The 1989 book "Topology via Logic" by Steven Vickers might be a good fit. Or a horribly bad one.

A subset of the unit interval is a complete sublattice if and only if the complement is a disjoint union of open intervals.

A subset of the unit interval is a complete lattice if and only if the complement is a disjoint union of open and half-open intervals.

Even though the letter does not exist in traditional French, the sound does: "Ouest"

There is a 2009 ET note by Weibull and Vielle that includes a log-utility example in which the multiple self model allows weird SPE even with exponential discounting.

At least for an example with common interest: Take an eq that doesn't lead to z*. By cont at infinity, there is a history on the z*-path such that all continuations are better than the eq. By backward induction, all choices leading to the history have continuation values better than the equilibrium.

There is also a backwards looking legitimate version of 1): You want to show that existing cases and models fit in the framework, or that existing results are subsumed.

I would not read {x_m: m in M} as an indexed set. Formally, an indexed set is a function whose domain is the index set. The set {x_m: m in M} is then the range of this function. Calling it linearly independent is fine. I would then write something like (x_m)_{m in M} for the indexed set itself.

One rule I've seen (N. Higham) is to write out numbers as adjectives but not as names or actual numbers (though 1 can be written out more often). "I talked to two mathematicians who convinced me that 2 is a prime number."

The proof is surprisingly easy: search.app/fHJSeaEV4zfY...

Good catch! The problem now reduces to showing there exists a finite set of states witnessing to an action being undominated.

... For each n, there is a probability distribution on Sn against which the original action is optimal by Pearce's result. But then it is optimal against a mixture of all these distributions, and this mixture has full support.

I think it goes through: Take an undominated action. For every other action, there must be a state at which the original action is better; collect these states in the finite set S1. Starting from S1, create an increasing sequence (Sn) of finite sets of states with a dense union...

With A finite, there can be at least no counterexample in which a unique best reply is not optimal under a full support probability if u is bounded and continuous and S Polish. This follows from weak convergence continuity of expected utility and full support probabilities being dense.

He apparently has changed his mind: "But birthright citizenship obviously doesn't apply in case of war or invasion. No one to my knowledge has ever argued that the children of invading aliens are entitled to birthright citizenship." reason.com/volokh/2024/...