Infinity-Borel Set - Incorrect Definition

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 containing all the open sets and closed under complementation and wellordered union. That is, one might wish to dispense with the ∞-Borel codes altogether and try a definition like this:

For each ordinal α define by transfinite recursion B_α as follows:

1. B₀ is the collection of all open subsets of .
2. For a given even ordinal α, B_α+1 is the union of B_α with the set of all complements of sets in B_α.
3. For a given even ordinal α, B_α+2 is the set of all wellordered unions of sets in B_α+1.
4. For a given limit ordinal λ, B_λ is the union of all B_α for α<λ

It follows from the Burali-Forti paradox that there must be some ordinal α such that B_β equals B_α for every β>α. For this value of α, B_α is the collection of "∞-Borel sets".

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.