Clearly log-odds aren't perfectly isomorphic with multiplicative probabilities, since clearly one allows probabilities of 0 and 1 and the other doesn't.
Bayes's theorem does assume nonzero probability, as you can observe by examining its proof.
You can't derive properties of probability from Russell's axioms, because these describe set theory and not probability. One standard way of deriving properties of probability is via Dutch Book arguments. These can only show that probabilities must be in the range [0,1] (including the endpoints). In fact, no finite sequence of bets you offer me can distinguish a credence of 1 from a credence of 1-epsilon for sufficiently small epsilon. (That is, for any epsilon, there's a bet that distinguishes 1-epsilon from 1, but for any sequence of bets, there's an 1-epsilon that is indistinguishable from 1).
Here is an analogy. The well-known formula D = RT describes the relationship between distance traveled, average speed, and time. You can also express this as log(D) = log(R) + log(T) if you like, or D/R = T. In either of these formulas, setting R=0 will be an error. This doesn't mean that there's no such thing as a speed of 0, and if you think your speed is 0 you are actually traveling at a speed of epsilon for some very small value of epsilon. It just means that when you passed to these (mostly equivalent) formulations, you lost the capability to discuss speeds of 0. In fact, when we set R to 0 in the original formula, we get a more useful description of what happens: D=0 no matter the value of T. In other words, 0 is a valid speed, but you can't travel a nonzero distance with an average speed of zero, no matter how much time you allow yourself.
What is the difference between log-odds and log-speeds, that makes the former an isomorphism and the latter an imperfect description?
Finally, do you really think that someone who thinks "0 and 1 are probabilities" is a statement LW is irrational about is unaware of the "0 and 1 are not probabilities" post?
Potholing that last sentence was mostly for fun.
By virtue of the definition of a logarithm, exp(log(x)) = x, we can derive that since the exponential function is well-defined for complex numbers, so is the logarithm. Taking the logarithm of a negative number nets you the logarithm of the absolute plus imaginary pi. The real part of any logarithm is a symmetric function, and there are probably a few other interesting properties of logarithms in complex analysis that I don't know of.
log(0) is undefined, as you note, but that does not mean the limit x -> 0...
I was very interested in the discussions and opinions that grew out of the last time this was played, but find digging through 800+ comments for a new game to start on the same thread annoying. I also don't want this game ruined by a potential sock puppet (whom ever it may be). So here's a non-sockpuppetiered Irrationality Game, if there's still interest. If there isn't, downvote to oblivion!
The original rules:
Enjoy!