Forcing, the Separation, the Reduction and theUniformization property
Stefan Hoffelner
WWU Münster

October 26, 2023

I We let N to denote the Baire space, i.e. N = ω ω together withthe topology generated by sets of the form

Os := {f ∈ ω ω | s ⊂ f }where s ∈ ω <ω is a finite sequence of natural numbers. As usual
whenever we talk about reals we in fact talk about the Bairespace.
111
I Σ1n , Πn and their boldface counterparts Σn and Πn ; theprojective subsets of the reals.

Definition
I A ⊂ N is Σ1 if there is a recursive set R ⊂1S∞n=0 (ωn × ωn)such that for all x ∈ Nx ∈ A ⇔ ∃y ∈ ω ω ∀n ∈ ωR(x  n, y  n)
1I A ⊂ N is Π1n if N \ A is Σn
1I A ⊂ N is Σ1n+1 if there is a Πn set B ⊂ N × N such thatx ∈ A ⇔ ∃yB(x, y)
The definition generalizes to yield the pointclass Σ1n (a) etc. for a ∈ N .
One finally defines the boldface projective hierarchy via[a∈NΣ1n (a) =: Σ1n

Given two disjoint sets of reals A1 and A2 , we say that a set Cseparates A1 and A2 iff A1 ⊂ C and A2 ⊂ C c . The problem
becomes interesting when considered through the lens of definability.

Definition
We say that the Σ1n -separation property holds iff every pair A1 and A2of disjoint Σ1n -sets has a separating set C such that C is both Σ1n andΠ1n .
The separation property is connected to two notions, the reductionand the uniformization property.

Definition
A projective pointclass Γ has the reduction property if for every pairA, B ∈ Γ there are reducing sets A0 and B 0 , the reducing sets for A
and B which must satisfy:
1. A0 ⊂ A and B 0 ⊂ B
2. A0 , B 0 ∈ Γ
3. A0 ∩ B 0 = ∅4. A ∪ B = A0 ∪ B 0 .

Given A ⊂ ω ω × ω ω , we say that f is a uniformization (or auniformizing function) of A if

dom(f ) = pr1 (A) = {x ∈ ω ω : ∃y((x, y) ∈ A}
and the graph of f is a subset of A.

Definition (Uniformization Property)
We say that the pointclass Γ has the uniformization property iff every
Γ-set in the plane admits a uniformization whose graph is in Γ, i.e. therelation (x, y) ∈ f is Γ.

A couple of classical results:

Theorem (Kondo,1938)
Π11 -uniformization is provable over ZFC. Hence Π11 -reduction andΣ11 -separation holds.
Theorem (Novikov, 1935)
The Γ-reduction property implies that the Γ-separation property must
fail. Hence the Γ and Γ̃-reduction property can not hold together. This
also implies that Γ and Γ̃-uniformization can not both hold.
TheoremIf Π1n -uniformization holds then so does Σ1n+1 uniformization.

Theorem (Levy, 1965)
There is a universe of ZFC over which a Π12 -set can not be uniformized
by an ordinal definable function. So in particular for each Γ projective
pointclass ofTheorem (Addison, 1958)
In Gödel’s constructible universe L, for every n ≥ 2 theΣ1n -uniformization property holds. So also Σ1n -reduction andΠ1n -separation hold.
Reason is the existence of a so-called good well-order of the reals
which is Σ12 -definable.

Theorem (Silver, 1968)
Let U be a measure, then in L[U ], for every n ≥ 2 theΣ1n -uniformization property holds.

Determinacy assumptions draw a very different (and more accurate)
picture:Theorem (Moschovakis, 1970)
Assume that every ∆12n -set is determined, then theΠ12n+1 -uniformization property holds. In particular, assuming thatevery projective set is determined (Axiom of projective determinacy)
the Π12n+1 - and Σ12n+2 -uniformization property holds for every n ∈ ω .
By the famous results of Woodin and Martin-Steel, projective
determinacy assumptions are large cardinal assumptions in disguise.

Theorem (Martin-Steel, 1989)
Assume there are n-many Woodin cardinals and one measurable
cardinal on top. Then Π1n+1 -determinacy holds. In particularassuming infinitely many Woodin cardinals, then projective
determinacy holds.

Theorem (Woodin)
Let n ≥ 1 and assume that Π1n+1 -determinacy holds. Then there isan inner model of ZFC which contains n-many Woodin cardinals.
So large cardinals on the one hand, and projectively definable, goodwell-orders of the reals on the other hand yield two different ways how
the uniformization property can behave within the projective sets.
But there are many basic and natural questions left open:

Question
I Do we need large cardinals in order to obtain e.g. theΣ13 -separation property?
I For even n, is, say, the Π1n -uniformization property evenconsistent?
I Are separation, reduction, and uniformization different concepts

after all?
I Are there other, global patterns of the behaviour of, say, theuniformization property possible?

Question (localized 12th Delfino Problem by Woodin)
Does ZFC together with the “Every Π13 -set is Lebesgue measurable
and has the Baire property” and together with the Π13 -uniformization
property already imply ∆12 -determinacy?

Theorem
There is an ω1 -preserving partial order P ∈ L such that for g ⊂ Pgeneric,L[g] |= “Σ13 -separation property holds true.”
So the Σ13 -separation property bears no large cardinal strength.
Additional work, building on the techniques developed for the proof of
the above theorem give:

Theorem
There is an ω1 -preserving partial order P ∈ L such that

L[g] |= “Σ13 -separation holds, but Π13 -reduction fails.”
So the separation property and the reduction property are different
concepts over ZFC.

Turning to Reduction:

Theorem
There is a partial order P, again ω1 -preserving, such that for g ⊂ P
generic over L
L[g] |= “Π13 -reduction holds.”
So Π13 -reduction bears no large cardinal strength as well. Also:

TheoremIn the above universe Π13 -uniformization fails. So reduction anduniformization are different concepts over ZFC.

Finally turning to Uniformization:

Theorem
There is a partial order P ∈ L such that

L[g] |= “Π13 -uniformization holds.”
The proof of the last theorem can be lifted (this requires some work) toobtain:

Theorem
Let Mn be the canonical inner model with n many Woodin cardinals.
There is a partial order P ∈ Mn such thatMn [g] |= “Π1n+3 -uniformization holds.”
Note that this in particular gives that the Π12n -uniformization property is
consistent relative to 2n-3 many Woodin cardinals. So in sharp
contrast to the picture drawn by determinacy assumptions on
projective sets, the uniformization property can hold for even Π-levels
as well.

Lifting the previous proofs to the Mn ’s, we also obtain:

Theorem
I Work over Mn , then there is a partial order P ∈ Mn such that forg ⊂ P generic,Mn [g] |= “Σ1n+3 -separation holds but Π1n+3 -reduction fails”.
I There is a partial order P ∈ Mn such thatMn [g] |= “Π1n+3 -reduction holds but Π1n+3 -uniformization fails”.

We very briefly sketch the idea how to prove the Π13 -uniformization
property over L. First we define a suitable ground model W ⊃ L, W
has no new reals and over W there is a Σ13 -formula Φ(v0 ) such that:
1. W |= ¬Φ(x), so there are no reals x ∈ L satisfying Φ.
2. For an arbitrary real x there is a proper (coding) forcing Px ∈ Wsuch that for gx ⊂ Px genericW [gx ] |= Φ(x)and for reals y 6= x still

W [gx ] |= ¬Φ(y)These forcings can be iterated and the iterations will preserve property
2, so we are in full control over the set of reals which satisfy Φ(x) in
the resulting generic extension. If we force with a forcing Px then we
say that the real x got coded.

Let us now consider one fixed Π13 -set A ⊂ R2 in the plane. Let x be
such that there is a y such that (x, y) ∈ A, i.e. x is in the projection of
A. The goal is now to single out exactly one real yx such that(x, yx ) ∈ A, which should be the value of our uniformizing function. Inorder to make the graph of the uniformizing function Π13 we iteratively
force such that for every y 6= yx , the pair (x, y) does satisfy Φ((x, y))(this can be done using an ℵ1 -length iteration in order to catch ourtail). Eventually (x, yx ) should become the unique pair of reals forwhich Φ((x, yx )) fails, thus it satisfies a Π13 -formula, as Φ was Σ13 .

If we can do this for all reals x in an iterative way, then, eventually, wewould succeed, but there is an obvious problem: The Π13 -set A willchange as we enlarge our universe. In particular it could be that ourchosen pair (x, yx ) will drop out of A as we proceed in our iteration.
Consequentially (x, yx ) is not a good value for our uniformizing
function and we need a new one. But maybe there are no new
potential values left anymore, as all (x, y) ∈ A got coded, i.e. wealready forced to make Φ((x, y)) to become true for each pair(x, y) ∈ A.
In that situation our strategy has failed, and we need to start all overagain, maybe picking a new value (x, yx0 ) for the uniformizing funtion
at x. But this choice could fail for the very same reasons.

To solve these issues, we rephrase the problem as afixed-point-problem for sets of iterations of our coding forcings i.e.
forcings of the form Px for x a real.
Given a set P of iterations of coding forcings and a bookkeeping
function F we define an iteration P of length ω1 using F and P byinduction as follows: Suppose we are at stage β of our iteration.
I If F (β) = (x, y) and (x, y) ∈ A and assume that for every

P ∈ P it holds that P (x, y) ∈ A, then define our uniformizing
function at x as f (x) = y , provided we have not chosen a valueof our uniformizing function f yet for x. Do not force at stage β inthis situation. If we do have a value f (x) = yx 6= y for x alreadydefined in an earlier stage of the iteration, force such that Φ(x, y)holds in the generic extension.
I If F (β) = (x, y) and (x, y) ∈ A and there is a P ∈ P such thatP(x, y) ∈/ A, then use one such P at stage β .

Now typically, the so defined iteration will not be an element of P itself,in other words the set of coding forcings P will typically not be a fixed
point for the operation, which at input P, F outputs an iteration ofcoding forcings.
But if we can ensure that the resulting P is itself an element of ourprevious fixed class of forcings P , then P would force theΠ13 -uniformization property. Indeed, for a real x, either we defined avalue f (x) = yx at some point of our iteration, and, as P ∈ P ensured
that P (x, yx ) ∈ A. Or we never defined a value, but then P will
force that Ax is empty and there is no need for a value yx anyway.
So how to find such a set of forcings P ?

The idea is to start with P0 being all possible countable iterations ofcoding forcings. As a first approximation we define P1 to be defined asfollows relative to P0 and some bookkeeping F :I If F (β) = (x, y) and for all P ∈ P0 we have thatP(x, y) ∈ A

and y is the least such real with this property (relative to somepreviously fixed well-order), then set f (x) = y and do not force
with P(x,y) , i.e. do nothing
I If on the other hand there is a P ∈ P such that

P(x, y) ∈/A

then we let the bookkeeping F decide whether to code (x, y) ornot.
The so defined set P1 is clearly a subset of P0 . And this wholeprocedure can now be done again with P0 and P1 together replacing
the role of P0 , yielding a set P2 which will be smaller than P1 .

This can be repeated transfinitely often, yielding for every α anon-empty set Pα . We will always have that Pα ⊂ Pβ for α > β . So
there must be a stage α at which this thinning out process starts tostabilize. This set of forcings is then the right set P to do the argument
described above.
The presented results are just the very first step towards a complete
understanding of separation, reduction and uniformization. There aremany natural problems which remain open, the most of them seem tobe quite difficult. To name a few:
I Can one force the Π1 -uniformization property over just L, instead
4of M1 ?
I Can one force the Π1 and the Π1 -uniformization property
3
5
simultaneously?
I What global patterns of the uniformization property are possibleat all?
Thank you!