It should probably be defined by calibration: do some people have a type of belief where they are always right?

You can phrase statements of logical deduction such that they have no premises and only conclusions. If we let S be the set of logical principles under which our logical system operates and T be some sentence that entails Y, then S AND T implies Y is something that I have absolute certainty in, even if this world is an illusion, because the premise of the implication contains all the rules necessary to derive the result.

A less formal example of this would be the sentence: If the rules of logic as I know them hold and the axioms of mathematics are true, then it is the case that 2+2=4

As someone who doesn't know much beyond basic statistics, in what way are 0 or 1 probabilities? Isn't it just axiomatic truth at that point? In that sense saying zero and one are probabilities is just saying 'certain' or 'impossible' as far as I understand it. Situations where an event will definitely or definitely not occur doesn't seem to be consistent with the idea of randomness which I've understood probability to revolve around.

I suppose the alternative would be that we'd have to assume every mathematical proof has infinite evidence if we wanted to get anywhere productive- after all axioms are assumed to be true. It doesn't make much sense to need evidence in that scenario- except perhaps the probability of error and mistake? That isn't particularly calculable and would actually change from person to person.

Using one and zero makes sense to me as a matter of assumed or proven truths, but I'm still unsure how that makes it a probability.

Formally, probability is defined via areas. The basic idea is that the probability of picking an element from a set A out of a set B is the ratio of the areas of A to B, where "area" can be defined not only for things like squares but also things like lines, or actually almost every* subset of R. So, lets say you want to randomly select a real number from the interval [0,1] and want to know the odds it falls in a set, S. The area of [0,1] is 1, so the answer is just the area of S.

If S={0}, then S has area zero. If S=[0,1), then S has area 1. Not only are both of these theoretical possibilities, they are practical ones too. There are real world examples of probability zero events (the only one that comes to mind involves QM though so I don't want to bother with the details).

Now, notice that this isn't the same thing as "impossible". Instead, it means more like "it won't happen I promise even by the time the universe ends". The way I tend to think about probability zero events is that they are so unlikely they are beyond the reach of the principle that as the number of trials increases, events become expected. For any nonzero probability, there is a number of trials, n, such that once you do it n times the expected value becomes greater than 1. That's not the case with probability zero events. Probability 1 events can then be thought of as the negation of probability 0 events.

*not actually "almost every" in a formal sense, but "almost any" in a "unless you go try to build a set that you can't measure it probably has a well defined area" sense

This article is largely incoherent. The main justification is the abuse of an invalid transformations: y=x/(1-x) is not the bijection that he asserts it is, because it's not a function that maps [0,1] onto R. It's a function that maps [0,1] onto [1,\intfy] as a subset of the topological closure of R. And that's okay, but you can't say "well I don't like the topological closure of R, so I'll just use R and claim that 1 is where the problem is."

Additionally, his discussion of log odds and such is perfectly fine, but ignores the fact that there are places where you do need to have an odds of 0:1, or a log odds of negative infinity. Probability theory stops working when you throw out 0 and 1, it's as simple as that.

Even if you don't want to handle tautologies or contradictions, there are other ways to get P(X)=0 or 1. The probability that a real number chosen uniformly from the real interval [0,1] is 0. It has to be. It's a provable fact under ZFC and to decide otherwise is to say that you're more attached to the idea of 0 and 1 not being probabilities than you are to the fact that mathematics is consistent and if you really believe that, well, there's absolutely nothing I have to say to you.

This is one of those situations where EY just demonstrates he knows very little mathematics.

That seems a solid enough explanation, but how can something of probability zero have a chance to occur? How then do you represent an impossible outcome? It seems like otherwise 'zero' is equivalent to 'absurdly low'. That doesn't quite jive with my understanding.

Impossible things also have a probability of zero. I totally understand that this seems a bit unintuitive, and the underlying structure (which includes things like infinities of different sizes) is generally pretty unintuitive at first. Which is kinda just saying "sorry, I can't explain the intuition," which is unfortunately true.