In set theory, a subset of a Polish space
Contents
Formal definition
More formally: we define by simultaneous transfinite recursion the notion of ∞-Borel code, and of the interpretation of such codes. Since
Now a set is ∞-Borel if it is the interpretation of some ∞-Borel code.
The axiom of choice implies that every set can be wellordered, and therefore that every subset of every Polish space is ∞-Borel. Therefore, the notion is interesting only in contexts where AC does not hold (or is not known to hold). Unfortunately, without the axiom of choice, it is not clear that the ∞-Borel sets are closed under wellordered union. This is because, given a wellordered union of ∞-Borel sets, each of the individual sets may have many ∞-Borel codes, and there may be no way to choose one code for each of the sets, with which to form the code for the union.
The assumption that every set of reals is ∞-Borel is part of AD+, an extension of the axiom of determinacy studied by Woodin.
Incorrect definition
It is very tempting to read the informal description at the top of this article as claiming that the ∞-Borel sets are the smallest class of subsets of
- B0 is the collection of all open subsets of
X . - For a given even ordinal α, Bα+1 is the union of Bα with the set of all complements of sets in Bα.
- For a given even ordinal α, Bα+2 is the set of all wellordered unions of sets in Bα+1.
- For a given limit ordinal λ, Bλ is the union of all Bα for α<λ
This set is manifestly closed under well-ordered unions, but without AC it cannot be proved equal to the ∞-Borel sets (as defined in the previous section). Specifically, it is instead the closure of the ∞-Borel sets under all well-ordered unions, even those for which a choice of codes cannot be made.
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[S,x] 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.