The purpose of this note is to prove the following.

arXiv:math/9212202v1 [math.LO] 2 Dec 1992A Large Π12 Set, Absolute for Set ForcingsSy D. Friedman*M.I.T.The purpose of this note is to prove the following.Theorem. Let κ be an L-cardinal, definable in L. Then there is a set of reals X,class-generic over L, such that(a) L(X) |= Card = CardL and X has cardinality κ.

(b) Some fixed Π12 formula defines X in all set-generic extensions of L(X).By L´evy-Shoenfield Absoluteness, any Π12 formula defining X in L(X) defines asuperset of X in each extension of L(X). The point of (b) is that this superset isjust X in set-generic extensions of L(X).

If O# exists then X as in the Theoremactually exists in V, though of course it will be only countable there.The basic idea of the proof comes from David [82]. In his paper a real R class-generic over L is produced so that {R} is Π12, uniformly for set-generic extensionsof L(R).

The added technique here is to use “diagonal supports” to take a largeproduct of David-style forcings.Here are some further applications of the Theorem and its proof.Corollary 1. Assume consistency of an inaccessible cardinal.

Then it is consistentfor the Perfect Set Property to hold for Σ∼12 sets yet fail for some Π12 set.Proof. Using the Theorem get a Π12 set X which has cardinality κ in L(X), κ = leastL-inaccessible, and which has a Π12-definition uniform for set-generic extensions.Then gently collapse κ to ω1 and add ω2 Cohen reals.

In this extension, ω1 > ωL(R)1for each real R and X is a Π12 set of cardinality ω1 < ω2 = 2ℵ0.⊣Corollary 2. Assume consistency of an inaccessible.

Then it is consistent that thePerfect Set Property holds for Σ∼12 sets and there is a Π12 well ordering of some setof reals of length ℵ1000. *Research supported by NSF contract #9205530-DMA.1

2The latter answers a question of Harrington.

3The Proof.We modify the construction of David [82] to suit our purposes. First we describethe α+-Souslin tree Tα in L, where α is a successor L-cardinal: Tα has a uniquenode on level 0 and exactly 2 immediate successors on level β + 1 to each node onlevel β, for β < α+.

If β < α+ is a limit of cofinality < α then level β assigns a topto each branch through the tree below level β. Now suppose β < α+ has cofinalityα.

Let P be the forcing consisting of pairs (γ, f) where γ < β and f is a functionfrom γ into the nodes at levels < β, with extension defined by (γ′, f ′) ≤(γ, f) iffγ′ ≥γ, f ′(δ) tree-extends f(δ) for each δ < γ. Choose G to be P-generic over Lβ∗where β∗= largest p.r.

closed β∗≥β such that β∗= β or Lβ∗|= card(β) > α.Then the nodes on level β are obtained by putting tops on the branches defined by{f(δ)|(γ, f) ∈G some γ} for δ < β. This completes the definition of the α+-Souslintree Tα.Now fix an L-definable cardinal κ and also fix an L-definable 1 −1 functionF : κ × ω × ORD −→Successor L-cardinals greater than κ.

The forcing P(γ, n),γ < κ and n < ω, is designed to produce a real R(γ, n) coding branches throughTα whenever α is of the form F(γ, n, δ) for some δ. This forcing is obtained bymodifying the Jensen coding of the empty class(see Beller-Jensen-Welch [82]) asfollows: In defining the strings s : [α, |s|) −→2 in Sα, require that Even (s) codea branch through Tα if α ∈Card(γ, n) = {F(γ, n, δ)|δ ∈ORD}.

Also use David’strick to create a Π12 condition implying that branches through the appropriate treesare coded: for any α, for s to belong to Sα require that for ξ ≤|s| and η > ξ, ifLη(s ↾ξ) |= ξ = α+ + ZF −+ Card = CardL then Lη(s ↾ξ) |= for some γ∗< κ∗,Even(s ↾ξ) codes a branch through T ∗α∗whenever α∗∈Card∗(γ∗, n), where κ∗, T ∗α∗,Card∗(γ∗, n) are defined in Lη as were κ, Tα, Card(γ, n) in L. The ≤α-distributivityof P(γ, n)α(= the forcing at and above α) is established as in David [82], with oneadded observation: if α′ ∈Card(γ, n) then we have to be sure that Even (pα′) codesa branch through Tα′, where p arises as the greatest lower bound to an α-sequenceconstructed to meet α-many open dense sets. There is no problem if α′ > α sincethen Tα′ is ≤α-closed.

If α′ = α then the property follows from the definitionof level |pα| of Tα, since we can arrange that Even (pα) is sufficiently generic forTα ↾(levels < |pα|). (In fact the latter genericity is a consequence of the usual

4construction of the α-sequence leading to p.)The forcing P(γ), γ < κ, is designed to produce a real R(γ) such that n ∈R(γ) iffR(γ) codes a branch through Tα for each α in Card (γ, n). A condition isp ∈QnP(γ, n) where p(n)(0) (a finite object) is (∅∅) for all but finitely many n.Extension is defined by q ≤p iffq(n) ≤p(n) in P(γ, n) unless n is not of the form2n03n1 or n = 2n03n1 where q(n0)0(n1) = 0, in which case there is no requirementon q(n).

A generic G can be identified with the real {2n3m|p(n)0(m) = 1 for somep ∈G} = R(γ). The forcing at or above α, P(γ)α, obeys “quasi-distributivity”: ifDi, i < α are predense below p then there are q ≤p and di ⊆Di, i < α such thateach di is countable and predense below q.

This is established as in David [82] by“guessing at ⟨p(n)(0)|n ∈ω⟩” and yields cardinal preservation.Our desired forcing P is the “diagonally supported” product of the P(γ), γ <κ. Specifically, a condition is p ∈Qγ<κP(γ) where for infinite cardinals α < κ,{γ|p(γ)(α) ̸= (∅, ∅)} has cardinality ≤α and in addition {γ|p(γ)(0) ̸= (∅, ∅)} isfinite.

Quasi-distributivity for Pα = forcing at or above α follows just as for P(γ)α.The point of the diagonal supports is that for infinite successor cardinals α, P factorsas Pα ∗PGα where Gα denotes the Pα-generic and PGα is α+ −CC. Thus we getcardinal-preservation.Now note that if ⟨R(γ)|γ < κ⟩comes from (and therefore determines) a P-generic then n ∈R(γ) −→R(γ) codes a branch through Tα for α in Card (γ, n).Conversely, if n /∈R(γ) then there is no condition on extension of conditions in P(γ)to cause R(γ) to code a branch through such Tα.

In fact, by the quasi-distributivityargument for Pα, given any term τ for a subset of α+ and any condition p, we canfind β < α+ of cofinality α and q ≤p such that q forces τ ∩β to be one of α-manypossibilities, each constructed before β∗, where β = |qα|. Thus q forces that τ is nota branch through Tα, so we get: n ∈R(γ) iffR(γ) codes a branch through each Tα,α ∈Card(γ, n) iffR(γ) codes a branch through some Tα, α ∈Card(γ, n).

The codingis localized in the sense that if n ∈R(γ) then whenever Lη(R(γ)) |= ZF −+Card =CardL, there is γ∗< κ∗such that Lη(R(γ)) |= R(γ) codes a branch through T ∗α∗whenever α∗∈Card∗(γ∗, n), where κ∗, T ∗α∗, Card∗(γ∗, n) are defined in Lη just asκ, Tα, Card(γ∗, n) are defined in L. The latter condition on R(γ) is sufficient toknow that R(γ) is equal to one of the intended R(γ), γ < κ, even if we restrict

5ourselves to countable η. With that restriction we get a Π12 condition equivalentto membership in X = {R(γ)|γ < κ}.

Since set-forcing preserves the Souslin-nessof trees at sufficiently large cardinals, the above Π12 definition of X works in anyset-generic extension of L(X). This completes the proof of the Theorem.Proof of Corollary 2.

As in the proof of Corollary 1 we can obtain X = {R(γ)|γ <κ}, κ = 999th cardinal after the least L-inaccessible, which has a Π12 definitionuniform for set-generic extensions of L(X), where CardL(X) = CardL . We canguarantee that Y = {⟨R(0), R(γ1), R(γ2)⟩|0 < γ1 ≤γ2 < κ} also has such a uniformΠ12 definition, using the following trick: Design R(0) so that u ∈R(0) ⇐⇒Even(R(0)) codes a branch through Tα for each α in Card (0, n), and so that Odd(R0)almost disjointly codes {⟨R(γ1), R(γ2)⟩|0 < γ1 ≤γ2 < κ}.

Thus, for R ∈L(X), R∗is almost disjoint from Odd(R0) iffR = ⟨R(γ1), R(γ2)⟩for some 0 < γ1 ≤γ2 < κ,where R∗= {n|n codes a finite initial segment of R}. The former requires only avery small modification to the definition of the P(0) forcings.

The latter requiresonly a small modification to the definition of P : take the diagonally-supportedproduct as before, but restrain p(0) for p ∈P so as to affect the desired almostdisjoint coding. These finite restraints do not interfere with the quasi-distributivityargument for P.Now we have the desired Π12 definition for Y = {⟨R(0), R(γ1), R(γ2)⟩|0 < γ1 ≤γ2 < κ} : R belongs to Y iffR = ⟨R0, R1, R2⟩where R0 = R(0) and ⟨R1, R2⟩∗isalmost disjoint from R0 and R1, R2 belong to X.

Since R(0) is uniformly definableas a Π12-singleton in set-generic extensions of L(X), this is the desired definition.Of course, using Y we obtain a Π12 well-ordering of length κ. Finally as in the proofof Corollary 1, gently collapse κ to ω1 and we have ω1 > ωL(R)1for each real R witha Π12 well-ordering of length ℵ1000.⊣Remarks.The same proof gives length ℵα for any L-definable α.

We can alsoadd Cohen reals so that the continuum is as large as desired, without changing themaximum length of a Π12 well-ordering.It is possible to show that if O# exists then there is a Π12 set X such that X haslarge cardinality in L(X). But this requires the more difficult technique of Friedman[90].

6References[82] Beller-Jensen-Welch, Coding The Universe, Cambridge University Press. [82] David, A Very Absolute Π12-Singleton, Annals of Pure and Applied Logic 23.

[90] Friedman, The Π12-Singleton Conjecture, Journal of the American Mathematical Society 3,Number 4. [77] Harrington, Long Projective Well orderings, Annals of Mathematical Logic 12.

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