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 the range of Buchholz's psi function and Feferman's theta function. It was named by David Madore, after Gaisi Takeuti, Solomon Feferman and Wilfried Buchholz. It is written as using Buchholz's psi function, an ordinal collapsing function invented by Wilfried Buchholz, and in Feferman's theta function, an ordinal collapsing function invented by Solomon Feferman. It is the proof-theoretic ordinal of several formal theories: , a subsystem of second-order arithmetic -comprehension + transfinite induction IDω, the system of ω-times iterated inductive definitions
Definition
Let represent the smallest uncountable ordinal with cardinality . Let represent the th epsilon number, equal to the th fixed point of Let represent Buchholz's psi function
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