[PDF] - Duality for r.e. sets with applications V. Yu. Shavrukov PDF Document

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Duality for r.e. sets with applications

V. Yu. Shavrukov

v.yu.shavrukov@gmail.com Dagstuhl 15441, 2015–10

Duality for r.e. sets with applications

V. Yu. Shavrukov

v.yu.shavrukov@gmail.com Dagstuhl 15441, 2015–10

Duality for r.e. sets with applications

Good morning.
First of all, I have to offer my apologies to those members of the

audience who have heard fragments of my talk on previous occasions.

We are going to talk about the lattice of r.e. sets and how classical

Priestley duality for bounded distributive lattice applies to it.

Another important ingredient will be non-standard models arithmetic —

we use those to represent individual prime filters on the lattice.

Rather than opt for a comprehensive survey, I am going to briefly cover

some prerequisite basics and focus on a particular application.

That application, in my view, supports the thesis that duality can play

more than just decorative role with respect to traditional questions in the theory of r.e. sets.

You can see how a point like that can be important on some kind of

personal level to a proponent of duality methods.

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(E ∗)⋆

E = r.e. subsets of ω, ∪, ∩, ∅, ω) — the lattice of r.e. sets E ∗ = E /fin

(E ∗)⋆ = the set of prim(e filter)s on E ∗, ⊆, π — the Priestley dual

larger prime filters smaller prime filters

Let X be r.e. X⋆ is the picture of X. p ∋ X ⇐⇒ p ∈ X⋆ X⋆ is ↑-closed. R⋆ is -closed ⇐⇒ R is recursive.

Observation. Reduction Principle

=⇒ ⊆ is ↑-forestlike on (E ∗)⋆.

(E ∗)⋆ E = r.e. subsets of ω, ∪, ∩, ∅, ω) — the lattice of r.e. sets E ∗ = E /fin (E ∗)⋆ = the set of prim(e filter)s on E ∗, ⊆, π — the Priestley dual

larger prime filters smaller prime filters Let X be r.e. X⋆ is the picture of X. p ∋ X ⇐⇒ p ∈ X⋆ X⋆ is ↑-closed. R⋆ is -closed ⇐⇒ R is recursive.

Observation. Reduction Principle

=⇒ ⊆ is ↑-forestlike on (E ∗)⋆.

Duality for r.e. sets with applications The dual (E ∗)⋆

The lattice E of r.e. sets consists of all recursively enumerable subsets of ω together with the set

theoretical operations. Which makes it a bounded distributive lattice.

When you quotient this by finite differences between r.e. sets, you get the lattice called E ∗.
(E ∗)⋆ is then the classical Priestley space of E ∗. That’s the ordered topological space comprised by

the collection of all prime filters on the lattice of r.e. sets which we are going to just call primes or even points, together with the order relation of inclusion and an appropriate topology.

Note that the two asterisks in the notation for the dual are very different: The inner one says goodbye to

finite differences while the outer asterisk gets you from the lattice to the dual space.

When you attempt to visualize this space, you probably get a picture like this.
We think of the smaller primes going towards the bottom, and the larger primes, towards the top.
Given an r.e. set X, we can draw the picture of this set and call it X-star. It consists of all primes that

contain the set X. We tend to identify it with the r.e. set X.

We have an equivalence between a prime filter p containing X and the same prime filter, now thought

as a point in he dual space, lying within the picture of X. This is probably the key slogan of pictorial duality.

Thus one way to think of the dual space is as a canvas for a kind of Venn diagrams.
Observe that the picture of any r.e. set is upward closed, for once an r.e. set belongs to some prime

filter, it has to belong to all larger ones.

The picture of a recursive set is the both upward and downward closed, because the complement of a

recursive set is r.e. and must therefore also be upward closed.

Finally, let us point out that the inclusion ordering is upwards-forestlike — this is an easy consequence
f the Reduction Principle for r.e. sets.

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Maximal, r-maximal, and hyperhypersimple r.e. sets

M⋆ M is maximal ⇐⇒ M⋆ is a singleton Q is r-maximal ⇐⇒ min Q⋆ is a singleton. Q⋆ H⋆ H is hhsimple ⇐⇒ H⋆ ⊆ min(E ∗)⋆

Maximal, r-maximal, and hyperhypersimple r.e. sets M⋆ M is maximal ⇐⇒ M⋆ is a singleton Q is r-maximal ⇐⇒ min Q⋆ is a singleton. Q⋆ H⋆ H is hhsimple ⇐⇒ H⋆ ⊆ min(E ∗)⋆

Duality for r.e. sets with applications The dual Maximal, r-maximal, and hyperhypersimple r.e. sets

We are now going to look at pictures of typical representatives of some familiar classes of

r.e. sets.

An r.e. set M is maximal if any r.e. superset of M only differs finitely from either ω or M
itself. This is equivalent to the complement of the picture of M only having a single

element — you cannot split that complement by any r.e. set into two infinite halves — that single point is either in or out.

Maximality can be generalized in two orthogonal directions.
An r.e. set Q is r-maximal if its picture covers all minimal points of the dual space except
ne, which is then called the heel of that set.
The original definition is that no recursive set splits the complement of Q into two infinite

halves — the complement is the white area in the picture. I.o.w., that complement is r-cohesive. You cannot split it by any picture that is both upwards and downwards closed. If there was more than one point at the bottom of the complement, you could easily split those apart.

Finally, an r.e. set H is hyperhypersimple if the complement of its picture consists of

minimal points of the dual space only. The complement thus forms a single-element-wide strip at the bottom of the dual space.

Hyperhypersimple sets were introduced by Emil Post, as the next step after simple and

hypersimple sets. These are classes of sets with smaller and smaller complements, and I guess we should be thankful that it was not him who invented the maximal sets.

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R.e. prime powers

[p] = (Pp/≡p, 0, 1, +, ×) where p ∈ (E ∗)⋆

Pp is the set of all partial recursive functions f : ω → ω with dom f ∈ p

f ≡p g ⇐⇒ f and g agree on some set in p Σ1 Łoś Lemma. For Σ1 formulas σ(x), [p] |= σ([ f]p) ⇐⇒

i ∈ ω

|= σ( f (i))

∈ p.

[p] |= TA2 |− I∆0 + exp, IΣ−

1

but [p] | IΣ1, BΣ1 (TA2 = Th∀∃ ) Equality between r.e. sets and between (partial) recursive functions is absolute for models of TA2. xp = [id]p — the distinguished generator X ∈ p ⇐⇒ [p] |= xp ∈ X (r.e. X) tp[p]

Σ1

xp = p [p] |= f (xp) = [ f]p ( p ∋ dom f) [p] |= f ([g]p) = [ f ◦ g]p

R.e. prime powers [p] = (Pp/≡p, 0, 1, +, ×) where p ∈ (E ∗)⋆ Pp is the set of all partial recursive functions f : ω → ω with dom f ∈ p f ≡p g ⇐⇒ f and g agree on some set in p Σ1 Łoś Lemma. For Σ1 formulas σ(x), [p] |= σ([ f]p) ⇐⇒

Duality for r.e. sets with applications R.e. prime powers R.e. prime powers

So let p be a prime filter on E ∗. We are now going to associate to it a model of arithmetic.
The r.e. prime power correponding to p is formed by all partial recursive functions that are defined on a set in the filter p together with

coordinate-wise arithmetical operations. Two such functions are identified if they agree on a set in p.

For r.e. prime powers, we have a restricted version of Łoś’ Lemma: a Σ1 formula holds in the prime power if and only if it holds in on a set of

coordinates which is large in the sense of the prime filter.

It follows that each r.e. prime power is a model of the true forall-exists arithmetic, that is, the 1st order theory axiomatized by all forall-exists

sentences true in the natural numbers. This in particular implies that it is a model of elementary arithmetic or even parameter-free Σ1 induction.

On the downside, full Σ1 induction and even Σ1 collection fail in every r.e. prime power.
Another nice feature of TA2-models is that equality between r.e. sets and partial recursive functions (with standard index) is absolute between

models of that theory. This often allows to treat r.e. set or partial recursive function as constants, without specifying which particular r.e. or computable index is selected.

With an r.e. prime power, the boldface x (perhaps with an appropriate subscript) denotes the distinguished generator, namely the equivalence

class of the identity function. Just as with usual ultrapowers, it must be non-standard.

In view of Łoś’ Lemma, the filter p can be recovered from the r.e. prime power as the collection of all r.e. sets that the distinguished generator

belongs to. I.o.w., the Σ1 type of the generator is exactly the original prime filter.

The value of a partial recursive function at the recursive generator is the equivalence class of that function. Incidentally, f (xp) is defined if and
nly if the domain of f is large in the sense of the prime filter p.
More generally, the value of a partial recursive f at an arbitrary element of the prime power given by some other partial recursive function g,

if defined, is the equivalence class of the composition — clearly, this is the only reasonable way for these functions to behave.

Two subclasses of r.e. prime powers have been studied earlier — I will explain in the next slide.
The reason I like r.e. prime powers is that here we have the prime filter p realized as the Σ1 type of an element of a model in a minimalist

fashion — all the elements of the prime power have a direct relation to that type. Sure, you can also have such a realization in a model of full true arithmetic, but there is no minimal or canonical such model, so in any model of true aithmetic you also get lots of elements whose presence and properties are of accidental nature.

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Recursive and r.e. ultrapowers

Definition. M |= TA2 is existentially complete (among TA2-models)

⇐⇒ M is Σ1-elementary in any TA2-extension p maximal ⇐⇒ [p] existentially complete (r.e. ultrapower)

minimal primes recursive ultrapowers (Hirschfeld) maximal primes (Hirschfeld & Wheeler) e.c. r.e. ultrapowers

(E ∗)⋆

larger primes smaller primes

Let u be an ultrafilter on R ∗. u ⊆ ¯ u = { r.e. X | ∃R ∈ u R ⊆ X } ∈ min(E ∗)⋆ All minimal primes are of the form ¯ u [u] = (T /≡u, 0, 1, +, ×) — recursive ultrapower (aka Nerode semiring) where T are total recursive functions, ≡u is agreement mod u [u] ∼ = [¯ u] In [u], each partial recursive value of xu is a total recursive value of xu

Recursive and r.e. ultrapowers

Definition. M |= TA2 is existentially complete (among TA2-models)

⇐⇒ M is Σ1-elementary in any TA2-extension p maximal ⇐⇒ [p] existentially complete (r.e. ultrapower) minimal primes recursive ultrapowers (Hirschfeld) maximal primes (Hirschfeld & Wheeler) e.c. r.e. ultrapowers (E ∗)⋆ larger primes smaller primes Let u be an ultrafilter on R ∗. u ⊆ ¯ u = { r.e. X | ∃R ∈ u R ⊆ X } ∈ min(E ∗)⋆ All minimal primes are of the form ¯ u [u] = (T /≡u, 0, 1, +, ×) — recursive ultrapower (aka Nerode semiring) where T are total recursive functions, ≡u is agreement mod u [u] ∼ = [¯ u] In [u], each partial recursive value of xu is a total recursive value of xu

Duality for r.e. sets with applications R.e. prime powers Recursive and r.e. ultrapowers

We now address previously investigated subclasses of r.e. prime powers and the position of the corresponding primes in the dual space.
Let u be an ultrafilter on the Boolean algebra of recursive sets, again, modulo finite differences. Let u-bar be the collection of all r.e. sets that

have a recursive subset which is an element of u. This is the least filter on E ∗ containing u.

u-bar happens to be a prime filter. It is a minimal prime in the sense of inclusion ordering. Furthermore, all minimal primes are obtained in this

way — just take the collection of all recursive members of that prime and apply the bar. I.o.w., the minimal prime filters are exactly those that have a base of recursive sets.

We can use u to define a power-like model [u] taking all total recursive functions and identifying two functions when they agree on some set

in u.

The result is called a recursive ultrapower or Nerode semiring. Recursive ultrapowers were defined under that name by Joram Hirschfeld who is

responsible for their pioneering study although they briefly cropped up earlier in Nerode’s work on regressive isols.

The inclusion of total recursive functions into partial ones translates into isomorphism between the recursive ultrapower of u and the r.e. prime

power of ¯ u.

In a recursive ultrapower, the value of any partial recursive function at x, if it exists, is equal to the value of some total recursive function at x

(this does not generalize to other elenments of recursive ultrapowers) — just take the appropriate recursive subset of the domain of the function, and put the values to 0 outside that recursive subset, obtaining an equivalent total recursive function.

A model M of forall-exists arithmetic is called existentially complete (among TA2-models) if that model is a Σ1-elementary submodel of any

extension which is also a model of TA2.

A prime p is maximal in the dual space if and only if the corresponding prime power is existentially complete. This should be clear because the

maximal primes are exactly those that cannot be extended to accomodate membership of the distinguished generator in any more r.e. sets, and membership of the distinguished generator in some r.e. set is the universal form of Σ1 formulas, all other elements of the model being partial recursive values of the generator.

The powers of maximal primes were introduced by Hirschfeld & Wheeler under the name of simple models. They are also called

r.e. ultrapowers — maximal prime filters on a lattice are sometimes referred to as ultrafilters.

The top and the bottom of the Priestley dual do have a non-empty intersection, so this tells us that there exist existentially complete recursive

ultrapowers.

As it happens, recursive ultrapowers and r.e. prime powers are fully representative of finitely generated models of TA2: Any TA2-model which

is finitely generated w.r.t. total recursive functions is (isomorphic to) a recursive ultrapower. Any TA2-model which is finitely generated w.r.t. partial recursive functions is an r.e. prime power.

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Effectively finite-to-one functions

Definition. Suppose f and R are recursive and dom f ⊇ R.

f is ef21 on R ⇐⇒ ∃ total recursive G ∀y ∈ ω G(y) = f |R −1(y) (G(y) is the canonical code of a finite set)

Effectively finite-to-one functions

Definition. Suppose f and R are recursive and dom f ⊇ R.

f is ef21 on R ⇐⇒ ∃ total recursive G ∀y ∈ ω G(y) = f |R −1(y) (G(y) is the canonical code of a finite set)

Duality for r.e. sets with applications Application Effectively finite-to-one functions

At this point we are going to break the natural order of exposition and

immediately state a theorem which is an application of duality and r.e. prime powers to r.e. sets. After that, we shall see how much time there’s left to review the main steps of the argument.

First, we require a definition.
Suppose we have us a recursive set R and a recursive function f which is

defined on the set R.

f is said to be effectively finite-to-one on R if, when restricted to R, it is

not just finite-to-one, but you also have a recursive function capital-G that gives you the finite set of pre-images for any given number y.

We require that this finite set is given by its canonical code. This means

that it is a finite listing of all elements, rather than just an enumeration algorithm for a finite set.

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Recursive images of complements of r-maximal and hhsimple sets

Q R H E F

Theorem. (a) If Q is r-maximal and f is total recursive, then

there is recursive R ⊇∗ Q s.t. f is constant or ef21 on R. (b) If H is hhsimple and f is total recursive, then there are recursive F and E s.t. F ∪ E ⊇ H, f[F] is finite and f is ef21 on E.

Corollary. If X is r-maximal or hhsimple and f is total recursive, then

f[X] is either finite or the complement of an r.e. set, r-maximal or hhsimple resp.

Recursive images of complements of r-maximal and hhsimple sets Q R H E F

Theorem. (a) If Q is r-maximal and f is total recursive, then

there is recursive R ⊇∗ Q s.t. f is constant or ef21 on R. (b) If H is hhsimple and f is total recursive, then there are recursive F and E s.t. F ∪ E ⊇ H, f[F] is finite and f is ef21 on E.

Corollary. If X is r-maximal or hhsimple and f is total recursive, then

f[X] is either finite or the complement of an r.e. set, r-maximal or hhsimple resp.

Duality for r.e. sets with applications Application Recursive images of complements of r-maximal and hhsimple sets

We are now ready to state the theorem. It comes in two somewhat similar parts, one for r-maximal, and the other for hyperhypersimple r.e. sets.
Given an r-maximal set Q and a total recursive function f , there is a recursive set R which contains the complement of Q up to finitely many

elements and such that f is either constant or effectively finite-to-one on R. (Incidentally, effective finite-to-oneness, unlike constancy, tolerates adding a finite number of elements to the set R.)

When together with a total recursive f , we are given a hyperhypersimple set H, we have the following partition effect:
There are recursive sets F and E that between them cover the complement of H and such that f only takes finitely many values on F, and f is

effectively finite-to-one on E.

There’s a corollary to this:
The image of the complement of an r-maximal or a hyperhypersimple set under any total recursive function must either be finite, or must itself

be a complement to an r.e. set, r-maximal or hhsimple respectively.

Here the part that requires the new theorem is “complement to an r.e. set”, the part about r-maximality or hhsimplicity being easy.
Here is why the complement to the image of the complement must be r.e.: To determine if a given number is outside the image of the

complement, just take a look at the finite set of pre-images and see if all of them are in the original r.e. set. This makes the complement of the image recursively enumerable (and also shows that ef21 functions are a bit like one-to-one recursive functions with recursive inverse). The finitely many values at blue arguments in the hyperhypersimple case cause no trouble at all.

Generally, the image of a co-r.e. set under a recursive function is only guaranteed to be Σ0
2. The moral of the corollary probably is that the

complements r-maximal and hyperhypersimple sets are really really small.

Observe that both the theorem and the corollary are stated in a language that a recursion theorist not familiar with duality nor with recursive

ultrapowers will understand.

At first I thought something like this must have been known for a long time already — you kind of expect that all simple things about r-maximal
r hhsimple sets are already known. But, after extensive asking around, this appears not to be the case. Please do tell me if you know otherwise.

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Skies and 0′′

Let a, b ∈ M |= TA2. b ≤ ≤ a ⇐⇒ ∃ total recursive f M |= f (a) ≥ b ≤ ≤ is a linear preorder; equivalence classes are (total recursive) skies

Theorem. If M |= TA2 has at least two distinct non-standard skies, then

the skies of M are densely ordered (including )

Definition. Let < M |= TA2 and X ⊆ ω.

M codes X ⇐⇒ ∃a ∈ M ∀n ∈ ω n ∈ X ↔ M |= (a)n = 0 Theorem (with J. Schmerl). Let < M |= TA2. M codes 0′′ ⇐⇒ M is multi-sky

Skies and 0′′ Let a, b ∈ M |= TA2. b ≤ ≤ a ⇐⇒ ∃ total recursive f M |= f (a) ≥ b ≤ ≤ is a linear preorder; equivalence classes are (total recursive) skies

Theorem. If M |= TA2 has at least two distinct non-standard skies, then

the skies of M are densely ordered (including )

Definition. Let < M |= TA2 and X ⊆ ω.

M codes X ⇐⇒ ∃a ∈ M ∀n ∈ ω n ∈ X ↔ M |= (a)n = 0 Theorem (with J. Schmerl). Let < M |= TA2. M codes 0′′ ⇐⇒ M is multi-sky

Duality for r.e. sets with applications Dichotomy Skies and 0”

Some of our considerations pertain to all models of TA2 rather than just prime powers.
Given a model M of forall-exists arithmetic, we introduce a preorder on the elements of M.
Say that b is at most total recursively larger than a if you can get from a past b by some total recursive
function. Incidentally, when we say ‘recursive function’, total or partial, we mean an absolutely

standard recursive function, that is, one with a standard index — it just operates in a non-standard model.

The equivalence classes of this preorder are called (total recursive) skies. All numbers that are at at

most total recursive distance from one another go into the same sky. Skies are convex and probably look somewhat like this. The standard numbers form the lowermost sky.

Generally, with other classes of models, one could consider 1st order definable or, say, provably

recursive skies, or skies determined by all standard functions regardless of their complexity — that’s what they do in non-standard analysis — but total recursive ones are appropriate for our present setup.

It turns out that we have an important dichotomy:
Either our model M has a single non-standard sky, or all the skies of M, incuding the standard one, are

densely ordered, so there are infinitely many skies.

There is a useful criterion to distinguish between the two cases. First, a definition:
Suppose M is a non-standard model of forall-exists arithmetic, and X is some subset of ω.
We say that M codes X if there is an element a of M, the code of X, such that for all standard n, the nth

element of a viewed as a (non-standardly) finite sequence is equal to 0 if and only if n is an element

f X. This is an absolutely classical definition.
Our theorem established together with James Schmerl says that a TA2-model M has infinitely many

skies if and only if M codes the Turing degree 0′′ — the collection of sets coded in a given model is closed under Turing reducibility, so one can talk about M coding a given Turing degree.

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Single sky and ef21

[p] is single-sky =⇒ p ∈ min(E ∗)⋆ =⇒ p = ¯ u, some u ∈ (R ∗)⋆ [u] is single-sky ⇐⇒ ∀[ f] ∈ [u] [ f] ∈ or ∃g ∈ T [u] |= g([ f]) ≥ x ⇐⇒ ∀f ∈ T ∃n ∈ ω f ≡u n or ∃g ∈ T g ◦ f ≥u id

Lemma. Suppose R and f are recursive. Then

∃g ∈ T ∀x ∈ R g ◦ f (x) ≥ x ⇐⇒ f is ef21 on R.

Proof. (⇐)

∃G ∈ T ∀y ∈ ω G(y) = ( f |R)−1(y) . Put g(y) = max G(y). (⇒) G(y) = { x ≤ g(y) | x ∈ R & f (x) = y }, for R ∋ x > g(y) ⇒ y f (x). ∃g ∈ T g ◦ f ≥u id ⇐⇒ ∃R ∈ u ∃g ∈ T ∀x ∈ R g ◦ f (x) ≥ x ⇐⇒ ∃R ∈ u ( f is ef21 on R )

Moral. [u] is single-sky

⇐⇒ ∀f ∈ T ∃R ∈ u ( f is constant or ef21 on R).

Single sky and ef21 [p] is single-sky =⇒ p ∈ min(E ∗)⋆ =⇒ p = ¯ u, some u ∈ (R ∗)⋆ [u] is single-sky ⇐⇒ ∀[ f] ∈ [u] [ f] ∈ or ∃g ∈ T [u] |= g([ f]) ≥ x ⇐⇒ ∀f ∈ T ∃n ∈ ω f ≡u n or ∃g ∈ T g ◦ f ≥u id

Lemma. Suppose R and f are recursive. Then

∃g ∈ T ∀x ∈ R g ◦ f (x) ≥ x ⇐⇒ f is ef21 on R.

Proof. (⇐)

∃G ∈ T ∀y ∈ ω G(y) = ( f |R)−1(y) . Put g(y) = max G(y). (⇒) G(y) = { x ≤ g(y) | x ∈ R & f (x) = y }, for R ∋ x > g(y) ⇒ y f (x). ∃g ∈ T g ◦ f ≥u id ⇐⇒ ∃R ∈ u ∃g ∈ T ∀x ∈ R g ◦ f (x) ≥ x ⇐⇒ ∃R ∈ u ( f is ef21 on R )

Moral. [u] is single-sky

⇐⇒ ∀f ∈ T ∃R ∈ u ( f is constant or ef21 on R).

Duality for r.e. sets with applications Single sky powers Single sky and ef21

We are now going to focus on single-sky r.e. prime powers.
First we note that any sigle-sky power must correspond to a minimal prime filter, because in the powers corresponding to larger prime filters

there is, first, the non-standard sky corresponding to the distinguished generator x, and, second, there must be another, higher, sky in which the generator enters some new r.e. set.

Hence the power is in fact a recursive ultrapower corresponding to some ultrafilter u on the recursive sets.
We would now like to understand what the single-sky condition means in terms of the underlying ultrafilter. Consider a single-sky recursive

ultrapower [u].

The single sky condition holds if each element of the ultrapower — it has the form of the equivalence class of some total recursive function —

either belongs to the standard sky, or to the same sky as the generator. In which case there should exist a total recursive g such that g( f (x)) is at least as large as x.

This means that any total recursive f is either constant modulo the ultrafilter u, or the composition of f with some total recursive g exceeds,

modulo u, the identity.

We now claim that the existence of such a pseudo-inverse g to f which works on some recursive set R of arguments is equivalent to effective

finite-to-oneness of f on that set R.

Let us show the equivalence: From right to left, the green condition rewrites by definition as the existence of a total recursive capital-G which

gives the f -pre-image on R. So given a capital-G, it suffices to put g-small(y) equal to the maximum of the set capital-G(y).

For the opposite direction, capital-G(y) is defined as the code of the set of those x bounded by g-small(y) that belong to R and such that x goes

to y under f . This is clearly finite and recursive.

To see that we don’t have to look for pre-images beyond the bound g(y), note that if an element x of R is larger than g(y) then y cannot be the

image of x under f — just substitute f (x) for y and observe that this contradicts the blue condition that g ◦ f must exceed the identity on R.

So the yellow condition rewrites as the existence of a recursive set R in the ultrafilter u on which there is a pseudo-inverse g-small to f .
By the Lemma, this is the same as f being effectively finite-to-one on R.
We summarize the present situation by restating the single-sky condition as follows: The ultrafilter u must restrict each total recursive functon to

a constant ot to an effectively finite-to-one function.

This finally connects sky counting to effective finite-to-oneness of functions.

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r-maximality and single skies

Re =

n ∈ ω

∀k ≤ n {e}(k)↓ & {e}(n) = 0

ix u =

e ∈ ω

Re ∈ u

(u ∈ (R ∗)⋆)

Q u Lemma (Lerman–Shore–Soare). Q is r-maximal =⇒ ix u is ∆0

3.

Proof. e ∈ ix u

⇐⇒ Re ∈ u ⇐⇒ Re ∪ Q =∗ ω e ix u ⇐⇒ Re u ⇐⇒ Re ⊆∗ Q

Lemma. [u] does NOT code ix u for any u ∈ (R ∗)⋆.
Corollary. Q is r-maximal

=⇒ [u] is single-sky.

Proof. [u] multi-sky ⇒ [u] codes 0′′ ⇒ [u] codes ix u

⇒ ⊥

r-maximality and single skies Re =

n ∈ ω ∀k ≤ n {e}(k)↓ & {e}(n) = 0

ix u =

e ∈ ω Re ∈ u

(u ∈ (R ∗)⋆)

Q u Lemma (Lerman–Shore–Soare). Q is r-maximal =⇒ ix u is ∆0

3.

Proof. e ∈ ix u

⇐⇒ Re ∈ u ⇐⇒ Re ∪ Q =∗ ω e ix u ⇐⇒ Re u ⇐⇒ Re ⊆∗ Q

Lemma. [u] does NOT code ix u for any u ∈ (R ∗)⋆.
Corollary. Q is r-maximal

=⇒ [u] is single-sky.

Proof. [u] multi-sky ⇒ [u] codes 0′′ ⇒ [u] codes ix u

⇒ ⊥

Duality for r.e. sets with applications Single sky powers r-maximality and single skies

Now we are going to show that the heels of r-maximal sets are always single-sky.
We fix one of the conventional enumerations Re of all recursive sets — for such an enumeration, the membership of a number in the eth set is a

recursively enumerable binary relation. It cannot be recursive on pain of diagonalization.

The index set of an ultrafilter u on the recursive set is the collection of those R-indices whose recursive sets lie in the ultrafilter.
Recall that an r.e. set Q is r-maximal if its picture covers all minimal points of the dual space except one, which is then called the heel of that set.
The key observation here is one made by Lerman, Shore and Soare that the R-index set of the recursive ultrafilter corresponding to the heel must

lie in 0′′. ∆0 3 is just a synonym for 0′′.

Indeed, Re is an element of u if and only if the point u lies in the picture of Re which will then have to cover the whole complement of Q

because pictures are upwards closed. This is equivalent to the union of Re and Q being cofinite.

If Re lies outside u, that is, the picture of Re does not cover the point u, then the whole picture of Re is disjoint from that of the complement
f Q. That’s because the picture of any recursive set is closed downwards as well, so no recursive set splits the complement of Q non-trivially.

Disjointness is equivalent to the picture of Re lying inside that of Q, i.o.w., Re being a subset of Q modulo some finite set.

We see that both these conditions are Σ0

3, so this proves the lemma.

The next lemma says that a recursive ultrapower never codes the R-index set of the ultrafilter. It’s one of those situations where an object cannot

describe itself: Its proof employs the 2nd Recursion Theorem.

We are now ready to see why the heel of an r-maximal set has to be single-sky.
If there were many skies, then the ultrapower would code 0′′. The index set of u will then also be coded, as 0′′ is another name for ∆0

3. Now this contradicts the previous lemma.

So here we have an example of complexity influencing structure.
Now all that remains is to note that we are done with the r-maximal part of our application.

SLIDE 11

dual powers ef21 skies 1sky ciao

r-maximality and ef21

Q

f constant

r ef21

u

r-maximality and ef21

Q

f constant

r ef21

u

Duality for r.e. sets with applications Single sky powers r-maximality and ef21

Going back to the statement, given a total recursive f, the

heel u of the r-maximal set restricts f to a constant or an ef21 function on some recursive set in the ultrafilter.

The picture of that recursive set covers the heel, and

therefore must also cover the whole of the complement of (the picture of) the r-maximal set Q.

SLIDE 12

dual powers ef21 skies 1sky ciao

Hyperhypersimplicity and ef21

H E F

Proposition. H is hhsimple

=⇒ all h ∈ H⋆ are single-sky. Consider any total recursive f. h ∈ H⋆ =⇒ there is a recursive Rh ∈ h on which f is constant or ef21.

R⋆

h

h∈H⋆ is a clopen cover of H⋆

(Priestley topology). H⋆ is clopen in the compact space (E ∗)⋆. Select a finite subcover

R⋆

h

h∈I :

I ⊆ H⋆ finite,

h∈I

R⋆

h ⊇ H⋆.

F =

h∈I

Rh

f is constant on Rh

;

E =

h∈I

Rh

f is ef21 on Rh

f[F] is finite;

f is ef21 on E.

Hyperhypersimplicity and ef21 H E F

Proposition. H is hhsimple

=⇒ all h ∈ H⋆ are single-sky. Consider any total recursive f. h ∈ H⋆ =⇒ there is a recursive Rh ∈ h on which f is constant or ef21. R⋆

h

h∈H⋆ is a clopen cover of H⋆

(Priestley topology). H⋆ is clopen in the compact space (E ∗)⋆. Select a finite subcover R⋆

h

h∈I :

I ⊆ H⋆ finite,

h∈I

R⋆

h ⊇ H⋆.

F =

h∈I Rh

f is constant on Rh ; E =

h∈I Rh

f is ef21 on Rh

f[F] is finite;

f is ef21 on E.

Duality for r.e. sets with applications Single sky powers Hyperhypersimplicity and ef21

We now look at an arbitrary hyperhypersimple set H.
Just as in the r-maximal case we have that all primes in the complement of H are single-sky.
The proofs of this proposition — I have two proofs that probably are different — are nothing like the
ne for the r-maximal case. They do not use 2nd Recursion, but they also show the failure to code 0′′

by any of the powers corresponding to primes outside the picture of H.

We now ask ourselves what can be said about the behaviour of an arbitrary total recursive function f on

the complement of H.

Given a prime h in the complement of H, there is, by single-skyness, a recursive set Rh on which the

function f is either constant or effectively finite-to-one.

The collection of Rh for all primes h in the complement forms a clopen cover of the picture of that

complement in the Priestley topology. (Recall that the picture of any r.e. set is a clopen subset of the dual space.)

The complement of H, being a closed or even clopen subset of a compact space, is itself compact.

There is now only one way this argument can go: From the open cover we select a finite subcover that still covers the complement.

What you get from that finite cover is this: On each piece of the cover, the function f is either

constant a.e. or ef21. Distribute the pieces into two categories accordingly and take the two unions.

You get recursive sets F and E whose union covers the complement and such that f only takes finitely

many values on F, and f is ef21 on E. Note that the ef21 property is preserved by unions: if a function is ef21 on two pieces then it also is ef21 on their union.

This settles the hyperhypersimple clause of the Theorem and concludes our story.

SLIDE 13

Thank You!

Duality for r.e. sets with applications Thank You!

That’s it for today.
Thank you very much for your attention.