Infinity-Borel Set - Alternative Characterization

Alternative Characterization

For subsets of Baire space or Cantor space, there is a more concise (if less transparent) alternative definition, which turns out to be equivalent. A subset A of Baire space is ∞-Borel just in case there is a set of ordinals S and a first-order formula φ of the language of set theory such that, for every x in Baire space,

where L is Gödel's constructible universe relativized to S and x. When using this definition, the ∞-Borel code is made up of the set S and the formula φ, taken together.