Wadge hierarchy and Veblen hierarchy. I: Borel sets of finite rank (Q2732268)

Assume \(\Lambda\) is a set of size less than some uncountable regular cardinal \(\kappa\). Given \(A , B \subseteq \Lambda^\omega\), say that \(A\) is Wadge reducible to \(B\), \(A \leq_W B\) in symbols, if there is a continuous function \(f: \Lambda^\omega \to \Lambda^\omega\) such that \(A = f^{-1} (B)\). If \(A \leq_W B \leq_W A\), call \(A\) and \(B\) Wadge equivalent and write \(A \equiv_W B\). Martin proved \(\leq_W\) is wellfounded on the Borel sets. Recursively define the Wadge degree of a Borel set by \(d_W (\emptyset) = d_W (\omega^\omega) = 0\) and \(d_W (A) = \sup \{ d_W (B) + 1 ; B <_W A \}\). Wadge's Lemma says two Borel sets \(A, B\) have the same degree iff either \(A \equiv_W B\) or \(A \equiv_W \Lambda^\omega \setminus B\). A set \(A\) is self-dual if \(A \equiv_W \Lambda^\omega \setminus A\). The Wadge degree is a measure of the complexity of a Borel set much finer than Borel rank, and height in the Wadge hierarchy is related to the Veblen hierarchy of fast increasing ordinal functions. It is known that a Borel set has finite rank iff its Wadge degree is less than \({}^\kappa \epsilon_0\), the first fixpoint of the exponentiation with base \(\kappa\). NEWLINENEWLINENEWLINEThe author provides a recursive construction of a canonical Borel set of degree \(\alpha\) where \(\alpha < {}^\kappa \epsilon_0\). This is achieved by concentrating on non self-dual sets, because a self-dual set of a given degree can be easily obtained from non self-dual sets of smaller degree, and by defining analogues, for Borel sets of finite rank, of the three operations ordinal addition, multiplication by an ordinal \(< \kappa\), and exponentiation with base \(\kappa\). The arguments make heavy use of the Wadge game.