User:Binary198/Comparison between ordinal notations

An ordinal notation is a partial function from a finite alphabet to a countable set of ordinals. A Gödel numbering is a function from the set of well-formed formulae to the natural numbers. There are many such schemes of ordinal notations, including schemes by Wilhelm Ackermann, Heinz Bachmann, Wilfried Buchholz, Georg Cantor, Solomon Feferman, Gerhard Jäger and Wolfram Pohlers. This article is a comparison between various schemes of ordinal notation.

On the other hand, an ordinal collapsing function is a technique for defining certain large countable ordinals. Ordinal collapsing functions are typically denoted using some variation of either the Greek letter psi or theta. They are inextricably intertwined with the theory of ordinal analysis, as they are used to describe the strength of certain formal systems. Ordinal notations and ordinal collapsing functions are often confused: they are not the same, so in this article their strength will be compared.

Anyone who views this article, feel free to add your own notation/function.

Comparison

Comparison between ordinal notations and ordinal collapsing functions
Ordinal	ξ notation	Cantor normal form	Veblen φ	Bachmann ψ_Ω (Rathjen recast)	Feferman θ	Bird θ	Binary198 +
0	0						+
1	ξ00	ω⁰1	φ00	ψ_Ω(0)	θ₀(0)	θ(0)	+0
2	ξ0ξ00	ω⁰2					+(00)
ω	ξξ000	ω¹1	φ01	ψ_Ω(1)	θ₀(1)	θ(1)	+00
ω+1	ξ0ξξ000	ω¹1 + ω⁰1					+0(00)
ω+2	ξ0ξ0ξξ000	ω¹1 + ω⁰2					+0(000)
ω2	ξξ00ξ00	ω¹2					+0(00\|00)
ω3	ξξ00ξ0ξ00	ω¹3					+0(00\|00\|00)
ω²	ξξ00ξξ000	ω²1	φ02	ψ_Ω(2)	θ₀(2)	θ(2)	+0(00↑00)
ω³	ξξ00ξξ00ξξ000	ω³1	φ03	ψ_Ω(3)	θ₀(3)	θ(3)	+0(00↑00↑00)
ω^ω	ξξ0ξ000	$\omega ^{\omega ^{1}1}1$	φ0φ01	ψ_Ω(ψ_Ω(1))	θ₀(θ₀(1))	θ(θ(1))	+000
$\omega ^{\omega ^{\omega }}$	ξξξ0000	$\omega ^{\omega ^{\omega ^{1}1}1}1$	φ0φ0φ0φ01	ψ_Ω(ψ_Ω(ψ_Ω(1)))	θ₀(θ₀(θ₀(1)))	θ(θ(θ(1)))	+0000
ε₀			φ10	ψ_Ω(Ω)	θ₁(0)	θ(Ω)	+0+0
ε₁			φ11	ψ_Ω(Ω2)	θ₁(1)	θ(Ω+1)
ζ₀			φ20	ψ_Ω(Ω²)	θ₂(0)	θ(Ω2)	+0+0+0
ζ₁			φ21	ψ_Ω(Ω²2)	θ₂(1)	θ(Ω2+1)
η₀			φ30	ψ_Ω(Ω³)	θ₃(0)	θ(Ω3)	+0+0(+0+0)
η₁			φ31	ψ_Ω(Ω³2)	θ₃(1)	θ(Ω3+1)
Γ₀ (FSO)			φ100	ψ_Ω(Ω^Ω)	θ_Ω(0)	θ(Ω²)	+0+0+0+0
Γ₁			φ101	ψ_Ω(Ω^Ω2)	θ_Ω(1)	θ(Ω²+1)
Ackermann			φ1000	$\psi _{\Omega }(\Omega ^{\Omega ^{2}})$	$\theta _{\Omega ^{2}}(0)$	θ(Ω²2)	+0+0+0+(00)
SVO				$\psi _{\Omega }(\Omega ^{\Omega ^{\omega }})$	$\theta _{\Omega ^{\omega }}(0)$	θ(Ω²ω)	+0+0+0(+00)
LVO				$\psi _{\Omega }(\Omega ^{\Omega ^{\Omega }})$	$\theta _{\Omega ^{\Omega }}(0)$	θ(Ω³)	+0+0+0+0+0
BHO				$\psi _{\Omega }(\varepsilon _{\Omega +1})$	$\theta _{\varepsilon _{\Omega +1}}(0)$	θ(Ω^ω)	+0+(00)
${\mathsf {PTO(U(ID_{1}))}}$				$\psi _{\Omega }(\Gamma _{\Omega +1})$	$\theta _{\Gamma _{\Omega +1}}(0)$	θ(Ω^ω+1)
BO				$\psi _{\Omega }(\Omega _{\omega })$	$\theta _{\Omega _{\omega }}(0)$		+0(+00)
${\mathsf {PTO(W-ID_{\omega })}}$				$\psi _{\Omega }(\Omega _{\omega }\varepsilon _{0})$	$\theta _{\Omega _{\omega }\varepsilon _{0}}(0)$
TFBO				$\psi _{\Omega }(\varepsilon _{\Omega _{\omega }+1})$	$\theta _{\varepsilon _{\Omega _{\omega }+1}}(0)$		+0(+0(00))
${\mathsf {PTO(ID_{<\omega ^{\omega }})}}$				$\psi _{\Omega }(\Omega _{\omega ^{\omega }})$	$\theta _{\Omega _{\omega ^{\omega }}}(0)$		+0(+000)
${\mathsf {PTO(ID_{<\varepsilon _{0}})}}$				$\psi _{\Omega }(\Omega _{\varepsilon _{0}})$	$\theta _{\Omega _{\varepsilon _{0}}}(0)$		+0(+0+0)

References

Hilbert Levitz, Transfinite Ordinals and Their Notations: For The Uninitiated, expository article (8 pages, in PostScript)
Pohlers, Wolfram (1989), Proof theory, Lecture Notes in Mathematics, vol. 1407, Berlin: Springer-Verlag, doi:10.1007/978-3-540-46825-7, ISBN 978-3-540-51842-6, MR 1026933
Schütte, Kurt (1977), Proof theory, Grundlehren der Mathematischen Wissenschaften, vol. 225, Berlin-New York: Springer-Verlag, pp. xii+299, ISBN 978-3-540-07911-8, MR 0505313
Takeuti, Gaisi (1987), Proof theory, Studies in Logic and the Foundations of Mathematics, vol. 81 (Second ed.), Amsterdam: North-Holland Publishing Co., ISBN 978-0-444-87943-1, MR 0882549
Smorynski, C. (1982), "The varieties of arboreal experience", Math. Intelligencer, 4 (4): 182–189, doi:10.1007/BF03023553 contains an informal description of the Veblen hierarchy.
Veblen, Oswald (1908), "Continuous Increasing Functions of Finite and Transfinite Ordinals", Transactions of the American Mathematical Society, 9 (3): 280–292, doi:10.2307/1988605, JSTOR 1988605
Miller, Larry W. (1976), "Normal Functions and Constructive Ordinal Notations", The Journal of Symbolic Logic, 41 (2): 439–459, doi:10.2307/2272243, JSTOR 2272243
Buchholz, Wilfried (1986), A New System of Proof-Theoretic Ordinal Functions, pp. 195–207, doi:10.1016/0168-0072(86)90052-7
Rathjen, Michael. "A history of ordinal representations (notes)" (PDF). Mathematics - University of Munich. Archived from the original (PDF) on 2021-06-09. Retrieved 2021-09-05. {{cite web}}: |archive-date= / |archive-url= timestamp mismatch; 2007-06-12 suggested (help)
Bachmann, Heinz. "The normal functions and the problem of the explicit sequences of ordinal numbers". ngzf. Retrieved 2021-09-05.{{cite web}}: CS1 maint: url-status (link)
Bird, Chris. "The Fast-Growing Hierarchy in Terms of Bird's Array Notations". Googology Wiki. Retrieved 2021-09-05.{{cite web}}: CS1 maint: url-status (link)