PI-1-1 NORMAL FORM IN A REGULAR CARDINAL 137 Lemma 3

A normal form is given for Π1 formulas over Vκ for a regular cardinal κ. Using it, a new characterization of weakly compact cardinals is given. AMS Subject Classification: 03E55

are bounded.For n ≥ 1 a Σ 0 n (Π 0 n ) formula is a ∆ 0 0 formula, preceded by n alternating blocks of first-order quantifiers, the first of which is ∃ (∀).
A ∆ 1 0 formula is a formula which is ∆ 0 0 , Σ 0 n for some n, or Π 0 n for some n.For n ≥ 1 a Σ 1 n (Π 1 n ) formula is a ∆ 1 0 formula, preceded by n alternating blocks of second-order quantifiers, the first of which is ∃ (∀).For j = 1, 2 the notation Σ js n is used for "Σ j 1 over L s ∈ ", etc.Let Pow(x) denote the power set of x.Let C denote Pow(V κ ).Formulas of L s ∈ may be considered as defining predicates in a Cartesian product of copies of V κ and C (a "product space").Let N denote (V κ ) Vκ .Formulas of L f ∈ may be considered as defining predicates in a Cartesian product of copies of V κ and N .
Say that a predicate is Σ j n for j = 1, 2 (Π j n ) if it is defined by a Σ j n (Π j n ) formula.By predicate logic: • Any of these classes is closed under ∧ and ∨.
n or Π j n predicate is both Σ j m and Π j m .Various other closure properties may be shown; see [3] for example.
The notion of a ∆ j n predicate is defined in the usual manner, and by a standard abuse of language, a ∆ j n formula is a Σ j n or Π j n formula which defines a ∆ j n predicate.

L s
Proof.In a standard manner, t = u or t ∈ u may be rewritten in a ∆ 0 1 form involving only atomic formulas of the form y = F (x), w = x, or w ∈ x.
Lemma 2. The predicate Func(X) stating that X is the graph of a function is Π 0s 2 .
Proof.The formula ∀x∃p∃y(p = x, y ∧ X(p)) states that X is total.The formula ∀x∀p 1 ∀p 2 ∀y 1 ∀y Further facts can be stated; see for example theorems 15.XXV and 15.XXVI of [6] in the case of ω.

Normal Form
As in the case of ω, a normal form is more useful for formulas of L f ∈ (see [5] for example).In this case Skolem functions can be expressed in a simple manner.The notation x is used to denote a sequence x 1 , . . ., x k of variables; ∃ x denotes ∃x 1 • • • ∃x k , etc.A formula φ with free variables or parameters among F , x may be denoted φ( F , x).
Proof.This is a refinement of the Skolem normal form theorem of Section 4.2 of [4].Indeed, by this theorem φ can be written as where in ψ 1 F i may have multiple arguments (where in fact ψ 1 can be taken as open).
This may be rewritten as , where , where the x j are those not occurring in x.The formula may be transformed to The models in which the theorem holds include V κ for κ a cardinal.
Proof.Transform the first order part to Skolem normal form, prepend the original existential function quantifiers, and proceed as in the proof of the theorem.
Suppose κ is a regular cardinal, φ( F ) is a ∆ 0f 0 formula over V κ (possibly with first order parameters), and u ∈ V κ .Let φ ↾u denote φ with each occurrence of F i replaced by F i ↾ u.This may be written in ∆ 0f 1 form in a standard manner, as For a ∆ 0f 0 formula ψ with free variables x, F a predicate Q ψ (u, x, F ) will be defined by recursion on ψ, which holds iff the arguments of function applications are all in u.For a term t let Q t is ∨ s s ∈ u where s ranges over subterms other than t.
Proof.The first claim follows by induction on ψ.The second claim does also, using regularity for the bounded quantifiers.

Downsets
The notion of a tree is central to descriptive set theory.The generalization to a cardinal κ is better behaved if κ is assumed to be regular.Let A = {f ∈ V κ : f is a function}; by the restriction on κ if a function f : A may be generalized in a well-known manner.Let where Dom(f ) denotes the domain of the function f ; Dom( f ) may be written for the common domain.A (k) may be ordered by "componentwise inclusion" f ⊆ c g, which holds iff f i ⊆ g i for all i.The definition of a downset is essentially unchanged, and a branch is a vector F which is a branch componentwise. If A ∆ 0f 0 formula ψ( F ) may be translated to a ∆ 1f 0 formula ψ ′ ( f ) in a wellknown manner; by abuse of notation ψ( f ) will be written for this formula.The notation " x ∈ c u" will be used for Theorem 8. Suppose κ is a regular cardinal.Suppose φ is a formula ∀ F ∃ xψ( x, F , G) where ψ is ∆ 0f 0 (and for brevity has no first order parameters).Let Then D is a downset, and in V κ , φ holds iff D @ G is unbranched.

Proof. It follows readily by induction on
It then follows that D is a downset.Using lemma 7 it follows readily that if φ is true then D @ G is unbranched; and conversely.

Weak Compactness
It is a well-known fact that a cardinal κ is weakly compact iff it is Π 1 1 -indescribable, that is, for any Π 1s 1 sentence φ, if |= Vκ φ then |= V λ φ for some regular cardinal λ < κ.
Theorem 9.For a regular cardinal κ the following are equivalent.a.For any Proof.a⇒b is immediate.Suppose b holds.Suppose φ is a Π 1s 1 formula ∀Xψ(X, Y ).Let φ ′ be ∀F ψ ′ (F, χ Y ) where ψ ′ is the translation of ψ obtained using lemma 3.That b⇒c follows, noting that χ Y ∩ V λ is total.
Say that a regular cardinal κ has the downset property iff, whenever D ⊆ N is an unbranched downset, there is a regular cardinal λ < κ such that D ∩ V λ is unbranched.
Theorem 10.If a regular cardinal κ has the downset property then κ is weakly compact.
Proof.Suppose φ is a Π 1f 1 formula, which may be assumed to be in normal form ∀ F ∃ xψ where ψ is ∆ 1f 1 , has no first order parameters, and for a second order parameter G and a regular cardinal κ G ∩ V λ is total.Suppose φ is true in V κ .Then the downset D @ G of theorem 8 is unbranched.By hypothesis D @ G ∩ V λ is unbranched for some regular cardinal λ < κ.By theorem 8 φ is true in V λ .
Theorem 11.If κ is a weakly compact cardinal then κ has the downset property.
Proof.There is a Π 1f 1 sentence in the parameter D which is true in V λ for a regular cardinal λ iff D is an unbranched downset.

Concluding Remarks
It is a question of considerable interest, whether the existence of weakly compact cardinals can be justified by postulating sufficiently long stationary set chains (see [2]).Although the results given here do not shed much light on the problem, they do indicate that methods from descriptive set theory can be adopted to some extent.This suggests that further research in this area might be of interest.