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On the Slowly Well Orderedness of ε o
Author(s) -
Arai Toshiyasu
Publication year - 2002
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/1521-3870(200201)48:1<125::aid-malq125>3.0.co;2-n
Subject(s) - mathematics , combinatorics , unary operation , inverse , function (biology) , base (topology) , natural number , logarithm , binary number , discrete mathematics , arithmetic , mathematical analysis , geometry , evolutionary biology , biology
For α < ε 0 , N α denotes the number of occurrences of ω in the Cantor normal form of α with the base ω. For a binary number‐theoretic function f let B ( K ; f ) denote the length n of the longest descending chain (α 0 , …, α n –1 ) of ordinals <ε 0 such that for all i < n , N α i ≤ f ( K , i ). Simpson [2] called ε 0 as slowly well ordered when B ( K ; f ) is totally defined for f ( K ; i ) = K · ( i + 1). Let | n | denote the binary length of the natural number n , and | n | k the k ‐times iterate of the logarithmic function | n |. For a unary function h let L ( K ; h ) denote the function B ( K ; h 0 ( K ; i )) with h 0 ( K , i ) = K + | i | · | i | h ( i ) . In this note we show, inspired from Weiermann [4], that, under a reasonable condition on h , the function L ( K ; h ) is primitive recursive in the inverse h –1 and vice versa.