Definable combinatorics at the first uncountable cardinal
Authors: Chan William , Jackson Stephen 
Published: September 2020

We work throughout in the theory

Z
F

\mathsf {ZF}

with the axiom of determinacy,

A
D

\mathsf {AD}

. We introduce and prove some club uniformization principles under

A
D

\mathsf {AD}

and

A
D

R

\mathsf {AD}_\mathbb {R}

. Using these principles, we establish continuity results for functions of the form

Φ
:
[

ω

1

]

ω

1

→

ω

1

\Phi \colon [{\omega _{1}}]^{\omega _{1}} \rightarrow {\omega _{1}}

and

Ψ
:
[

ω

1

]

ω

1

→

ω

1

ω

1

\Psi \colon [{\omega _{1}}]^{\omega _{1}} \rightarrow {}^{\omega _{1}}{\omega _{1}}

. Specifically, for every function

Φ
:
[

ω
1

]

ω
1

→

ω
1

\Phi \colon [\omega _1]^{\omega _1} \rightarrow \omega _1

, there is a club

C
⊆

ω
1

C \subseteq \omega _1

so that

Φ
↾
[
C

]
∗

ω
1

\Phi \upharpoonright [C]^{\omega _1}_*

is a continuous function. This has several consequences such as establishing the cardinal relation

|

[

ω

1

]

>

ω

1

|

>

|

[

ω

1

]

ω

1

|

|[{\omega _{1}}]^{>{\omega _{1}}}| > |[{\omega _{1}}]^{\omega _{1}}|

and answering a question of Zapletal by showing that if

⟨

X
α

:
α
>

ω
1

⟩

\langle X_\alpha : \alpha > \omega _1\rangle

is a collection of subsets of

[

ω
1

]

ω
1

[\omega _1]^{\omega _1}

with the property that

⋃

α
>

ω
1

X
α

=
[

ω
1

]

ω
1

\bigcup _{\alpha > \omega _1}X_\alpha = [\omega _1]^{\omega _1}

, then there is an

α
>

ω
1

\alpha > \omega _1

so that

X
α

X_\alpha

and

[

ω
1

]

ω
1

[\omega _1]^{\omega _1}

are in bijection.

We show that under

A
D

R

\mathsf {AD}_\mathbb {R}

everywhere

[

ω
1

]

>

ω
1

[\omega _1]^{>\omega _1}

-club uniformization holds which is the following statement: Let

c
l
u
b

ω
1

\mathsf {club}_{\omega _1}

denote the collection of club subsets of

ω
1

\omega _1

. Suppose

R
⊆
[

ω
1

]

>

ω
1

×

c
l
u
b

ω
1

R \subseteq [\omega _1]^{>\omega _1} \times \mathsf {club}_{\omega _1}

is

⊆
\subseteq

-downward closed in the sense that for all

σ
∈
[

ω
1

]

>

ω
1

\sigma \in [\omega _1]^{>\omega _1}

, for all clubs

C
⊆
D

C \subseteq D

,

R
(
σ
,
D
)

R(\sigma ,D)

implies

R
(
σ
,
C
)

R(\sigma ,C)

. Then there is a function

F
:

d
o
m

(
R
)
→

c
l
u
b

ω
1

F \colon {\mathrm {dom}}(R) \rightarrow \mathsf {club}_{\omega _1}

so that for all

σ
∈

d
o
m

(
R
)

\sigma \in {\mathrm {dom}}(R)

,

R
(
σ
,
F
(
σ
)
)

R(\sigma ,F(\sigma ))

.

We show that under

A
D

\mathsf {AD}

almost everywhere

[

ω

1

]

>

ω

1

[{\omega _{1}}]^{>{\omega _{1}}}

-club uniformization holds which is the statement that for every

R
⊆
[

ω

1

]

>

ω

1

×

c
l
u
b

ω

1

R \subseteq [{\omega _{1}}]^{>{\omega _{1}}} \times \mathsf {club}_{\omega _{1}}

which is

⊆
\subseteq

-downward closed, there is a club

C
C

and a function
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F colon normal d normal o normal m left-parenthesis upper R right-parenthesis intersection left-bracket upper C right-bracket Subscript asterisk Superscript greater-than omega 1 Baseline right-arrow normal c normal l normal u normal b Subscript omega 1 Baseline">

F
:

d
o
m

(
R
)
∩
[
C

]
∗

>

ω

1

→

c
l
u
b

ω

1

F \colon {\mathrm {dom}}(R) \cap [C]^{>{\omega _{1}}}_* \rightarrow \mathrm {club}_{\omega _{1}}

so that for all

σ
∈

d
o
m

(
R
)
∩
[
C

]
∗

>

ω

1

\sigma \in {\mathrm {dom}}(R) \cap [C]^{>{\omega _{1}}}_*

,

R
(
σ
,
F
(
σ
)
)

R(\sigma ,F(\sigma ))

.