Infinity-Borel Set - Formal Definition

Formal Definition

More formally: we define by simultaneous transfinite recursion the notion of ∞-Borel code, and of the interpretation of such codes. Since is Polish, it has a countable base. Let enumerate that base (that is, is the basic open set). Now:

• Every natural number is an ∞-Borel code. Its interpretation is .
• If is an ∞-Borel code with interpretation, then the ordered pair is also an ∞-Borel code, and its interpretation is the complement of, that is, .
• If is a length-α sequence of ∞-Borel codes for some ordinal α (that is, if for every β<α, is an ∞-Borel code, say with interpretation ), then the ordered pair is an ∞-Borel code, and its interpretation is .

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.