Takeuti–Feferman–Buchholz ordinal

In the mathematical fields of set theory and proof theory, the Takeuti–Feferman–Buchholz ordinal (TFBO) is a large countable ordinal, which acts as the limit of (largest number definable using) Buchholz’s psi function and Feferman’s theta function.[1][2] It was named by David Madore,[2] after Gaisi Takeuti, Solomon Feferman and Wilfried Buchholz. It is written as

ψ(εΩω+1){displaystyle psi (varepsilon _{Omega _{omega }+1})}

in Buchholz’s psi function,[3] an OCF invented by Wilfried Buchholz,[4][5][6] and

θεΩω+1(0){displaystyle theta _{varepsilon _{Omega _{omega }+1}}(0)}

in Feferman’s theta function, an OCF invented by Solomon Feferman.[7][8] It is the proof-theoretic ordinal of

Π11CA+BI{displaystyle Pi _{1}^{1}-CA+BI}

,[9] a subsystem of second-order arithmetic,

Π11{displaystyle Pi _{1}^{1}}

-comprehension + transfinite induction,[3]IDω, the system of ω-times iterated inductive definitions[10] and KPI, Kripke-Platek set theory with a recursively inaccessible universe.[10]

Despite being one of the largest large countable ordinals and recursive ordinals, it is still vastly smaller than the proof-theoretic ordinal of ZFC.[11]

. . . Takeuti–Feferman–Buchholz ordinal . . .

  • Let
    Ωα{displaystyle Omega _{alpha }}

    represent an uncountable ordinal with cardinality

    α{displaystyle aleph _{alpha }}

    .

  • Let
    εβ{displaystyle varepsilon _{beta }}

    represent the

    β{displaystyle beta }

    th epsilon number, equal to the

    1+β{displaystyle 1+beta }

    th fixed point of

    αωα{displaystyle alpha mapsto omega ^{alpha }}

  • Let
    ψ{displaystyle psi }

    represent Buchholz’s psi function

  • The TFBO is equal to
    ψ(εΩω+1){displaystyle psi (varepsilon _{Omega _{omega }+1})}

    .

In other words, the TFBO is the smallest ordinal which cannot be expressed from

0{displaystyle 0}

,

1{displaystyle 1}

,

ω{displaystyle omega }

and

Ω{displaystyle Omega }

using sums, products, exponentials, and the

ψ{displaystyle psi }

function itself, the latter of which only to previously constructed ordinals less than

εΩω+1{displaystyle varepsilon _{Omega _{omega }+1}}

.

. . . Takeuti–Feferman–Buchholz ordinal . . .

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. . . Takeuti–Feferman–Buchholz ordinal . . .