Buchholz's functions

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

ψ
ν

(
α
)

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

 introduced by german mathematician Wilfried Buchholz in 1986. These functions are a simplified version of the

θ

{\displaystyle \theta}

-functions, but nevertheless have the same strength as those. Later on this approach was extended by Jaiger and Schtitte.

## Definition

Buchholz defined his functions as follows:

* C
ν
0

(
α
# )

Ω
ν

{\displaystyle C\_\nu^0(\alpha) = \Omega\_\nu}

,
* C
ν

n
1

(
α
# )

C
ν
n

(
α
)
∪
{
γ

P
(
γ
)
⊆

C
ν
n

(
α
)
}
∪
{

ψ
μ

(
ξ
)

ξ
∈
α
∩

C
ν
n

(
α
)
∧
ξ
∈

C
μ

(
ξ
)
∧
μ
≤
ω
}

{\displaystyle C\_\nu^{n+1}(\alpha) = C\_\nu^n(\alpha) \cup \{\gamma | P(\gamma) \subseteq C\_\nu^n(\alpha)\} \cup \{\psi\_\mu(\xi) | \xi \in \alpha \cap C\_\nu^n(\alpha) \wedge \xi \in C\_\mu(\xi) \wedge \mu \leq \omega\}}

,
* C
ν

(
α
# )

⋃

n
<
ω

C
ν
n

(
α
)

{\displaystyle C\_\nu(\alpha) = \bigcup\_{n < \omega} C\_\nu^n (\alpha)}

,
* ψ
ν

(
α
# )
min
{
γ

γ
∉

C
ν

(
α
)
}

{\displaystyle \psi\_\nu(\alpha) = \min\{\gamma | \gamma \not\in C\_\nu(\alpha)\}}

,

where

Ω
ν

=

{

1
 if
# ν
0

ℵ
ν

 if
ν
>
0

{\displaystyle \Omega\_\nu=\left\{\begin{array}{lcr}
1\text{ if }\nu=0\\
\aleph\_\nu\text{ if }\nu>0\\
\end{array}\right.}

and

P
(
γ
# )
{

γ
1

,
.
.
.
,

γ
k

}

{\displaystyle P(\gamma)=\{\gamma\_1,...,\gamma\_k\}}

 is the set of additive principal numbers in form

ω

ξ

{\displaystyle \omega^{\xi}}

,

# P
{
α
∈
O
n
:
0
<
α
∧
∀
ξ
,
η
<
α
(
ξ
η
<
α
)
# }
{

ω
ξ

:
ξ
∈
O
n
}

{\displaystyle P=\{\alpha\in On: 0<\alpha \wedge \forall \xi, \eta < \alpha (\xi+\eta < \alpha)\}=\{\omega^\xi: \xi \in On\}}

,

the sum of which gives this ordinal

γ

{\displaystyle \gamma}

:

# γ

α
1

α
2

⋯

α
k

{\displaystyle \gamma=\alpha\_1+\alpha\_2+\cdots+\alpha\_k}

 where

α
1

≥

α
2

≥
⋯
≥

α
k

{\displaystyle \alpha\_1\geq\alpha\_2\geq\cdots\geq\alpha\_k}

 and

α
1

,

α
2

,
.
.
.
,

α
k

∈
P
(
γ
)

{\displaystyle \alpha\_1,\alpha\_2,...,\alpha\_k \in P(\gamma)}

.

Note: Greek letters always denotes ordinals.

O
n

{\displaystyle On}

 denotes the class of all ordinals.

The limit of this notation is Takeuti-Feferman-Buchholz ordinal.

## Properties

Buchholz showed following properties of those functions:

* ψ
ν

(
0
# )

Ω
ν

{\displaystyle \psi\_\nu(0)=\Omega\_\nu}

,
* ψ
ν

(
α
)
∈
P

{\displaystyle \psi\_\nu(\alpha)\in P}

,
* ψ
ν

(
α
1
# )
min
{
γ
∈
P
:

ψ
ν

(
α
)
<
γ
}
 if
α
∈

C
ν

(
α
)

{\displaystyle \psi\_\nu(\alpha+1)=\text{min}\{\gamma\in P: \psi\_\nu(\alpha)<\gamma\}\text{ if }\alpha\in C\_\nu(\alpha)}

,
* Ω
ν

≤

ψ
ν

(
α
)
<

Ω

ν
1

{\displaystyle \Omega\_\nu\le\psi\_\nu(\alpha)<\Omega\_{\nu+1}}

,
* α
≤
β
⇒

ψ
ν

(
α
)
≤

ψ
ν

(
β
)

{\displaystyle \alpha\le\beta\Rightarrow\psi\_\nu(\alpha)\le\psi\_\nu(\beta)}

,
* ψ
0

(
α
# )

ω
α

 if
α
<

ε
0

{\displaystyle \psi\_0(\alpha)=\omega^\alpha \text{ if }\alpha<\varepsilon\_0}

,
* ψ
ν

(
α
# )

ω

Ω
ν

α

 if
α
<

ε

Ω
ν

1

 and
ν
≠
0

{\displaystyle \psi\_\nu(\alpha)=\omega^{\Omega\_\nu+\alpha} \text{ if }\alpha<\varepsilon\_{\Omega\_\nu+1} \text{ and } \nu\neq 0}

,
* θ
(

ε

Ω
ν

1

,
0
# )

ψ
0

(

ε

Ω
ν

1

)

{\displaystyle \theta(\varepsilon\_{\Omega\_\nu+1},0)=\psi\_0(\varepsilon\_{\Omega\_\nu+1})}

 for

0
<
ν
≤
ω

{\displaystyle 0<\nu\le\omega}

.

## Explanation

Buchholz is working in Zermelo–Fraenkel set theory, that means every ordinal

α

{\displaystyle \alpha}

 is equal to set

{
β

β
<
α
}

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

. Then condition

C
ν
0

(
α
# )

Ω
ν

{\displaystyle C\_\nu^0(\alpha) = \Omega\_\nu}

 means that set

C
ν
0

(
α
)

{\displaystyle C\_\nu^0(\alpha)}

 includes all ordinals less than

Ω
ν

{\displaystyle \Omega\_\nu}

 in other words

C
ν
0

(
α
# )
{
β

β
<

Ω
ν

}

{\displaystyle C\_\nu^0(\alpha) = \{\beta|\beta<\Omega\_\nu\}}

.

The condition

C
ν

n
1

(
α
# )

C
ν
n

(
α
)
∪
{
γ

P
(
γ
)
⊆

C
ν
n

(
α
)
}
∪
{

ψ
μ

(
ξ
)

ξ
∈
α
∩

C
ν
n

(
α
)
∧
μ
≤
ω
}

{\displaystyle C\_\nu^{n+1}(\alpha) = C\_\nu^n(\alpha) \cup \{\gamma | P(\gamma) \subseteq C\_\nu^n(\alpha)\} \cup \{\psi\_\mu(\xi) | \xi \in \alpha \cap C\_\nu^n(\alpha) \wedge \mu \leq \omega\}}

 means that set

C
ν

n
1

(
α
)

{\displaystyle C\_\nu^{n+1}(\alpha)}

 includes:

1. all ordinals from previous set

C
ν
n

(
α
)

{\displaystyle C\_\nu^n(\alpha)}

,
2. all ordinals that can be obtained by summation the additively principal ordinals from previous set

C
ν
n

(
α
)

{\displaystyle C\_\nu^n(\alpha)}

,
3. all ordinals that can be obtained by applying ordinals less than

α

{\displaystyle \alpha}

 from the previous set

C
ν
n

(
α
)

{\displaystyle C\_\nu^n(\alpha)}

 as arguments of functions

ψ
μ

{\displaystyle \psi\_\mu}

, where

μ
≤
ω

{\displaystyle \mu\le\omega}

.

That is why we can rewrite this condition as:

C
ν

n
1

(
α
# )
{
β
γ
,

ψ
μ

(
η
)

β
,
γ
,
η
∈

C

ν

n

(
α
)
∧
η
<
α
∧
μ
≤
ω
}

{\displaystyle C\_\nu^{n+1}(\alpha) = \{\beta+\gamma,\psi\_\mu(\eta)|\beta, \gamma,\eta\in C\_{\nu}^n(\alpha)\wedge\eta<\alpha \wedge \mu \leq \omega\}}

.

Thus union of all sets

C
ν
n

(
α
)

{\displaystyle C\_\nu^n(\alpha)}

 with

n
<
ω

{\displaystyle n<\omega}

 i.e.

C
ν

(
α
# )

⋃

n
<
ω

C
ν
n

(
α
)

{\displaystyle C\_\nu(\alpha) = \bigcup\_{n < \omega} C\_\nu^n (\alpha)}

 denotes the set of all ordinals which can be generated from ordinals

<

ℵ
ν

{\displaystyle <\aleph\_\nu}

 by the functions + (addition) and

ψ

μ

(
ξ
)

{\displaystyle \psi\_{\mu}(\xi)}

, where

μ
≤
ω

{\displaystyle \mu\le\omega}

 and

ξ
<
α

{\displaystyle \xi<\alpha}

.

Then

ψ
ν

(
α
# )
min
{
γ

γ
∉

C
ν

(
α
)
}

{\displaystyle \psi\_\nu(\alpha) = \min\{\gamma | \gamma \not\in C\_\nu(\alpha)\}}

 is the smallest ordinal that does not belong to this set.

### Examples

Consider the following examples:

C
0
0

(
α
# )
{
0
# }
{
β
:
β
<
1
}

{\displaystyle C\_0^0(\alpha)=\{0\} =\{\beta:\beta<1\}}

,

C
0

(
0
# )
{
0
}

{\displaystyle C\_0(0)=\{0\}}

 (since no functions

ψ
(
η
<
0
)

{\displaystyle \psi(\eta<0)}

 and 0+0=0).

Then

ψ
0

(
0
# )
1

{\displaystyle \psi\_0(0)=1}

.

C
0

(
1
)

{\displaystyle C\_0(1)}

 includes

ψ
0

(
0
# )
1

{\displaystyle \psi\_0(0)=1}

 and all possible sums of natural numbers:

C
0

(
1
# )
{
0
,
1
,
2
,
.
.
.
,
googol
,
.
.
.
,
TREE(googol)
,
.
.
.
}

{\displaystyle C\_0(1)=\{0,1,2,...,\text{googol}, ...,\text{TREE(googol)},...\}}

.

Then

ψ
0

(
1
# )
ω

{\displaystyle \psi\_0(1)=\omega}

 - first transfinite ordinal, which is greater than all natural numbers by its definition.

C
0

(
2
)

{\displaystyle C\_0(2)}

 includes

ψ
0

(
0
# )
1
,

ψ
0

(
1
# )
ω

{\displaystyle \psi\_0(0)=1, \psi\_0(1)=\omega}

 and all possible sums of them.

Then

ψ
0

(
2
# )

ω
2

{\displaystyle \psi\_0(2)=\omega^2}

.

For

C
0

(
ω
)

{\displaystyle C\_0(\omega)}

 we have set

C
0

(
ω
# )
{
0
,
ψ
(
0
# )
1
,
.
.
.
,
ψ
(
1
# )
ω
,
.
.
.
,
ψ
(
2
# )

ω
2

,
.
.
.
,
ψ
(
3
# )

ω
3

,
.
.
.
}

{\displaystyle C\_0(\omega)=\{0,\psi(0)=1,...,\psi(1)=\omega,...,\psi(2)=\omega^2,...,\psi(3)=\omega^3,...\}}

.

Then

ψ
0

(
ω
# )

ω
ω

{\displaystyle \psi\_0(\omega)=\omega^\omega}

.

For

C
0

(
Ω
)

{\displaystyle C\_0(\Omega)}

 we have set

C
0

(
Ω
# )
{
0
,
ψ
(
0
# )
1
,
.
.
.
,
ψ
(
1
# )
ω
,
.
.
.
,
ψ
(
ω
# )

ω
ω

,
.
.
.
,
ψ
(

ω
ω

# )

ω

ω
ω

,
.
.
.
}

{\displaystyle C\_0(\Omega)=\{0,\psi(0)=1,...,\psi(1)=\omega,...,\psi(\omega)=\omega^\omega,...,\psi(\omega^\omega)=\omega^{\omega^\omega},...\}}

.

Then

ψ
0

(
Ω
# )

ε
0

{\displaystyle \psi\_0(\Omega)=\varepsilon\_0}

.

For

C
0

(
Ω
1
)

{\displaystyle C\_0(\Omega+1)}

 we have set

C
0

(
Ω
# )
{
0
,
1
,
.
.
.
,

ψ
0

(
Ω
# )

ε
0

,
.
.
.
,

ε
0

ε
0

,
.
.
.

ψ
1

(
0
# )
Ω
,
.
.
.
}

{\displaystyle C\_0(\Omega)=\{0,1,...,\psi\_0(\Omega)=\varepsilon\_0,...,\varepsilon\_0+\varepsilon\_0,...\psi\_1(0)=\Omega,...\}}

.

Then

ψ
0

(
Ω
1
# )

ε
0

# ω

ω

ε
0

1

{\displaystyle \psi\_0(\Omega+1)=\varepsilon\_0\omega=\omega^{\varepsilon\_0+1}}

.

ψ
0

(
Ω
2
# )

ε
1

{\displaystyle \psi\_0(\Omega2)=\varepsilon\_1}

,

ψ
0

(

Ω
2

# )

ζ
0

{\displaystyle \psi\_0(\Omega^2)=\zeta\_0}

,

φ
(
α
,
1
β
# )

ψ
0

(

Ω
α

β
)

{\displaystyle \varphi(\alpha,1+\beta)=\psi\_0(\Omega^\alpha\beta)}

,

ψ
0

(

Ω
Ω

# )

Γ
0

=
θ
(
Ω
,
0
)

{\displaystyle \psi\_0(\Omega^\Omega)=\Gamma\_0=\theta(\Omega,0)}

, using Feferman theta-function,

ψ
0

(

Ω

Ω

Ω

)

{\displaystyle \psi\_0(\Omega^{\Omega^{\Omega}})}

 is large Veblen ordinal,

ψ
0

(
Ω
↑↑
ω
# )

ψ
0

(

ε

Ω
1

# )
θ
(

ε

Ω
1

,
0
)

{\displaystyle \psi\_0(\Omega\uparrow\uparrow\omega)=\psi\_0(\varepsilon\_{\Omega+1})=\theta(\varepsilon\_{\Omega+1},0)}

.

Now let's research how

ψ
1

{\displaystyle
\psi\_1
}

 works:

C
1
0

(
α
# )
{
β
:
β
<

Ω
1

# }
{
0
,
ψ
(
0
# )
1
,
2
,
.
.
.
googol
,
.
.
.
,

ψ
0

(
1
# )
ω
,
.
.
.
,

ψ
0

(
Ω
# )

ε
0

,
.
.
.

{\displaystyle C\_1^0(\alpha)=\{\beta:\beta<\Omega\_1\}=\{0,\psi(0)=1,2,...\text{googol},...,\psi\_0(1)=\omega,...,\psi\_0(\Omega)=\varepsilon\_0,...}

.
.
.
,

ψ
0

(

Ω
Ω

# )

Γ
0

,
.
.
.
,
ψ
(

Ω

Ω
Ω

Ω
2

)
,
.
.
.
}

{\displaystyle ...,\psi\_0(\Omega^\Omega)=\Gamma\_0,...,\psi(\Omega^{\Omega^\Omega+\Omega^2}),...\}}

 i.e. includes all countable ordinals.

C
1

(
α
)

{\displaystyle C\_1(\alpha)}

 includes all possible sums of all countable ordinals. Then

ψ
1

(
0
# )

Ω
1

{\displaystyle \psi\_1(0)=\Omega\_1}

 first uncountable ordinal which is greater than all countable ordinal by its definition i.e. smallest number with cardinality

ℵ
1

{\displaystyle \aleph\_1}

.

C
1

(
1
# )
{
0
,
.
.
.
,

ψ
0

(
0
# )
ω
,
.
.
.
,

ψ
1

(
0
# )
Ω
,
.
.
.
,
Ω
ω
,
.
.
.
,
Ω
Ω
,
.
.
.
}

{\displaystyle C\_1(1)=\{0,...,\psi\_0(0)=\omega,...,\psi\_1(0)=\Omega,...,\Omega+\omega,...,\Omega+\Omega,...\}}

Then

ψ
1

(
1
# ω

ω

Ω
1

{\displaystyle \psi\_1(1)=\Omega\omega=\omega^{\Omega+1}}

.

Then

ψ
1

(
2
# )
Ω

ω
2

=

ω

Ω
2

{\displaystyle \psi\_1(2)=\Omega\omega^2=\omega^{\Omega+2}}

,

ψ
1

(

ψ
0

(
Ω
)
# )
Ω

ε
0

=

ω

Ω

ε
0

{\displaystyle \psi\_1(\psi\_0(\Omega))=\Omega\varepsilon\_0=\omega^{\Omega+\varepsilon\_0}}

,

ψ
1

(

ψ
0

(

Ω
Ω

)
# )
Ω

Γ
0

=

ω

Ω

Γ
0

{\displaystyle \psi\_1(\psi\_0(\Omega^\Omega))=\Omega\Gamma\_0=\omega^{\Omega+\Gamma\_0}}

,

ψ
1

(

ψ
1

(
0
)
# )

ψ
1

(
Ω
# )

Ω
2

=

ω

Ω
Ω

{\displaystyle \psi\_1(\psi\_1(0))=\psi\_1(\Omega)=\Omega^2=\omega^{\Omega+\Omega}}

,

ψ
1

(

ψ
1

(

ψ
1

(
0
)
)
# )

ω

Ω

ω

Ω
Ω

=

ω

Ω
⋅
Ω

=
(

ω

Ω

)
Ω

=

Ω
Ω

{\displaystyle \psi\_1(\psi\_1(\psi\_1(0)))=\omega^{\Omega+\omega^{\Omega+\Omega}}=\omega^{\Omega\cdot\Omega}=(\omega^{\Omega})^\Omega=\Omega^\Omega}

,

ψ
1
4

(
0
# )

Ω

Ω
Ω

{\displaystyle \psi\_1^4(0)=\Omega^{\Omega^\Omega}}

,

ψ
1

(

Ω
2

# )

ψ
1
ω

(
0
# )
Ω
↑↑
# ω

ε

Ω
1

{\displaystyle \psi\_1(\Omega\_2)=\psi\_1^\omega(0)=\Omega\uparrow\uparrow\omega=\varepsilon\_{\Omega+1}}

.

For case

ψ
(

Ω
2

)

{\displaystyle \psi(\Omega\_2)}

 the set

C
0

(

Ω
2

)

{\displaystyle C\_0(\Omega\_2)}

 includes functions

ψ
0

{\displaystyle \psi\_0}

 with all arguments less than

Ω
2

{\displaystyle \Omega\_2}

 i.e. such arguments as

0
,

ψ
1

(
0
)
,

ψ
1

(

ψ
1

(
0
)
)
,

ψ
1
3

(
0
)
,
.
.
.
,

ψ
1
ω

(
0
)

{\displaystyle 0, \psi\_1(0), \psi\_1(\psi\_1(0)), \psi\_1^3(0),..., \psi\_1^\omega(0)}

and then

ψ
0

(

Ω
2

# )

ψ
0

(

ψ
1

(

Ω
2

)
# )

ψ
0

(

ε

Ω
1

)

{\displaystyle \psi\_0(\Omega\_2)=\psi\_0(\psi\_1(\Omega\_2))=\psi\_0(\varepsilon\_{\Omega+1})}

.

In general case:

ψ
0

(

Ω

ν
1

# )

ψ
0

(

ψ
ν

(

Ω

ν
1

)
# )

ψ
0

(

ε

Ω
ν

1

# )
θ
(

ε

Ω
ν

1

,
0
)

{\displaystyle \psi\_0(\Omega\_{\nu+1})=\psi\_0(\psi\_\nu(\Omega\_{\nu+1}))=\psi\_0(\varepsilon\_{\Omega\_\nu+1})=\theta(\varepsilon\_{\Omega\_\nu+1},0)}

.

We also can write:

θ
(

Ω
ν

,
0
# )

ψ
0

(

Ω
ν

Ω
ν

)

{\displaystyle \theta(\Omega\_\nu,0)=\psi\_0(\Omega\_\nu^{\Omega\_\nu})}

 ( for

1
≤
ν
<
ω

{\displaystyle 1\le\nu<\omega}

).

## Extension

We rewrite Buchholz's definition as follows :

* C
ν
0

(
α
# )
{
β

β
<

Ω
ν

}

{\displaystyle C\_\nu^0(\alpha) = \{\beta|\beta<\Omega\_\nu\}}

,
* C
ν

n
1

(
α
# )
{
β
γ
,

ψ
μ

(
η
)

μ
,
β
,
γ
,
η
∈

C

ν

n

(
α
)
∧
η
<
α
}

{\displaystyle C\_\nu^{n+1}(\alpha) = \{\beta+\gamma,\psi\_\mu(\eta)|\mu,\beta, \gamma,\eta\in C\_{\nu}^n(\alpha)\wedge\eta<\alpha\}}

,
* C
ν

(
α
# )

⋃

n
<
ω

C
ν
n

(
α
)

{\displaystyle C\_\nu(\alpha) = \bigcup\_{n < \omega} C\_\nu^n (\alpha)}

,
* ψ
ν

(
α
# )
min
{
γ

γ
∉

C
ν

(
α
)
}

{\displaystyle \psi\_\nu(\alpha) = \min\{\gamma | \gamma \not\in C\_\nu(\alpha)\}}

,

where

Ω
ν

=

{

1
 if
# ν
0

smallest ordinal with cardinality

ℵ
ν

 if
ν
>
0

{\displaystyle \Omega\_\nu=\left\{\begin{array}{lcr}
1\text{ if }\nu=0\\
\text{smallest ordinal with cardinality }\aleph\_\nu \text{ if }\nu>0\\
\end{array}\right.}

and

ω

{\displaystyle \omega}

 is the smallest infinite ordinal.

There is only one little detail difference with original Buchholz definition: ordinal

μ

{\displaystyle \mu}

 is not limited by

ω

{\displaystyle \omega}

, now ordinal

μ

{\displaystyle \mu}

 belongs to previous set

C
n

{\displaystyle C\_n}

.

For example if

C
0
0

(
1
# )
{
0
}

{\displaystyle C\_0^0(1)=\{0\}}

 then

C
0
1

(
1
# )
{
0
,

ψ
0

(
0
# )
1
}

{\displaystyle C\_0^1(1)=\{0,\psi\_0(0)=1\}}

 and

C
0
2

(
1
# )
{
0
,
.
.
.
,

ψ
1

(
0
# )
Ω
}

{\displaystyle C\_0^2 (1)=\{0,...,\psi\_1(0)=\Omega\}}

 and

C
0
3

(
1
# )
{
0
,
.
.
.
,

ψ
Ω

(
0
# )

Ω
Ω

}

{\displaystyle C\_0^3(1)=\{0,...,\psi\_\Omega(0) = \Omega\_\Omega\}}

 and so on.

Limit of this notation must be omega fixed point

ψ
(

Ω

Ω

Ω

.
.
.

# )
ψ
(

ψ

ψ

.
.
.

(
0
)

(
0
)
)

{\displaystyle \psi(\Omega\_{\Omega\_{\Omega\_{...}}})=\psi(\psi\_{\psi\_{...}(0)}(0))}

, where

ψ

{\displaystyle \psi}

 without subscript denotes

ψ
0

{\displaystyle \psi\_0}

.

## Normal form and fundamental sequences

### Normal form

The normal form for 0 is 0. If

α

{\displaystyle \alpha}

 is a nonzero ordinal number

α
<
# Λ
min
{
β

ψ
β

(
0
# )
β
}

{\displaystyle \alpha<\Lambda=\text{min}\{\beta|\psi\_\beta(0)=\beta\}}

 then the normal form for

α

{\displaystyle \alpha}

 is

# α

ψ

ν
1

(

β
1

)

ψ

ν
2

(

β
2

)
⋯

ψ

ν
k

(

β
k

)

{\displaystyle \alpha=\psi\_{\nu\_1}(\beta\_1)+\psi\_{\nu\_2}(\beta\_2)+\cdots+\psi\_{\nu\_k}(\beta\_k)}

 where

k

{\displaystyle k}

 is a positive integer and

ψ

ν
1

(

β
1

)
≥

ψ

ν
2

(

β
2

)
≥
⋯
≥

ψ

ν
k

(

β
k

)

{\displaystyle \psi\_{\nu\_1}(\beta\_1)\geq\psi\_{\nu\_2}(\beta\_2)\geq\cdots\geq\psi\_{\nu\_k}(\beta\_k)}

 and each

ν
i

{\displaystyle \nu\_i}

,

β
i

{\displaystyle \beta\_i}

 are also written in normal form.

### Fundamental sequences

The fundamental sequence for an ordinal number

α

{\displaystyle \alpha}

 with cofinality

cof
(
α
# )
β

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

 is a strictly increasing sequence

(
α
[
η
]

)

η
<
β

{\displaystyle (\alpha[\eta])\_{\eta<\beta}}

 with length

β

{\displaystyle \beta}

 and with limit

α

{\displaystyle \alpha}

, where

α
[
η
]

{\displaystyle \alpha[\eta]}

 is the

η

{\displaystyle \eta }

-th element of this sequence. If

α

{\displaystyle \alpha}

 is a successor ordinal then

cof
(
α
# )
1

{\displaystyle \text{cof}(\alpha)=1}

 and the fundamental sequence has only one element

α
[
0
# ]
α
−
1

{\displaystyle \alpha[0]=\alpha-1}

. If

α

{\displaystyle \alpha}

 is a limit ordinal then

cof
(
α
)
∈
{
ω
}
∪
{

Ω

μ
1

μ
≥
0
}

{\displaystyle \text{cof}(\alpha)\in\{\omega\}\cup\{\Omega\_{\mu+1}|\mu\geq 0\}}

.

For nonzero ordinals

α
<
Λ

{\displaystyle \alpha<\Lambda}

, written in normal form, fundamental sequences are defined as follows:

1. If

# α

ψ

ν
1

(

β
1

)

ψ

ν
2

(

β
2

)
⋯

ψ

ν
k

(

β
k

)

{\displaystyle \alpha=\psi\_{\nu\_1}(\beta\_1)+\psi\_{\nu\_2}(\beta\_2)+\cdots+\psi\_{\nu\_k}(\beta\_k)}

 where

k
≥
2

{\displaystyle k \geq 2}

 then

cof
(
α
# )
cof
(

ψ

ν
k

(

β
k

)
)

{\displaystyle \text{cof}(\alpha)=\text{cof}(\psi\_{\nu\_k}(\beta\_k))}

 and

α
[
η
# ]

ψ

ν
1

(

β
1

)
⋯

ψ

ν

k
−
1

(

β

k
−
1

)
(

ψ

ν
k

(

β
k

)
[
η
]
)

{\displaystyle \alpha[\eta]=\psi\_{\nu\_1}(\beta\_1)+\cdots+\psi\_{\nu\_{k-1}}(\beta\_{k-1})+(\psi\_{\nu\_k}(\beta\_k)[\eta])}

,
2. If

# α

ψ

0

(
0
# )
1

{\displaystyle \alpha=\psi\_{0}(0)=1}

, then

cof
(
α
# )
1

{\displaystyle \text{cof}(\alpha)=1}

 and

α
[
0
# ]
0

{\displaystyle \alpha[0] = 0}

,
3. If

# α

ψ

ν
1

(
0
)

{\displaystyle \alpha=\psi\_{\nu+1}(0)}

, then

cof
(
α
# )

Ω

ν
1

{\displaystyle \text{cof}(\alpha)=\Omega\_{\nu+1}}

 and

α
[
η
# ]

Ω

ν
1

[
η
# ]
η

{\displaystyle \alpha[\eta]=\Omega\_{\nu+1}[\eta]=\eta}

,
4. If

# α

ψ

ν

(
0
)

{\displaystyle \alpha=\psi\_{\nu}(0)}

 and

cof
(
ν
)
∈
{
ω
}
∪
{

Ω

μ
1

μ
≥
0
}

{\displaystyle \text{cof}(\nu)\in\{\omega\}\cup\{\Omega\_{\mu+1}|\mu\geq 0\}}

, then

cof
(
α
# )
cof
(
ν
)

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

 and

α
[
η
# ]

ψ

ν
[
η
]

(
0
# )

Ω

ν
[
η
]

{\displaystyle \alpha[\eta]=\psi\_{\nu[\eta]}(0)=\Omega\_{\nu[\eta]}}

,
5. If

# α

ψ

ν

(
β
1
)

{\displaystyle \alpha=\psi\_{\nu}(\beta+1)}

 then

cof
(
α
# )
ω

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

 and

α
[
η
# ]

ψ

ν

(
β
)
⋅
η

{\displaystyle \alpha[\eta]=\psi\_{\nu}(\beta)\cdot \eta}

 (and note:

ψ
ν

(
0
# )

Ω
ν

{\displaystyle \psi\_\nu(0)=\Omega\_\nu}

),
6. If

# α

ψ

ν

(
β
)

{\displaystyle \alpha=\psi\_{\nu}(\beta)}

 and

cof
(
β
)
∈
{
ω
}
∪
{

Ω

μ
1

μ
<
ν
}

{\displaystyle \text{cof}(\beta)\in\{\omega\}\cup\{\Omega\_{\mu+1}|\mu<\nu\}}

 then

cof
(
α
# )
cof
(
β
)

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

 and

α
[
η
# ]

ψ

ν

(
β
[
η
]
)

{\displaystyle \alpha[\eta]=\psi\_{\nu}(\beta[\eta])}

,
7. If

# α

ψ

ν

(
β
)

{\displaystyle \alpha=\psi\_{\nu}(\beta)}

 and

cof
(
β
)
∈
{

Ω

μ
1

μ
≥
ν
}

{\displaystyle \text{cof}(\beta)\in\{\Omega\_{\mu+1}|\mu\geq\nu\}}

 then

cof
(
α
# )
ω

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

 and

α
[
η
# ]

ψ

ν

(
β
[
γ
[
η
]
]
)

{\displaystyle \alpha[\eta]=\psi\_{\nu}(\beta[\gamma[\eta]])}

 where

{

γ
[
0
# ]

Ω
μ

γ
[
η
1
# ]

ψ
μ

(
β
[
γ
[
η
]
]
)

{\displaystyle \left\{\begin{array}{lcr} \gamma[0]=\Omega\_\mu \\ \gamma[\eta+1]=\psi\_\mu(\beta[\gamma[\eta]])\\ \end{array}\right.}

.

If

# α
Λ

{\displaystyle \alpha=\Lambda}

 then

cof
(
α
# )
ω

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

 and

α
[
0
# ]
0

{\displaystyle \alpha[0] = 0}

 and

α
[
η
1
# ]

ψ

α
[
η
]

(
0
# )

Ω

α
[
η
]

{\displaystyle \alpha[\eta+1]=\psi\_{\alpha[\eta]}(0)=\Omega\_{\alpha[\eta]}}

.

## Sources

1. ↑ W. Buchholz. A New System of Proof-Theoretic Ordinal Functions. Annals of Pure and Applied Logic, vol. 32
2. ↑ Jager's psi-functions
3. ↑ Gerhard Jager. INACCESSIBLE ORDINALS, COLLAPSING FUNCTIONS AND A RECURSIVE NOTATION SYSTEM. Arch. math. Logik 24 (1984), 49-62
4. ↑ Denis Maksudov. The extension of Buchholz's function