The last formula in this post, the conservation of expected evidence, had a mistake which I've only just now fixed. Since I guess it's not obvious even to me, I'll put a reminder for myself here, which may not be useful to others. Really I'm just "translating" from the "law of iterated expectations" I learned in my stats theory class, which was:

$$E\left[E\right[X|Y\left]\right]=E\left[X\right]$$

This is using a notation which is pretty standard for defining conditional expectations. To define it you can first consider the expected value given a particular value of the random variable $Y$. Think of that as a function of that particular value: $$f\left(y\right)=E[X|Y=y]$$ Then we define conditional expectation as a random variable, obtained from plugging in the random value of $Y$: $$E\left[X|Y\right]=f\left(Y\right)$$ The problem with this notation is it gets confusing which capital letters are random variables and which are propositions, so I've bolded random variables. But it makes it very easy to state the law of iterated expectations.

The law of iterated expectations also holds when "relativized". That is, $$E\left[E\right[X|Y\left]|B\right]=E\left[X|B\right]$$ where $B$ is an event. If we wanted to stick to just putting random variables behind the conditional bar we could have used the indicator function of that event.

And this translates to the statement in my post. $X$ is an indicator for the event $H$, which makes a conditional expectation of it a conditional probability of $H$. So $E\left[X|Y\right]$ is $\Theta$. Our proposition $B$ is the background information $B$, I used the same symbol there. And the right hand side is another expectation of an indicator and therefore also a probability.

I really didn't want to define this notation in the post itself, but it's how I'm trained to think of this stuff, so for my own confidence in the final formula I had to write it out this way.

It would be nice if you had the sexes of the siblings, since it's supposedly only the older brothers that count, though I don't really expect that to change anything.

Really the important thing is just to separate birth order from family size. Usually the way I think of this is, we can look at number of older brothers, with a given number of older siblings. I like this setup because it looks like a randomized trial. I have two older siblings, so do you, meiosis randomizes their sexes.

But I guess with the data you have you can look at birth order with a given family size, so we don't have to worry about the effect of a larger or smaller family. I... don't think this is what you did? Did I misunderstand something? It seems like if cardinals come from smaller families, that would show up as lower birth orders.

With 9 million people I'd just split it into categories by number of siblings, with your data I'm not sure.

After reading this post, I handed over $200 for a month of ChatGPT pro and I don't think I can go back. o1-pro and Deep Research are next level. o1-pro often understands my code or what I'm asking about without a whole conversation of clarifying, whereas other models it's more work than it's worth to get them focused on the real issue rather than something superficially similar. And then I can use "Deep Research" to get links to webpages relevant to what I'm working on. It's like... smart Google, basically. Google that knows what I'm looking for. I never would have known this existed if I didn't hand over the $200.

Depends entirely on Cybercab. A driverless car can be made cheaper for a variety of reasons. If the self-driving tech actually works, and if it's widely legal, and if Tesla can mass produce it at a low price, then they can justify that valuation. Cybercab is a potential solution to the problem that they need to introduce a low priced car to get their sales growing again but cheap electric cars is a competitive market now without much profit margin. But there's a lot of ifs.

Yeah, just went through this whole same line of evasion. Alright, the Collatz conjecture will never be "proved" in this restrictive sense—and neither will the Steve conjecture or the irrationality of √2—do we care? It may still be proved according to the ordinary meaning.

Alright, so Collatz will be proved, and the proof will not be done by "staying in arithmetic". Just as the proof that there do not exist numbers p and q that satisfy the equation p² = 2 q² (or equivalently, that all numbers do not satisfy it) is not done by "staying in arithmetic". It doesn't matter.

Not off the top of my head, but since a proof of Collatz does not require working under these constraints, I don't think the distinction has any important implications.

We can eliminate the concept of rational numbers by framing it as the proof that there are no integer solutions to p² = 2 q²... but... no proof by contradiction? If escape from self-reference is that easy, then surely it is possible to prove the Collatz conjecture. Someone just needs to prove that the existence of any cycle beyond the familiar one implies a contradiction.

Yes, the proof that there's no rational solution to x²=2. It doesn't require real numbers.