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I find your distinction between solvability and decidability somewhat interesting. I have not heard the distinction drawn that way. It has been a few years since I last looked at a book on this subject (and I gave away most of my math books about 4 years ago, so I don't even have one to look this up in right now), but I would think I would have remembered something like that.

So I did a quick Google search. And the funny thing is that I found lots of presentations online that disagreed with your proposed usage. I found none that agreed. For a random instance this lecture states directly that the Halting problem is not solvable.

And even when I look at your links, they say nothing about finite versus infinite. The classification they draw goes like this. A decision problem is a problem whose answers are all yes/no. A decidable problem a decision problem that is solvable. Solvable problems differ from decision problems in that solvable problems are not just limited to yes/no answers.

The fact that infinite answers are not accepted either for definitions of decidability or solvability is underscored by this lecture. They are very explicit about the fact that solvability and decidability both require that all of the computations involved halt. In other words you need to get an answer in due time.

Now you wouldn't just be making up your finite/infinite distinction, would you? Do you have a verifiable source you are willing to quote showing that someone else out there agrees with the usage you are claiming?

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