We briefly discussed relative constructibility and compared the models where is exclusively treated as a predicate with the models where is an element. In particular, is a model of choice but may fail to be.
An amusing application of the fact that is that the result of Exercise 3 from Homework 7 holds in , although the proof I wrote there uses choice. Namely, work in and consider two well-orderings of a set . We can assume that is an ordinal and the first well-ordering is . Let be the second well-ordering. Then (since is a set of ordered pairs of ordinals). In , where choice holds, (and therefore also in ) there is a subset of of the same size as and where coincides with .
Question. Find a `choice-free’ argument for Exercise 3.
The main example we will consider of a model of the form is , due to its connection with determinacy.
We introduced the setting to discuss determinacy, namely infinite 2-person games with perfect information. We proved the Gale-Stewart theorem that open games are determined and discussed Martin’s extension to Borel games. A nice reference for the proof of Martin’s result (using the idea of `unraveling‘, which reduces any Borel game to an open game in a different space) is Kechris’s book on descriptive set theory: