Omega places a button in front of you. he promises that each press gives you an extra year of life, plus whatever your discounting factor is. If you walk away, the button is destroyed. Do you press the button forever and never leave?

They are different concepts, either you use statistical significance or you do Bayesian updating (ie. using priors):

If you are using a 5% threshold roughly speaking this means that you will accept a hypothesis if the chance of getting equally strong data if your hypothesis is false is 5% or less.

If you are doing Bayesian updating you start with a probability for how likely a statement is (this is your prior) and update based on how likely your data would be if your statement was true or false.

here is an xkcd which highlights the difference: https://xkcd.com/1132/

In particular, I intuitively believe that "my beliefs about the integers are consistent, because the integers exist". That's an uncomfortable situation to be in, because we know that a consistent theory can't assert its own consistency.

That is true, however you don't appear to be asserting the consistency of your beliefs, you are asserting the consistency of a particular subset of your beliefs which does not contain the assertion of its consistency. This is not in conflict with Gödel's incompleteness theorem which implies that no theory may consistently assert its own consistency. Gödel's incompleteness theorem does not forbid proofs of consistency by more powerful theories: for example there are proofs of the consistency of Peano arithmetic

so, what are the risks ? Is it secret ?

From chapter 100:

the blood must come from a live unicorn and the unicorn must die in the drinking

Quirrell doesn't have a very large window in which to drink the blood.

More to the point, wouldn't particulate matter, other fluids, other bits of the unicorn pollute the blood as a result of the transfiguration? I could see the blood itself fixing that issue, but in the case of another fluid in a similar situation, I could see the drinker getting sick (if not to the degree that the animal did).

I may not be understanding how transfiguration sickness works exactly.

EDIT: formatting

This chapter confirms earlier speculations that horcruxes work by making backup copies of brain states (with the caveat that actually using the horcrux will merge its memories and personality with those of its host body, resulting in a hybrid entity). The theory that Harry James Potter-Evans-Verres is an instance of Tom Riddle (or, rather, a hybrid of Tom Riddle and the original Harry Potter) now seems very, very probable. It explains why Harry is as smart as Tom's other instances (the original Riddle/Monroe/Voldemort and the Quirrell/Riddle hybrid), and why the remember-ball glowed like the sun (Harry forgot Riddle's memories because he was too young to remember them).

I learned of the horcrux sspell ssince long ago.

Parseltongue has a word for "horcrux"?

Death iss not truly gainssaid. Real sself is losst, as you ssay. Not to my pressent tasste. Admit I conssidered it, long ago.

Voldemort used horcruxes, obviously (that's what the five hidden items in the elemental pattern are), but between the missing memories and the hybridization Quirrellmort doesn't consider them to be worth the trouble. Keep in mind that there is a nice theory about not being able to lie in parseltongue.

Not like certain people living in certain countries, who were, it was said, as human as anyone else; who were said to be sapient beings, worth more than any mere unicorn. But who nonetheless wouldn't be allowed to live in Muggle Britain. On that score, at least, no Muggle had the right to look a wizard in the eye. Magical Britain might discriminate against Muggleborns, but at least it allowed them inside so they could be spat upon in person.

I suppose open borders and unrestricted immigration are in-keeping with Harry's character as a utilitarian who tries to assign equal value to each and every human life.

Also, won't Quirrell die of transfiguration sickness if he drinks the blood of transfigured Rarity?

What's the deal with spells and age? If Harry is really so far ahead of his class and can already cast spells nobody else can, why is it just now that he can cast "second-year" spells effortlessly?

Canon or not, this reminds me too much of the public school system of a certain country where kids are verboten to use words "they shouldn't know yet".

I am going to predict that you are talking about cvlx hc

I have invented a wormhole with ends separated by ten seconds in time. Unfortunately the power requirements scale exponentially with size so its not practical for anything larger than photons, but it does mean I can send information back in time. How would you exploit this?

Is it reasonable to assign P(X) = P(willbeproven(X)) / (P(willbeproven(X)) + P(willbedisproven(X))) ?

No I don't think so, consider the following example:

I flip a coin. If it comes up heads I take two green marbles, else I take one red and one green marble. Then I offer to let you see a random marble and I destroy the other one without showing you.

Then, suppose you wish to test whether my coin came up tails. if the marble is red, you have proven the coin came up tails and the chance of tails being disproven is zero, so your expression is 1, but it should be 0.5.