Number Theory and the Bachmann/Howard Ordinal

Publisher Summary This chapter discusses number theory and the Bachmann/Howard ordinal. The work of Gentzen and Hilbert/Bernays in the 1930s ɛ o is considered to be the characteristic ordinal of classical number theory. It is the least upper bound on the derivability of transfinite induction in number theory and the consistency of this theory can be proved by transfinite induction up to ɛ o included. Gentzen's technique of establishing the consistency of a formal system by transfinite induction up to a certain ordinal, and of showing transfinite induction to be derivable within the system for each smaller ordinal has been successfully applied to many other theories—especially to various subsystems of analysis. The characteristic ordinal found in this way is often interpreted as a measure of the proof-theoretical strength of the considered system. The proof of Girard's conjecture is based on the fact that the induction principles have recursion counterparts in fast- and slow-growing hierarchies of number-theoretic functions.