A famous result in logic is the mathematical incompleteness of Peano arithmetic by Paris and Harrington. Using analytical combinatorics and logical reasoning we classify the largeness conditions in the Paris Harrington assertion which lead to incompleteness.
We further apply multiplicative number theory to obtain asymptotic bounds for count functions for elementary recursive ordinal notations which are defined via a Schütte style prime number coding. These bounds together with logical methods yield sharp bounds for the resulting Friedman style combinatorial well-foundedness principles.