Bachmann psi function

Bachmann's psi function is an ordinal collapsing function with a limit of the Bachmann-Howard ordinal.

C
0

(
α
,
β
# )
β
∪
{
0
,
Ω
}

C

i
1

(
α
,
β
# )
{
γ
δ
,

ω
γ

,
ψ
(
η
)
:
γ
,
δ
,
η
∈

C
i

(
α
,
β
)
∧
η
<
α
}

C
(
α
,
β
# )

⋃

i
∈

N

C
i

(
α
,
β
)

ψ
(
α
# )
min
(
{
β
:
C
(
α
,
β
)
∩
Ω
⊆
β
}
)

{\displaystyle \begin{align}&C\_0(\alpha,\beta)=\beta\cup\{0,\Omega\} \\ &C\_{i+1}(\alpha,\beta)=\{\gamma+\delta,\omega^\gamma,\psi(\eta):\gamma,\delta,\eta\in C\_i(\alpha,\beta)\land\eta<\alpha\} \\ &C(\alpha,\beta)=\bigcup\_{i\in\mathbb{N}}C\_i(\alpha,\beta) \\ &\psi(\alpha)=\text{min}(\{\beta:C(\alpha,\beta)\cap\Omega\subseteq\beta\}) \end{align}}

## Values

* ψ
(
0
# )

ε
0

{\displaystyle \psi(0)=\varepsilon\_0}
* ψ
(
1
# )

ε
1

{\displaystyle \psi(1)=\varepsilon\_1}
* ψ
(
ω
# )

ε
ω

{\displaystyle \psi(\omega)=\varepsilon\_\omega}
* ψ
(
ψ
(
0
)
# )

ε

ε
0

{\displaystyle \psi(\psi(0))=\varepsilon\_{\varepsilon\_0}}
* ψ
(
ψ
(
ψ
(
0
)
)
# )

ε

ε

ε
0

{\displaystyle \psi(\psi(\psi(0)))=\varepsilon\_{\varepsilon\_{\varepsilon\_0}}}
* ψ
(
Ω
# )

ζ
0

{\displaystyle \psi(\Omega)=\zeta\_0}
* ψ
(
Ω
1
# )

ε

ζ
0

1

{\displaystyle \psi(\Omega+1)=\varepsilon\_{\zeta\_0+1}}
* ψ
(
Ω
ω
# )

ε

ζ
0

ω

{\displaystyle \psi(\Omega+\omega)=\varepsilon\_{\zeta\_0+\omega}}
* ψ
(
Ω
ψ
(
Ω
)
# )
ψ
(
Ω

ζ
0

# )

ε

ζ
0

2

{\displaystyle \psi(\Omega+\psi(\Omega))=\psi(\Omega+\zeta\_0)=\varepsilon\_{\zeta\_02}}
* ψ
(
Ω
ψ
(
Ω
ψ
(
Ω
)
)
# )

ε

ε

ζ
0

2

{\displaystyle \psi(\Omega+\psi(\Omega+\psi(\Omega)))=\varepsilon\_{\varepsilon\_{\zeta\_02}}}
* ψ
(
Ω
2
# )

ζ
1

{\displaystyle \psi(\Omega2)=\zeta\_1}
* ψ
(

ω

Ω
1

# )

ζ
ω

{\displaystyle \psi(\omega^{\Omega+1})=\zeta\_\omega}
* ψ
(

ω

Ω
1

Ω
# )

ζ

ω
1

{\displaystyle \psi(\omega^{\Omega+1}+\Omega)=\zeta\_{\omega+1}}
* ψ
(

ω

Ω
1

2
# )

ζ

ω
2

{\displaystyle \psi(\omega^{\Omega+1}2)=\zeta\_{\omega2}}
* ψ
(

ω

Ω
2

# )

ζ

ω
2

{\displaystyle \psi(\omega^{\Omega+2})=\zeta\_{\omega^2}}
* ψ
(

ω

Ω
ω

# )

ζ

ω
ω

{\displaystyle \psi(\omega^{\Omega+\omega})=\zeta\_{\omega^\omega}}
* ψ
(

ω

Ω

ε
0

# )

ζ

ε
0

{\displaystyle \psi(\omega^{\Omega+\varepsilon\_0})=\zeta\_{\varepsilon\_0}}
* ψ
(

ω

Ω
ψ
(
Ω
)

# )

ζ

ζ
0

{\displaystyle \psi(\omega^{\Omega+\psi(\Omega)})=\zeta\_{\zeta\_0}}
* ψ
(

ω

Ω
ψ
(

ω

Ω
ψ
(
Ω
)

)

# )

ζ

ζ

ζ
0

{\displaystyle \psi(\omega^{\Omega+\psi(\omega^{\Omega+\psi(\Omega)})})=\zeta\_{\zeta\_{\zeta\_0}}}
* ψ
(

ω

Ω
2

# )

η
0

{\displaystyle \psi(\omega^{\Omega2})=\eta\_0}
* ψ
(

ω

Ω
2

Ω
# )

ζ

η
0

1

{\displaystyle \psi(\omega^{\Omega2}+\Omega)=\zeta\_{\eta\_0+1}}
* ψ
(

ω

Ω
2

Ω
2
# )

ζ

η
0

2

{\displaystyle \psi(\omega^{\Omega2}+\Omega2)=\zeta\_{\eta\_0+2}}
* ψ
(

ω

Ω
2

ω

Ω
1

# )

ζ

η
0

ω

{\displaystyle \psi(\omega^{\Omega2}+\omega^{\Omega+1})=\zeta\_{\eta\_0+\omega}}
* ψ
(

ω

Ω
2

ω

Ω
1

Ω
# )

ζ

η
0

ω
1

{\displaystyle \psi(\omega^{\Omega2}+\omega^{\Omega+1}+\Omega)=\zeta\_{\eta\_0+\omega+1}}
* ψ
(

ω

Ω
2

ω

Ω
1

2
# )

ζ

η
0

ω
2

{\displaystyle \psi(\omega^{\Omega2}+\omega^{\Omega+1}2)=\zeta\_{\eta\_0+\omega2}}
* ψ
(

ω

Ω
2

ω

Ω
2

# )

ζ

η
0

ω
2

{\displaystyle \psi(\omega^{\Omega2}+\omega^{\Omega+2})=\zeta\_{\eta\_0+\omega^2}}
* ψ
(

ω

Ω
2

ω

Ω
ψ
(

ω

Ω
2

)

# )

ζ

η
0

2

{\displaystyle \psi(\omega^{\Omega2}+\omega^{\Omega+\psi(\omega^{\Omega2})})=\zeta\_{\eta\_02}}
* ψ
(

ω

Ω
2

ω

Ω
ψ
(

ω

Ω
2

ω

Ω
ψ
(

ω

Ω
2

)

)

# )

ζ

ζ

η
0

2

{\displaystyle \psi(\omega^{\Omega2}+\omega^{\Omega+\psi(\omega^{\Omega2}+\omega^{\Omega+\psi(\omega^{\Omega2})})})=\zeta\_{\zeta\_{\eta\_02}}}
* ψ
(

ω

Ω
2

2
# )

η
1

{\displaystyle \psi(\omega^{\Omega2}2)=\eta\_1}
* ψ
(

ω

Ω
2
1

# )

η
ω

{\displaystyle \psi(\omega^{\Omega2+1})=\eta\_\omega}
* ψ
(

ω

Ω
2
1

ω

Ω
2

# )

η

ω
1

{\displaystyle \psi(\omega^{\Omega2+1}+\omega^{\Omega2})=\eta\_{\omega+1}}
* ψ
(

ω

Ω
2
2

# )

η

ω
2

{\displaystyle \psi(\omega^{\Omega2+2})=\eta\_{\omega^2}}
* ψ
(

ω

Ω
2
ψ
(

ω

Ω
2

)

# )

η

η
0

{\displaystyle \psi(\omega^{\Omega2+\psi(\omega^{\Omega2})})=\eta\_{\eta\_0}}
* ψ
(

ω

Ω
3

# )
φ
(
4
,
0
)

{\displaystyle \psi(\omega^{\Omega3})=\varphi(4,0)}
* ψ
(

ω

Ω
n

γ
0

β
0

# )
φ
(
n
,
ω

γ
0

β
0

)

{\displaystyle \psi(\omega^{\Omega n+\gamma\_0}\beta\_0)=\varphi(n,\omega\gamma\_0+\beta\_0)}

 with

n
<
ω

{\displaystyle n<\omega}
* ψ
(

ω

ω

Ω
1

# )
φ
(
ω
,
0
)

{\displaystyle \psi(\omega^{\omega^{\Omega+1}})=\varphi(\omega,0)}
* ψ
(

ω

ω

Ω
2

# )

Γ
0

{\displaystyle \psi(\omega^{\omega^{\Omega2}})=\Gamma\_0}
* ψ
(

ω

ω

Ω
2

2
# )

Γ
1

{\displaystyle \psi(\omega^{\omega^{\Omega2}}2)=\Gamma\_1}
* ψ
(

ω

ω

Ω
2

Ω

# )
φ
(
1
,
1
,
0
)

{\displaystyle \psi(\omega^{\omega^{\Omega2}+\Omega})=\varphi(1,1,0)}
* ψ
(

ω

ω

Ω
2

Ω
2

# )
φ
(
1
,
2
,
0
)

{\displaystyle \psi(\omega^{\omega^{\Omega2}+\Omega2})=\varphi(1,2,0)}
* ψ
(

ω

ω

Ω
2

2

# )
φ
(
2
,
0
,
0
)

{\displaystyle \psi(\omega^{\omega^{\Omega2}2})=\varphi(2,0,0)}
* ψ
(

ω

ω

Ω
3

# )
φ
(
1
,
0
,
0
,
0
)

{\displaystyle \psi(\omega^{\omega^{\Omega3}})=\varphi(1,0,0,0)}
* ψ
(

ω

ω

ω

Ω
1

# )
SVO

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