Talk:Veblen function

Where is the definition?

I hope someone improves this article so that it defines the Veblen function. Right now it simply seems to state a bunch of properties of this function, without asserting that some set of properties characterizes it. John Baez (talk) 06:08, 4 November 2012 (UTC)

In the lead it gives the general definition, "If φ₀ is any normal function, then for any non-zero ordinal α, φ_α is the function enumerating the common fixed points of φ_β for β<α.". In the first section, it defines "Veblen hierarchy" by specifying that "In the special case when φ₀(α)=ω^α, this family of functions is known as the Veblen hierarchy.". JRSpriggs (talk) 10:15, 4 November 2012 (UTC)

Formal definitions

For background see: First-order predicate calculus, Zermelo-Frankel set theory with the axiom of choice, Ordinal number#Von Neumann definition of ordinals, and Ordinal arithmetic.

For the binary Veblen hierarchy, a formal definition using recursion on α and β is:

$\phi _{\alpha }(\beta )=\gamma \,\Leftrightarrow \,\operatorname {Ord} (\alpha )\land \operatorname {Ord} (\beta )\land \operatorname {Ord} (\gamma )\land 0<\gamma \land \forall \delta <\gamma (\delta +\delta <\gamma )\land \forall \delta <\alpha (\phi _{\delta }(\gamma )=\gamma )\land \forall \delta <\beta (\phi _{\alpha }(\delta )<\gamma )\land \lnot \exists \rho <\gamma \left(0<\rho \land \forall \delta <\rho (\delta +\delta <\rho )\land \forall \delta <\alpha (\phi _{\delta }(\rho )=\rho )\land \forall \delta <\beta (\phi _{\alpha }(\delta )<\rho )\right)$.

OK? JRSpriggs (talk) 23:13, 15 July 2022 (UTC)

Assuming the previous definition, the Γ function may be defined by:

$\Gamma _{\alpha }=\gamma \,\Leftrightarrow \,\operatorname {Ord} (\alpha )\land \phi _{\gamma }(0)=\gamma \land \forall \delta <\alpha (\Gamma _{\delta }<\gamma )\land \lnot \exists \rho <\gamma (\phi _{\rho }(0)=\rho \land \forall \delta <\alpha (\Gamma _{\delta }<\rho ))$.

OK? JRSpriggs (talk) 07:15, 3 August 2022 (UTC)

If Κ is an uncountable regular cardinal, then the finitary Veblen hierarchy on Κ maps Κ^ω to Κ according to:

$\phi (\mu )=\nu \,\Leftrightarrow \,\mu <\mathrm {K} ^{\omega }\land \nu <\mathrm {K} \land \exists \alpha <\omega \,\exists \sigma <\mathrm {K} ^{\omega }\,\exists \beta <\mathrm {K} \,\exists \gamma <\mathrm {K} (\mu =\mathrm {K} ^{\alpha +2}\cdot \sigma +\mathrm {K} ^{\alpha +1}\cdot \beta +\gamma \land (0<\beta \lor (\alpha =0\land \sigma =0\land \beta =0))\land$

$0<\nu \land \forall \delta <\nu (\delta +\delta <\nu )\land \forall \delta <\beta (\phi (\mathrm {K} ^{\alpha +2}\cdot \sigma +\mathrm {K} ^{\alpha +1}\cdot \delta +\mathrm {K} ^{\alpha }\cdot \nu )=\nu )\land \forall \delta <\gamma (\phi (\mathrm {K} ^{\alpha +2}\cdot \sigma +\mathrm {K} ^{\alpha +1}\cdot \beta +\delta )<\nu )\land$

$\lnot \exists \rho <\nu \left(0<\rho \land \forall \delta <\rho (\delta +\delta <\rho )\land \forall \delta <\beta (\phi (\mathrm {K} ^{\alpha +2}\cdot \sigma +\mathrm {K} ^{\alpha +1}\cdot \delta +\mathrm {K} ^{\alpha }\cdot \rho )=\rho )\land \forall \delta <\gamma (\phi (\mathrm {K} ^{\alpha +2}\cdot \sigma +\mathrm {K} ^{\alpha +1}\cdot \beta +\delta )<\rho )\right))$.

OK? JRSpriggs (talk) 13:02, 5 August 2022 (UTC)

If Κ is an uncountable regular cardinal, then the transfinitary Veblen hierarchy on Κ maps Κ^Κ to Κ according to:

$\phi (\mu )=\nu \,\Leftrightarrow \,\mu <\mathrm {K} ^{\mathrm {K} }\land \nu <\mathrm {K} \land \exists \alpha <\mathrm {K} \,\exists \sigma <\mathrm {K} ^{\mathrm {K} }\,\exists \beta <\mathrm {K} \,\exists \gamma <\mathrm {K} (\mu =\mathrm {K} ^{\alpha +1}\cdot \sigma +\mathrm {K} ^{\alpha }\cdot \beta +\gamma \land ((0<\alpha \land 0<\beta )\lor (\alpha =1\land \sigma =0\land \beta =0))\land$

$0<\nu \land \forall \delta <\nu (\delta +\delta <\nu )\land \forall \delta <\beta \,\forall \eta <\alpha (\phi (\mathrm {K} ^{\alpha +1}\cdot \sigma +\mathrm {K} ^{\alpha }\cdot \delta +\mathrm {K} ^{\eta }\cdot \nu )=\nu )\land \forall \delta <\gamma (\phi (\mathrm {K} ^{\alpha +1}\cdot \sigma +\mathrm {K} ^{\alpha }\cdot \beta +\delta )<\nu )\land$

$\lnot \exists \rho <\nu \left(0<\rho \land \forall \delta <\rho (\delta +\delta <\rho )\land \forall \delta <\beta \,\forall \eta <\alpha (\phi (\mathrm {K} ^{\alpha +1}\cdot \sigma +\mathrm {K} ^{\alpha }\cdot \delta +\mathrm {K} ^{\eta }\cdot \rho )=\rho )\land \forall \delta <\gamma (\phi (\mathrm {K} ^{\alpha +1}\cdot \sigma +\mathrm {K} ^{\alpha }\cdot \beta +\delta )<\rho )\right))$.

OK? JRSpriggs (talk) 21:34, 6 August 2022 (UTC)

The small Veblen ordinal (SVO) is φ(Κ^ω) and the large Veblen ordinal (LVO) is $\sup\{\phi (\mathrm {K} ^{\omega }),\phi (\mathrm {K} ^{\phi (\mathrm {K} ^{\omega })}),\phi (\mathrm {K} ^{\phi (\mathrm {K} ^{\phi (\mathrm {K} ^{\omega })})}),\ldots \}$. JRSpriggs (talk) 21:54, 6 August 2022 (UTC)

The Feferman–Schütte ordinal, Γ₀, is the smallest nonzero ordinal which is neither the sum of smaller ordinals nor a value of the binary Veblen hierarchy applied to smaller ordinals. It contains ω and all finite ordinals. It is also closed under: multiplication, exponentiation, and the function enumerating the epsilon-numbers. Other values of Γ_α are also closed under the same functions.

SVO = Γ_SVO has those closure properties and is also closed under the mapping by the finitary Veblen hierarchy of functions of finite support from ω to SVO (regarded as elements of Κ^ω) to Κ.

LVO = Γ_LVO has the above closure properties and is also closed under the mapping by the transfinitary Veblen hierarchy of functions of finite support from LVO to LVO (regarded as elements of Κ^Κ) to Κ.

OK? JRSpriggs (talk) 21:39, 9 August 2022 (UTC)