⃗F is an Abstract Skolem System.Note that P ∗−→P, in view of Lemma 9.

We define an L-like principle to bea statement ϕ which is true in L[ ⃗F] whenever ⃗F is a Standard Skolem System. ByLemma 9 and the Characterization Theorem, P ∗implies all L-like principles.

Butunfortunately P ∗is not much weaker than V = L :

11Theorem 3.3. P ∗holds iffV = L[A], A ⊆ω1 where A is L-reshaped (α < ω1 −→α < ω1 in L[A ∩α]).Proof.

Suppose V = L[ ⃗F] for some Abstract Skolem System ⃗F. By Lemma 3.1 andthe Characterization Theorem, we may assume that ⃗F is a Standard Skolem Systemfor some real R. Now suppose that α is countable in L[R].

If α < λ limit, eJRλ |= αuncountable then ⃗F ↾λ can be recovered inductively from ⃗F ↾α, using continuityand condensation for Abstract Skolem Systems. We can also recover F λn for alln > 0 for such λ.

Thus if λ is least so that α is countable in eJRλ+ω, we see that α iscountable in L[ ⃗F ↾α]. So ⃗F ↾ω1 is L-reshaped.

The same argument shows that ⃗Fis definable over L[ ⃗F ↾ω1] so we have the desired conclusion.For the converse note that for L-reshaped A ⊆ω1 we can define the CanonicalSkolem System ⃗F A for A as we defined ⃗F R for reals R, provided we replace thehierarchy eJRδ , δ ∈ORD by eJAδ , δ ∈ORD and we assume that for λ < ω1, A∩[λ, ˆλ] =∅where ˆλ is the least limit so that eJA↾λˆλ|= λ is countable. Then L[ ⃗F A] = L[A]and ⃗F A satisfies the axioms for an Abstract Skolem System.

(In fact ⃗F A = ⃗F R forsome generic real R coding A. )⊣Though P ∗does not therefore have models which are very far from L, we hopethat its analogue in the context of core models will lead to an interesting class of“K-like” models.ReferencesJensen, R.B.

[72] The Fine Structure of the Constructible Hierarchy, Annals ofMathematical Logic.Jensen, R.B. [89] Handwritten notes on the Σ∗Theory.Jensen, R.B.

and Solovay, R.M. [68] Some Applications of Almost Disjoint Sets,in Mathematical Logic and the Foundations of Set Theory, Bar-Hillel, Editor.

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