Validación de la propuesta

The set WF is Π1

1. IfA⊆ N is Π11, we know thatA≤wWF. We will show that

the reduction f can be chosen computable. There is a recursive tree T, such that

x∈A⇔ ∀y (x, y)6∈[T]⇔T(x)∈WF. The functionx7→T(x) is computable.

A similar construction gives rise to a Π1

1-complete (for≤m) subset ofN.

Let O = {e ∈ N : φe is the characteristic function of a Well-founded tree

Te⊆N<ω}. Thene∈O if and only if

i)∀σ φe(σ)↓

ii)∀σ∈N<ω _∀τ∈N<ω _{((σ⊆τ∧φ}

e(τ) = 1)→φe(σ) = 1).

iii)∀f :N→N<ω_∃n(φ

e(f(n)) = 0∨φe(f(n+ 1)) = 0∨f(n)6⊂f(n+ 1)).

Conditions i) and ii) are Π0

2 while iii) is Π11. ThusO is Π11.

Proposition 5.35 O isΠ1

1-complete.

Proof We will argue that N\O is Σ1

1-complete. Suppose A∈ Σ11. There is B ⊆N× N in Π0

1 such that n ∈ A if and only if ∃x (n, x) ∈ B. There is a recursive treeT ⊆N×N<ω _{such that (n, x)∈Aif and only if (n, x|m)∈T for}

allm∈N. There is a recursivef :N→Nsuch thatφf(n) is the characteristic

function of{σ: (n, σ)∈T}. Thenφf(n) is the characteristic function of a tree

Tnand

n∈A⇔Tn 6∈WF⇔f(n)6∈O.

O will play a very important role in effective descriptive set theory. As a first example, we will show how once we know the complexity of a set, we can say find relatively simple elements of the set.

Lemma 5.36 SupposeT ⊆N<ω _{is a recursive tree. If[T]6=∅, there isx∈[T]}

withx≤T O.

Proof There is a recursive function f such that φf(σ) is the characteristic function ofTσ for allσ∈N<ω. We build∅=σ0⊂σ1. . .withσi ∈T such that

[Tσi]6=∅. Givenσi. Letn∈Nbe least such thatσibn∈T andf(σi)6∈O. Corollary 5.37 (Kleene Basis Theorem) If A ⊆ N is Σ1

1 and nonempty,

there isx∈Awith x≤T O. Proof There is a Π0

1-setB ⊆ N ×Nsuch thatx∈Aif and only if∃y(x, y)∈B. By the previous lemma there is (x, y)∈B with (x, y)≤T O. Clearlyx≤T O.

Using the Uniformization Theorem, we can find definable elements of Π1 1- sets.

Proposition 5.38 If A⊆ N isΠ1

1, there isx∈A such thatx∈∆12.

Proof Uniformizing {0} ×A, we find x ∈ N such that B = {(0, x)}is Π1 1. Then

x(n) =m ⇔ ∃y ((0, y)∈B∧y(n) =m) ⇔ ∀y ((0, y)6∈B∨y(n) =m) The first definition is Σ1

2, while the second is Π12.

We next need to understand the possible heights of recursive trees.

Definition 5.39 We say that an ordinalα isrecursive if there is a recursive setA⊆Nand≺a recursive linear order of Asuch that (A,≺)∼= (α, <).

Lemma 5.40 a) If αis a recursive ordinal and β < α, then β is a recursive ordinal.

b) Ifαis a recursive ordinal, thenα+ 1 is a recursive ordinal.

c) Suppose f :N→N, g:N→Nare recursive functions such that Pf(n) is

a program to compute the characteristic function ofAn, Pg(n) is a program that

computes the characterisitic function of≺n a well-order ofAn and(An,≺n)has

order-typeαn. Thensupαn is a recursive ordinal.

Proof a) and b) are routine. For c) we show that PAn is a recursive well-

order. Let A={(n, m) :Mf(n)(m) = 1}and let (n, m)≺(n0, m0) if and only

ifn < n0 _{or n=n}0 _{and m≺}

nm0. Then (A,≺) is a recursive well-order. Letα

be the order type ofA. Then αn ≤αfor all n. Since supαn ≤α, supαn is a

recursive ordinal.

There are only countably many recursive well-orders. Thus there are only countably many recursive ordinals.

Definition 5.41 Letωck

1 be the least non-recursive ordinal. We call this ordinal theChurch–Kleene ordinal.

More generally for anyx we letωx

1 be the least ordinal not recursive inx. We need to be able to compare ordinals with trees.

Definition 5.42 Forσ, τ ∈N<ω _{we sayσ / τ ifτ⊂σor there is annsuch that}

σ(n)6= τ(n), but σ(m) = τ(m) for allm < n. We call /the Kleene–Brower order.

Exercise 5.43 a)/is a linear order ofN<ω_.

b) If T ⊆ N<ω _{is a tree, then T is well-founded if and only if (T, /) is a}

well-order.[Hint: Ifσ0, σ1, . . .is an infinite descending sequence in (T, /), define xinductively byx(n) = leastmsuch that (x(0), . . . , x(n−1), m)/ σi for some

i. Prove thatx∈[T].] c) Prove thatωck

1 = sup{ρ(T) :T ⊆N<ω a recursive well founded tree}. The proof of 5.15 actually shows the following.

Theorem 5.44 i) The set{(S, T) :ρ(S)≤ρ(T)}isΣ1 1.

ii) There is R∈Σ1

1(N × N) such that ifT ∈WF, then{S: (S, T)∈R}= {S:ρ(S)< ρ(T)}.

Corollary 5.45 (EffectiveΣ1

1-Bounding) i) If A ⊆ O is Σ11, then there is α < ωck

1 such that ρ(T)< α for allT ∈A.

ii) IfA⊆WF andA∈Σ1

1. Thenρ(T)< ωck1 for allT ∈A.

Proof If either i) or ii) fails, then O={e:φe is the characteristic function of

a recursive tree∃T ∀σ∈N<ω_{((σ∈T ↔φ}

e(σ) = 1) and ∃S∈A ρ(T)≤ρ(S))}

is Σ1

1, a contradiction.

Exercise 5.46 Prove that ifA⊆ N is ∆1