It is the proof-theoretic ordinal of several mathematical theories, such as arithmetical transfinite recursion.
The Feferman–Schütte ordinal can be defined as the smallest ordinal that cannot be obtained by starting with 0 and using the operations of ordinal addition and the Veblen functions φα(β).
This ordinal is sometimes said to be the first impredicative ordinal,[2][3] though this is controversial, partly because there is no generally accepted precise definition of "predicative".
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