I'm very confused. Of course if φ is provable then it's true. That's the whole point of using proofs.

I'm very confused about something related to the Halting Problem. I discussed this on the IRC with some people, but I couldn't get across what I meant very well. So I wrote up something a bit longer and a bit more formal.

The gist of it is, the halting problem lets us prove that, for a specific counter example, there can not exist any proof that it halts or not. A proof that it does or does not halt, causes a paradox.

But if it's true that there doesn't exist a proof that it halts, then it will run forever searching for one. Therefore I've proved that the program will not halt. Therefore a proof that it doesn't halt does exist (this one), and it will eventually find it. Creating a paradox.

Just calling the problem undecidable doesn't actually solve anything. If you can prove it's undecidable, it creates the same paradox. If no Turing machine can know whether or not a program halts, and we are also Turing machines, then we can't know either.

I have to be honest: I hadn't considered that angle yet (I tend to create ideas first, then hone them and remove issues).

The first point is that this was just an example, the first one to occur to me, and we can certainly find safer examples or improve this one.

The second is that torture is very unlikely - death, maybe painful death, but not deliberate torture.

The third is that I know some people who might be willing to go through with this, if it cured cancer through the world.

But I will have to be more careful in these issues in future, thanks.

If the excellent simulation of a human with cancer is conscious, you've created a very good torture chamber, complete with mad vivisectionist AI.

Do you still get to do some science?

I think the point of the quote is that in the first case you have five methods you can use to attack different problems. In the second case you only have one method, and you have to hope every problem is a nail.

It is better to solve one problem five different ways, than to solve five problems one way

George Pólya, or at least attributed to him, as I am unable to find the exact source, despite its being widely quoted in texts related to mathematics education or problem solving in general.

That assertion isn't actually true, in the strong form in which he intends it. Even if you rely on the vagueness of "morals", it's certainly not true for legislation.

I got a new job! Which pays better than the old one.

There are different levels of impossible.

Imagine a universe with an infinite number of identical rooms, each of which contains a single human. Each room is numbered outside: 1, 2, 3, ...

The probability of you being in the first 100 rooms is 0 - if you ever have to make an expected utility calculation, you shouldn't even consider that chance. On the other hand, it is definitely possible in the sense that some people are in those first 100 rooms.

If you consider the probability of you being in room Q, this probability is also 0. However, it (intuitively) feels "more" impossible.

I don't really think this line of thought leads anywhere interesting, but it definitely violated my intuitions.