Reverse mathematics and ordinal exponentiation
The paper is devoted to the study of Simpson's claim that ``\(\text{ATR}_ 0\) is the weakest set of axioms which permits the development of a decent theory of countable ordinals''. An empirical support for this thesis is given. It is shown in particular that Cantor's Normal Form Theorem and Sherman's Inequality for countable well-orderings are both equivalent to \(\text{ATR}_ 0\). To obtain the proofs, ordinal exponentiation was developed and the comparability result of Friedman and Hirst was strengthened.
Sherman's Inequality for countable well- orderings
scientific article; zbMATH DE number 513338