Hyperstationary subsets of P ( )Martha Catalina Torres Pachón
RIMS Set Theory Workshop 2023 - Large Cardinals and the Continuum
Thursday 26th October, 2023

Contents

1 Introduction
2 Hyperstationary subsets of 3 Hyperstationary subsets of P (A)4 Further Work and Open Questions

Martha Catalina Torres Pachón

RIMS Set Theory Workshop 2023 - Large Cardinals and the Continuum

2 / 31

Introduction

Contents

1 Introduction
2 Hyperstationary subsets of 3 Hyperstationary subsets of P (A)4 Further Work and Open Questions

Martha Catalina Torres Pachón

RIMS Set Theory Workshop 2023 - Large Cardinals and the Continuum

3 / 31

Introduction

Introduction

The study of stationary subsets of cardinals of uncountable cofinality, and stationaryreflection, has a long history and it is perhaps one of the most studied notions ofcombinatorial set theory.
In 2015, Bagaria, Magidor and Sakai in the paper “Reflection And Indescribability In
The Constructible Universe” developed this notion, defining an iterated version ofstationary reflection yielding in L a characterisation of ⇧1n indescribability in terms ofhyperstationary sets.

Martha Catalina Torres PachónRIMS Set Theory Workshop 2023 - Large Cardinals and the Continuum

4 / 31

Introduction

Introduction

Later on (2018 - “Derived Topologies On Ordinals And Stationary Reflection”),
Bagaria expanded the results founded in 1 defining the iterations of stationaryreflection and indescribability for all ordinals. He studied the iterated topologies onthe ordinals obtained through the Cantor’s derivative operator and characterised
non-isolated points of this topologies in terms of ⇠-simultaneous reflecting ordinals.

1 Bagaria, Magidor, Sakai. Reflection And Indescribability In The Constructible Universe
Martha Catalina Torres PachónRIMS Set Theory Workshop 2023 - Large Cardinals and the Continuum

5 / 31

Introduction

What we do
We study the notion of n-stationarity in P (A) := {X ✓ A : |X| < } where  is an
uncountable regular limit cardinal and  ✓ A, proposed by Hazel Brickhill, Sakaé
Fuchino and Hiroshi Sakai on the slides “On generalized notions of hyperstationarity”. We develop some of the consequences of this definitions and we look atwhich of the results obtained by Bagaria in his article “Derived Topologies on
Ordinals and Stationary Reflection” can be obtained within the context of P (A).

Martha Catalina Torres PachónRIMS Set Theory Workshop 2023 - Large Cardinals and the Continuum

6 / 31

Introduction

Why is this interesting ?
While the consistency strength of hyperstationarity is rather low in the large-cardinalhierarchy (below a measurable cardinal), its generalisation to P (A) is posibly muchstronger. Thus, the formulation of the appropriate generalisation of hyperstationarity
for P (A) and the development of its theory, in analogy with the notion of
hyperstationarity for cardinals should allow more interesting applications at a muchhigher level, in terms of consistency strength.

Martha Catalina Torres Pachón

RIMS Set Theory Workshop 2023 - Large Cardinals and the Continuum

7 / 31Hyperstationary subsets of Contents

1 Introduction
2 Hyperstationary subsets of 3 Hyperstationary subsets of P (A)4 Further Work and Open Questions

Martha Catalina Torres Pachón

RIMS Set Theory Workshop 2023 - Large Cardinals and the Continuum

8 / 31Hyperstationary subsets of Hyperstationary subsets of 

Understanding the idea of hyperstationarity and why it is indeed a generalisation ofthe notion of stationarity.
Definition of a Stationary subset of  : S ✓  is stationary if and only if for all
club C ✓ , S \ C 6= ?.
Well known stationary reflection: We say that a stationary set S ✓  reflects if
there is < , such that S \ is a stationary subset of .

Martha Catalina Torres PachónRIMS Set Theory Workshop 2023 - Large Cardinals and the Continuum

9 / 31Hyperstationary subsets of Hyperstationary subsets of An equivalent notion of statationarity in .

S is a stationary subset of  if and only if for all unbounded subset T of  there issome 2 S such that T \ is unbounded in .

Martha Catalina Torres Pachón

RIMS Set Theory Workshop 2023 - Large Cardinals and the Continuum

10 / 31Hyperstationary subsets of Hyperstationary subsets of An equivalent notion of statationarity in .

S is a stationary subset of  if and only if for all unbounded subset T of  there issome 2 S such that T \ is unbounded in .(T reflects as an unbounded set!)
Well known stationary reflection: We say that a stationary set S ✓  reflects if
there is < , such that S \ is a stationary subset of .

Martha Catalina Torres Pachón

RIMS Set Theory Workshop 2023 - Large Cardinals and the Continuum

11 / 31Hyperstationary subsets of Hyperstationary subsets of S ✓  is 0-stationary if S is unbounded in .
S is 1-stationary if every unbounded subset T ✓  reflects at some point of S.
(If and only if S is stationary)
S is 2-stationary if every 2-stationary (stationary) subset T ✓  reflects at somepoint of S. (Stationary reflection for all subsets of , with witnesses in S.)

Martha Catalina Torres PachónRIMS Set Theory Workshop 2023 - Large Cardinals and the Continuum

12 / 31Hyperstationary subsets of 

Definition (Bagaria2 )We say that S ✓  is ⇠-stationary in  if and only if for every ⇣ < ⇠, if T ✓  is⇣-stationary in , there is 2 S, such that T \ is ⇣-stationary in .
S ✓  club ! S ⇠-stationary ! S ⇣-stationaryfor all ⇣ < ⇠Definition (Bagaria2 )We say that S ✓  is ⇠-simultaneously-stationary in  (⇠-s-stationary) if andonly if for every ⇣ < ⇠, every pair of subsets T1 , T2 ✓  that are ⇣-s-stationary in ,there is <  such that T1 , T2 are both ⇣-s-stationary in the same

2 Derived Topologies On Ordinals And Stationary Reflection
Martha Catalina Torres PachónRIMS Set Theory Workshop 2023 - Large Cardinals and the Continuum

13 / 31Hyperstationary subsets of Some of the result found on 3 and 4 we aim to generalise:
Cantor’s Derivative Operator: d⇠ : P( ) ! P( ), where d⇠ (A) := { <point of A in the topology ⌧⇠ }.

:is limit

If ⌧0 is the order topology on , there is an ascending sequence of topologies on
being ⌧⇠+1 the topology generated by B⇠+1 := B⇠ [ {d⇠ (S) : S ✓ }.
d⇠ (S) = { < : S is ⇠-s-stationary in }, “the set of points where S ⇠-reflects”.
Thus, the topology ⌧⇠ is non-discrete if and only if there is some ⇠-s-stationary.

3 Bagaria, Magidor, Sakai. Reflection And Indescribability In The Constructible Universe
4 Bagaria. Derived Topologies On Ordinals And Stationary Reflection
Martha Catalina Torres Pachón

RIMS Set Theory Workshop 2023 - Large Cardinals and the Continuum

14 / 31Hyperstationary subsets of The set I⇠ = {X ✓  : X is not ⇠-s-stationary in } is an ideal over . In thecase ⇠ = 1, this ideal is the dual ideal to the club filter on , Club().
For every ⇠, an ordinal  is ⇠-s-stationary if and only if I⇠ is a proper ideal.
There is a characterization of the dual filter of I⇠ in terms of d⇠ : F⇠ ={X ✓  : 9⇣ < ⇠ and T1 , T2 ✓ , ⇣-s-stationary such that d⇣ (T1 )\d⇣ (T2 )\ ✓ X}.

Martha Catalina Torres PachónRIMS Set Theory Workshop 2023 - Large Cardinals and the Continuum

15 / 31Hyperstationary subsets of If  is ⇧1⇠ -indescribable, then  is ⇠ + 1-s-stationary.
(V = L) If ⇠ > 0 and  is a regular (⇠ + 1)-s-stationary cardinal, then  is⇧1⇠ -indescribable.

Martha Catalina Torres Pachón

RIMS Set Theory Workshop 2023 - Large Cardinals and the Continuum

16 / 31Hyperstationary subsets of P (A)Contents

1 Introduction
2 Hyperstationary subsets of 3 Hyperstationary subsets of P (A)4 Further Work and Open Questions

Martha Catalina Torres Pachón

RIMS Set Theory Workshop 2023 - Large Cardinals and the Continuum

17 / 31Hyperstationary subsets of P (A)Hyperstationary subsets of P (A)The “struggle” of defining reflection on P (A)In 1971 Thomas Jech presented a generalisation of the concepts of closed unbounded
and stationary set for P (A). These were straightforward generalisations of case .
To define reflection of a stationary subset of P (A), we should keep in mind that werequire  to be an uncountable limit regular cardinal.
In 1999 Jech and Shelah defined a first form of stationary reflection on P (A) on “OnReflection Of Stationary Sets In P ( )” for a general (but inaccessible) .
Martha Catalina Torres Pachón

RIMS Set Theory Workshop 2023 - Large Cardinals and the Continuum18 / 31

Hyperstationary subsets of P (A)Definition (H. Brickhill, S. Fuchino and H. Sakai).
Let  be a regular cardinal,  ✓ A and n < !.

S ✓ P (A) is 0-stationary in P (A) i↵ S is unbounded in P (A).
2 S ✓ P (A) is n-stationary in P (A) i↵ for all m < n and for all T ✓ P (A)m-stationary in P (A), there is B 2 S and µ <  regular cardinal such that1

- µ = B \ ,
- T \ Pµ (B) is m-stationary in Pµ (B)3P (A) is n-stationary if it is n-stationary in P (A) as a subset of P (A).
S ✓ P (A) n-stationary ! S m-stationary

Martha Catalina Torres Pachónfor all m < n.

RIMS Set Theory Workshop 2023 - Large Cardinals and the Continuum

19 / 31Hyperstationary subsets of P (A)Proposition (H. Brickhill, S. Fuchino and H. Sakai).
P (A) is 1-stationary in P (A), then  is weakly Mahlo.
Fix a club C, and consider T := {X 2 P (A) : 9↵ 2 C s.t. X \  ( ↵  |X|}.
T is unbounded in P (A)There is B 2 P (A) such that (1)µ := B \  is a regular cardinal. And (2)T \ Pµ (B) is 0-stationary in Pµ (B).
C \ µ is unbounded in µ < . And so, µ 2 C.

Martha Catalina Torres PachónRIMS Set Theory Workshop 2023 - Large Cardinals and the Continuum

20 / 31Hyperstationary subsets of P (A)Proposition
If  is weakly Mahlo, then P (A) is 1-stationary in P (A).
For T ✓ P (A) 0-stationary, construct a transfinite unbounded sequence
hX : < i of “suitable” elements of T .

Look at the projection of the cardinalities of elements of hX : < i on ,U := {↵ <  : 9 <  s.t. |X | = ↵}. U is unbounded in  and so U \ µ isunbounded in µ.
Construct a “suitable” subsequence hX ↵ : ↵ < µi of hX :elements has cardinality less than µ.
SLet B := ↵<µ X ↵ 2 P (A).
Martha Catalina Torres Pachón< i, such that its

RIMS Set Theory Workshop 2023 - Large Cardinals and the Continuum

21 / 31Hyperstationary subsets of P (A)PropositionIf  is weakly Mahlo, then C ✓ P (A) club implies C is 1-stationary in P (A).
Proposition
If  is weakly Mahlo, then C ✓ P (A) 1.stationary implies C is stationary in P (A).
PropositionP (A) is 2-stationary in P (A), then  is 2-weakly Mahlo.

Martha Catalina Torres PachónRIMS Set Theory Workshop 2023 - Large Cardinals and the Continuum

22 / 31Hyperstationary subsets of P (A)Therefore we have a complete characterisation of weakly Mahloness in terms ofreflection of unbounded sets in P (A). Namely,  is weakly Mahlo if and only ifP (A) is 1-stationary in P (A).
We also have thatS ✓ P (A) club ! S 1-stationary ! S stationary ! S unbounded .
A natural question: Is 1-stationarity equivalent to stationarity as in the case of ?
The answer is unfortunately No.

Martha Catalina Torres PachónRIMS Set Theory Workshop 2023 - Large Cardinals and the Continuum

23 / 31Hyperstationary subsets of P (A)Therefore we have a complete characterisation of weakly Mahloness in terms ofreflection of unbounded sets in P ( ). Namely,  is weakly Mahlo if and only ifP (A) is 1-stationary in P (A).
We also have thatS ✓ P (A) club ! S 1-stationary •! S stationary ! S unbounded .
A natural question: Is 1-stationarity equivalent to stationarity as in the case of ?
The answer is unfortunately No.

Martha Catalina Torres PachónRIMS Set Theory Workshop 2023 - Large Cardinals and the Continuum

24 / 31Hyperstationary subsets of P (A)Proposition
Stationarity does Not implies 1-stationarity in P ( )C0 = {B 2 P ( ) : B \  is a cardinal} is a club subset of P ( ).
S = {B 2 P ( ) : B \  is singular cardinal} is a stationary subset of P ( ).
S cannot be 1-stationary, because any posible reflecting point B in S, does notsatisfy B \  regular. ⇤Martha Catalina Torres PachónRIMS Set Theory Workshop 2023 - Large Cardinals and the Continuum

25 / 31Hyperstationary subsets of P (A)1The set N S,A= {S ✓ P (A) : S is not 1-stationary in P (A)} is an ideal overP (A).

Definition
Let P (A) be n-stationary. Given an m-stationary subset S ✓ P (A) with m < n, letdm (S) := {X 2 P (A) : S m-reflects at X}.
n is the dual filter of N S n , then we have F n = {X ✓ P (A) : 9T ✓ P ( )If F,A
X
,A,Am-stationary for some m < n, such that dm (TX ) ✓ X}.

Martha Catalina Torres PachónRIMS Set Theory Workshop 2023 - Large Cardinals and the Continuum

26 / 31Hyperstationary subsets of P (A)In the direction of finding the consistency strength of n-stationary subsets of P ( ).
Lemma 3.2.11. The formula 'n (S) : “S ✓ P () is n-stationary in P ()” is ⇧1nover hV , 2, Si.
Theorem 3.2.12.
Let n < !. If  is ⇧1n indescribable, then P () is n + 1 stationary.
Theorem 3.2.14. (H. Brickhill, S. Fuchino and H. Sakai)
If  is -supercompact andMartha Catalina Torres Pachón< =then P ( ) is n-stationary for any n < !.

RIMS Set Theory Workshop 2023 - Large Cardinals and the Continuum

27 / 31Further Work and Open Questions

Contents

1 Introduction
2 Hyperstationary subsets of 3 Hyperstationary subsets of P (A)4 Further Work and Open Questions

Martha Catalina Torres PachónRIMS Set Theory Workshop 2023 - Large Cardinals and the Continuum

28 / 31Further Work and Open Questions

Open Questions

What is the least condition we have to demand on  to guarantee P (A) isn-stationary ?
Which are the suitable topologies ⌧n for P ( ) so that dn (S) corresponds tolimit point of S in ⌧n .
How does discreteness of ⌧n in P (A) relate with reflection of n-stationary sets inP (A)?

Martha Catalina Torres PachónRIMS Set Theory Workshop 2023 - Large Cardinals and the Continuum

29 / 31Further Work and Open Questions

Does the converse of Theorem 3.12.14. holds? i.e. If < = , does P ( ) beingn-stationary for any n < ! implies that  is -supercompact ?
(H. Brickhill, S. Fuchino and H. Sakai) Is P ( ) n-stationary for all n < !
assuming  is -strongly compact?

Martha Catalina Torres Pachón

RIMS Set Theory Workshop 2023 - Large Cardinals and the Continuum

30 / 31Further Work and Open Questions

Thank you for your attention.

Martha Catalina Torres Pachón

RIMS Set Theory Workshop 2023 - Large Cardinals and the Continuum

31 / 31