Here is a list of open problems that I am interested in. Some problems are posted with a small prize to them. Just to prevent this from being eternally binding, the statute of limitations is exactly ten years from the date of posting (unless specifeid otherwise). Exact nature of the prize may change depending on unforeseen circumstances, your mileage may vary.
Please send me updates on these problems, in case you know anything. I will add below each problem any progress that was made towards solving it.
You may also send me problems to post here. However, external problems must come with a prize, and I am not responsible for any disputes as to the nature of the prize between the winners and whomever suggested the problem.
1. The Partition Principle
The Partition Principle states: suppose that there is a surjective map \(f\colon A\to B\), then there an injective function \(g\colon B\to A\).
Well. Assuming the axiom of choice, we can prove there is even \(g\) which splits \(f\). And if we require the injection to actually split \(g\), then we can actually prove the axiom of choice. However, without assuming the axiom of choice, we cannot prove a whole lot about the partition principle. We know it implies \(\DC\), and therefore not provable from \(\ZF\) itself. But not much more.
Question: Is the Partition Principle equivlaent to the Axiom of Choice over \(\ZF\)?
Prize: A bottle of whisky aged 15 or older for a negative answer; a fine gin/vodka for a positive answer.
(April 8th, 2018)
2. Iterations of Symmetric Extensions on non-finite supports.
In my Ph.D. thesis I have developed a method to iterate symmetric extensions transfinitely. However, a significant part of the definition relies on the finiteness of the support of the iteration. This means that at limit steps countable choice principles are likely to fail.
Question: Are there sufficiently flexible limitations that let us define the iterations for countable, mixed, Easton, or otherwise-of-interest supports? Particularly, are there limitations which still allow adding reals, and thus obtain simpler constructions of choiceless models of \(\ZF+\DC\) where certain choice-consequences fail for the reals?
(April 8th, 2018)