Jäger's psi-functions

Jäger's psi-functions are a hierarchy of single-argument ordinal functions

ψ
π

(
α
)

{\displaystyle \psi\_\pi(\alpha)}

 introduced by german mathematician Gerhard Jäger in 1984. This is an extension of Buchholz's notation.

### Basic Notions

Let

M
0

{\displaystyle M\_0}

 be the least Mahlo cardinal,

ω

{\displaystyle \omega}

 be the first transfinite ordinal, other small Greek letters denote ordinals less than

M
0

{\displaystyle M\_0}

. Each ordinal

α

{\displaystyle \alpha}

 is identified with the set of its predecessors

# α
{
β

β
<
α
}

{\displaystyle \alpha=\{\beta|\beta<\alpha\}}

.

Every ordinal

α

{\displaystyle \alpha}

 is either zero, or a successor (if

# α
β
1

{\displaystyle \alpha=\beta+1}

), or a limit (if for all

β
<
α

{\displaystyle \beta<\alpha}

 there exists an ordinal

γ

{\displaystyle \gamma}

 such that

β
<
γ
<
α

{\displaystyle \beta<\gamma<\alpha}

). Let

S

{\displaystyle S}

 denote the set of all successor ordinals less than

M
0

{\displaystyle M\_0}

 and

L

{\displaystyle L}

 denote the set of all limit ordinals less than

M
0

{\displaystyle M\_0}

.

An ordinal

α

{\displaystyle \alpha}

 is an additive principal number if

α
>
0

{\displaystyle \alpha>0}

 and

ξ
η
<
α

{\displaystyle \xi+\eta<\alpha}

 for all

ξ
,
η
<
α

{\displaystyle \xi,\eta<\alpha}

. Let

P

{\displaystyle P}

 denote the set of all additive principal numbers less than

M
0

{\displaystyle M\_0}

.

α

=

N
F

α

1

⋯

α

n

:⇔
# α

α

1

⋯

α

n

∧

α

1

≥
⋯
≥

α

n

∧

α

1

,
.
.
.
,

α

n

∈
P

{\displaystyle \alpha=\_{NF}\alpha \_{1}+\cdots +\alpha \_{n}:\Leftrightarrow \alpha =\alpha \_{1}+\cdots +\alpha \_{n}\wedge \alpha \_{1}\geq \cdots \geq \alpha \_{n}\wedge \alpha \_{1},... ,\alpha \_{n}\in P}

Cofinality

cof
(
α
)

{\displaystyle \text{cof}(\alpha)}

 of an ordinal

α

{\displaystyle \alpha}

 is the least

β

{\displaystyle \beta}

 such that there exists a
function

f
:
β
→
α

{\displaystyle f:\beta\rightarrow\alpha}

 with

sup
{
f
(
ξ
)

ξ
<
β
# }
α

{\displaystyle \text{sup}\{f(\xi )|\xi <\beta \}=\alpha}

. An ordinal

α

{\displaystyle \alpha}

 is regular, if

α

{\displaystyle \alpha}

 is a limit ordinal and

cof
(
α
# )
α

{\displaystyle \text{cof}(\alpha)=\alpha}

. Let

R

{\displaystyle R}

 denote the set of all regular ordinals

∈
{
α

ω
<
α
<

M
0

}

{\displaystyle \in\{\alpha|\omega<\alpha<M\_0\}}

.

An ordinal

α

{\displaystyle \alpha}

 is (weakly) inaccessible if

α

{\displaystyle \alpha}

 is a regular limit cardinal larger than

ω

{\displaystyle \omega}

.

### Veblen function

φ
0

(
α
# )

ω

α

{\displaystyle \varphi\_0(\alpha)=\omega^{\alpha}}

.

φ
α

(
β
# )
min
{
γ
∈
P

(
∀
δ
<
α
,
# γ

φ
δ

(
γ
)
)
∧
(
∀
δ
<
β
,
γ
>

φ
α

(
δ
)
)
}

{\displaystyle \varphi\_\alpha(\beta)=\text{min}\{\gamma\in P|(\forall\delta<\alpha, \gamma=\varphi\_\delta(\gamma))\wedge(\forall\delta<\beta, \gamma>\varphi\_\alpha(\delta))\}}

Normal form

α

=

N
F

φ
β

(
γ
)
:⇔
# α

φ
β

(
γ
)
∧
β
,
γ
<
α

{\displaystyle \alpha=\_{NF}\varphi\_\beta(\gamma):\Leftrightarrow\alpha=\varphi\_\beta(\gamma)\wedge\beta,\gamma<\alpha}

An ordinal

α

{\displaystyle \alpha}

 is a strongly critical if

φ
(
α
,
0
# )
α

{\displaystyle \varphi(\alpha,0)=\alpha}

. Let

D

{\displaystyle D}

 denote the set of
all strongly critical ordinals less than

M
0

{\displaystyle M\_0}

.

Definition of

D
(
γ
)

{\displaystyle D(\gamma)}

 for arbitrary

γ

{\displaystyle \gamma}

.

D
(
γ
# )
{
γ
}

{\displaystyle D(\gamma)=\{\gamma\}}

 if

γ
∈
D
∪
{
0
}

{\displaystyle \gamma\in D\cup\{0\}}

D
(
γ
# )
{

α
1

,
.
.
.
,

α
n

}

{\displaystyle D(\gamma)=\{\alpha\_1,...,\alpha\_n\}}

 if

γ

=

N
F

α
1

⋯

α
n

∉
P

{\displaystyle \gamma=\_{NF}\alpha\_1+\cdots+\alpha\_n\notin P}

D
(
γ
# )
{
α
,
β
}

{\displaystyle D(\gamma)=\{\alpha,\beta\}}

 if

γ

=

N
F

φ
α

(
β
)
∉
D

{\displaystyle \gamma=\_{NF}\varphi\_\alpha(\beta)\notin D}

###

ρ

{\displaystyle \rho}

-Inaccessible Ordinals

An ordinal is

ρ

{\displaystyle \rho}

-inaccessible if it is a regular cardinal and limit of

α

{\displaystyle \alpha}

-inaccessible ordinals for all

α
<
ρ

{\displaystyle \alpha<\rho}

. So the 0-inaccessible ordinals are exactly the regular cardinals

>
ω

{\displaystyle >\omega}

, the 1-inaccessible ordinals are the inaccessible ordinals. Functions

I
ρ

:

M
0

→

M
0

{\displaystyle I\_\rho:M\_0 \rightarrow M\_0}

 enumerate the

ρ

{\displaystyle \rho}

-inaccessible ordinals less than

M
0

{\displaystyle M\_0}

 and their limits.

I
α

(
β
# )
min
{
γ
∈
R

(
∀
δ
<
α
,
# γ

I
δ

(
γ
)
)
∧
(
∀
η
<
β
,
γ
>

I
α

(
η
)
)
}

{\displaystyle I\_\alpha(\beta)=\text{min}\{\gamma\in R|(\forall\delta<\alpha,\gamma=I\_\delta(\gamma))\wedge(\forall\eta<\beta,\gamma>I\_\alpha(\eta))\}}

 if

β
∉
L

{\displaystyle \beta\notin L}

I
α

(
β
# )
sup
{

I
α

(
γ
)

γ
<
β
}

{\displaystyle I\_\alpha(\beta)=\text{sup}\{I\_\alpha(\gamma)|\gamma<\beta\}}

 if

β
∈
L

{\displaystyle \beta\in L}

Normal form

α

=

N
F

I
β

(
γ
)
:⇔
# α

I
β

(
γ
)
∧
γ
∉
L

{\displaystyle \alpha=\_{NF}I\_\beta(\gamma):\Leftrightarrow\alpha=I\_\beta(\gamma)\wedge\gamma\notin L}

Definition of

γ

−

{\displaystyle \gamma^{-}}

 for

γ
∈
R

{\displaystyle \gamma\in R}

.

γ

−

=
0

{\displaystyle \gamma^{-}=0}

 if

γ

=

N
F

I
α

(
0
)

{\displaystyle \gamma=\_{NF}I\_\alpha(0)}

γ

−

=

I
α

(
β
)

{\displaystyle \gamma^{-}=I\_\alpha(\beta)}

 if

γ

=

N
F

I
α

(
β
1
)

{\displaystyle \gamma=\_{NF}I\_\alpha(\beta+1)}

Properties

ρ

{\displaystyle \rho}

φ
α

(
β
)
∈
P

{\displaystyle \varphi\_\alpha(\beta)\in P}

I
α

(
0
)
,

I
α

(
β
1
)
∈
R

{\displaystyle I\_\alpha(0), I\_\alpha(\beta+1)\in R}

γ
<
α
⇒

φ
γ

(

φ
α

(
β
)
# )

φ
α

(
β
)

{\displaystyle \gamma<\alpha\Rightarrow\varphi\_\gamma(\varphi\_\alpha(\beta))=\varphi\_\alpha(\beta)}

γ
<
α
⇒

I
γ

(

I
α

(
β
)
# )

I
α

(
β
)

{\displaystyle \gamma<\alpha\Rightarrow I\_\gamma(I\_\alpha(\beta))=I\_\alpha(\beta)}

β
<
γ
⇒

φ
α

(
β
)
<

φ
α

(
γ
)

{\displaystyle \beta<\gamma\Rightarrow\varphi\_\alpha(\beta)<\varphi\_\alpha(\gamma)}

β
<
γ
⇒

I
α

(
β
)
<

I
α

(
γ
)

{\displaystyle \beta<\gamma\Rightarrow I\_\alpha(\beta)<I\_\alpha(\gamma)}

α
<
β
⇒

φ
α

(
0
)
<

φ
β

(
0
)

{\displaystyle \alpha<\beta\Rightarrow\varphi\_\alpha(0)<\varphi\_\beta(0)}

α
<
β
⇒

I
α

(
0
)
<

I
β

(
0
)

{\displaystyle \alpha<\beta\Rightarrow I\_\alpha(0)<I\_\beta(0)}

### The Ordinal Functions

ψ
κ

{\displaystyle \psi\_\kappa}

Every

ψ
κ

{\displaystyle \psi\_\kappa}

 is a function from

M
0

{\displaystyle M\_0}

 to

κ

{\displaystyle \kappa}

 which "collapses" the elements of

M
0

{\displaystyle M\_0}

 below

κ

{\displaystyle \kappa}

. By the Greek letters

κ

{\displaystyle \kappa}

 and

π

{\displaystyle \pi }

 we shall denote uncountable regular cardinals less than

M
0

{\displaystyle M\_0}

.

Inductive Definition of

C
κ

(
α
)

{\displaystyle C\_\kappa(\alpha)}

 and

ψ
κ

(
α
)

{\displaystyle \psi\_\kappa(\alpha)}

.

{

κ

−

}
∪

κ

−

⊂

C
κ
n

(
α
)

{\displaystyle \{\kappa^{-}\}\cup\kappa^{-}\subset C\_\kappa^n(\alpha)}

D
(
γ
)
⊂

C
κ
n

(
α
)
⇒
γ
∈

C
κ

n
1

(
α
)

{\displaystyle D(\gamma)\subset C\_\kappa^n(\alpha)\Rightarrow\gamma\in C\_\kappa^{n+1}(\alpha)}

β
,
γ
∈

C
κ
n

(
α
)
⇒

I
β

(
γ
)
∈

C
κ

n
1

(
α
)

{\displaystyle \beta,\gamma\in C\_\kappa^n(\alpha)\Rightarrow I\_\beta(\gamma)\in C\_\kappa^{n+1}(\alpha)}

γ
<
π
<
κ
∧
π
∈

C
κ
n

(
α
)
⇒
γ
∈

C
κ

n
1

(
α
)

{\displaystyle \gamma<\pi<\kappa\wedge\pi\in C\_\kappa^n(\alpha)\Rightarrow \gamma\in C\_\kappa^{n+1}(\alpha)}

γ
<
α
∧
γ
,
π
∈

C
κ
n

(
α
)
∧
γ
∈

C
π

(
γ
)
⇒

ψ
π

(
γ
)
∈

C
κ

n
1

(
α
)

{\displaystyle \gamma<\alpha\wedge\gamma,\pi\in C\_\kappa^n(\alpha)\wedge\gamma\in C\_\pi(\gamma)\Rightarrow \psi\_\pi(\gamma)\in C\_\kappa^{n+1}(\alpha)}

C
κ

(
α
# )
∪
{

C
κ
n

(
α
)

n
<
ω
}

{\displaystyle C\_\kappa(\alpha)=\cup\{C\_\kappa^n(\alpha)|n<\omega\}}

ψ
κ

(
α
# )
min
{
ξ

ξ
∉

C
κ

(
α
)
}

{\displaystyle \psi\_\kappa(\alpha)=\text{min}\{\xi|\xi\notin C\_\kappa(\alpha)\}}

## Sources

1. ↑ Jäger's psi-functions
2. ↑ Gerhard Jäger. INACCESSIBLE ORDINALS, COLLAPSING FUNCTIONS AND A RECURSIVE NOTATION SYSTEM. Arch. math. Logik 24 (1984), 49-62