The consistency strength of the theory
ZFC + “every universally Baire set has the
perfect set property”
Trevor Wilson
(joint work with Ralf Schindler)
Miami University, Ohio
February 17, 2017
Definition
A set of reals B has the perfect set property (PSP) if it is
countable or it has a perfect subset.
Remark
There is a set of reals without PSP. (Bernstein)
Remark
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Every analytic set of reals has PSP. (Suslin)
The statement “every coanalytic set of reals has PSP”
is independent of ZFC. (Solovay, Gödel)
Theorem (Solovay, Specker)
The
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2.
3.
following statements are equiconsistent modulo ZFC.
There is an inaccessible cardinal.
Every coanalytic set of reals has PSP.
Every set of reals in L(R) has PSP.
Remark
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If (1) is witnessed by κ, then (3) holds in V Col(ω,<κ) .
If (3) holds, then (2) holds.
If (2) holds, then (1) holds in L, witnessed by ω1V .
What about PSP for other sets of reals?
Which other sets of reals might possibly be well-behaved?
Definition
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A tree on ω × γ,Swhere γ is an ordinal, is a downward
closed subset of n<ω (ω n × γ n ).
The projection of a tree T on ω × γ is the set of reals
p[T ] = {x ∈ ω ω : ∃f ∈ γ ω ∀n < ω (x n, f n) ∈ T }.
Remark
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Every analytic set of reals is the projection of some tree
on ω × ω, and vice versa.
Every coanalytic set of reals is the projection of some
tree on ω × ω1 .
Under AC, every set of reals is the projection of some
tree on ω × c.
Definition (Feng–Magidor–Woodin)
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A set of reals B ⊂ ω ω is η-universally Baire, where η is a
cardinal, if there are trees T and T̃ such that
B = p[T ] = ω ω \ p[T̃ ], and every generic extension by a
poset of cardinality less than η satisfies p[T ] = ω ω \ p[T̃ ].
A set of reals B ⊂ ω ω is universally Baire if it is
η-universally Baire for every cardinal η.
Example
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Every analytic set of reals is universally Baire.
Every coanalytic set of reals is universally Baire.
Universally Baire sets of reals have some regularity
properties, but not necessarily PSP.
Question
What is the consistency strength of ZFC + “every universally
Baire set of reals has the perfect set property”?
Known bounds
Lower bound: the existence of an inaccessible cardinal.
Upper bound: the existence of a Woodin cardinal.
I If there is a Woodin cardinal, then every universally Baire
set of reals is weakly homogeneously Suslin. (Woodin)
I Every weakly homogeneously Suslin set of reals has PSP.
(Proved using the unfolded perfect set game.)
Main theorem (Schindler-W.)
The
1.
2.
3.
following statements are equiconsistent modulo ZFC.
There is a cardinal that is Shelah for remarkability.
Every universally Baire set of reals has PSP.
Every set of reals in L(R, uB) has PSP, where uB is the
collection of all universally Baire sets of reals.
Remark
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If (1) is witnessed by κ, then (3) holds in V Col(ω,<κ) .
If (3) holds, then (2) holds.
If (2) holds, then (1) holds in L, witnessed by ω1V .
Before defining “Shelah for remarkability,” we recall some
other large cardinal axioms.
Schindler showed that strong cardinals are remarkable, which
is an example of a “virtual” large cardinal property.
Definition
A cardinal κ is strong if for every λ > κ there is a transitive
class M and an elementary embedding j : V → M such that
crit(j) = κ and Vλ ⊂ M.
Definition (Schindler)
A cardinal κ is remarkable if for every λ > κ there is a λ̄ < κ
such that in some generic extension there is an elementary
embedding j : Vλ̄ → Vλ with j(crit(j)) = κ.
Similarly to showing that strong cardinals are remarkable, one
can show that Shelah cardinals are Shelah for remarkability.
Definition
A cardinal κ is Shelah if for every function f : κ → κ there is
a transitive class M and an elementary embedding j : V → M
such that crit(j) = κ and Vj(f )(κ) ⊂ M.
Definition (Schindler–W.)
A cardinal κ is Shelah for remarkability if for every function
f : κ → κ there are λ > κ and λ̄ < κ such that in some
generic extension there is an elementary embedding
j : Vλ̄ → Vλ with j(crit(j)) = κ, λ̄ ≥ f (crit(j)) and f ∈ ran(j).
These “virtual” large cardinal axioms can hold in L.
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If κ is remarkable then κ is remarkable in L.
(Schindler)
If κ is Shelah for remarkability then κ is Shelah for
remarkability in L.
They are weaker than the existence of 0] .
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If 0] exists then every indiscernible is remarkable in L.
(Schindler)
If 0] exists then every indiscernible is Shelah for
remarkability in L.
We now outline the proof of the main theorem, which is the
equiconsistency of the following statements modulo ZFC.
1. There is a cardinal that is Shelah for remarkability.
2. Every universally Baire set of reals has PSP.
3. Every set of reals in L(R, uB) has PSP, where uB is the
collection of all universally Baire sets of reals.
Clearly 3 implies 2, so it remains to prove:
I Con(1) =⇒ Con(3), the forcing argument, and
I Con(2) =⇒ Con(1), the inner model argument.
Forcing argument
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Assume that κ is Shelah for remarkability and let
G ⊂ Col(ω, <κ) be a V -generic filter.
Claim: every uB set of reals A in V [G ] comes from a
κ-uB set of reals in V [G α] for some α < κ.
If not, we can get a contradiction using the “Shelah for
remarkability” property for f : κ → κ where f (α) is the
least β + 3 such that A ∩ V [G β] does not come from
any β-uB set of reals in V [G α].
So every uB set of reals in V [G ] is in the symmetric
model V (RV [G ] ), and L(R, uB)V [G ] is contained in
V (RV [G ] ), where every set of reals has PSP by Solovay.
Inner model argument
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Assume that κ = ω1V is not Shelah for remarkability in L.
Then there is a function f : κ → κ in L such that for all
λ > κ and λ̄ < κ there is no elementary embedding
j : Lλ̄ → Lλ in any generic extension with j(crit(j)) = κ,
λ̄ ≥ f (crit(j)) and f ∈ ran(j).1
For example if κ is accessible in L, say κ = γ +L , take
f (α) to be least β > α such that Lβ projects to γ.
Now work in V . For each α < ω1 , choose a real xα
coding Lf (α) . Define the set of reals A = {xα : α < ω1 },
which is uncountable and has no perfect subset.
For every cardinal η, we can define a tree T̃ that projects
to the complement of A in every generic extension by a
poset of size less than η. (All new reals are in p[T̃ ].)
Note added 2017 Feb. 23: I’m not sure if this bullet point is correct.
It does not affect the validity of the main theorem, however.