A more general case is that a choice is the outcome of picking a subset C out of a set S of possible choices. Picking the empty set as the subset C may also be a choice depending on how we look at it.
Oh, it's more general alright. The problem is, it's now so general that it's not describing a choice; and, it's also so general that it loses meaning (what is a "possibility" in your description?)
Let's consider your sandwich example. To resolve ambiguity in such a way as to describe this scenario, I'm going to presume it is a particular sandwich we're talking about (as opposed to making and eating a sandwich).
Case 1a: Let's presume the sandwich isn't actually there; I somehow got a mistaken notion that it was. But as it so happens, I decided not to have the sandwich.
Case 1b: Again the sandwich isn't actually there. But as it happens, I decided I want to eat the sandwich. So I go to where it was supposed to be (say, the fridge), and look for it, discovering that it's not there.
Case 2: Let's presume the sandwich
is there. However, in this case, I do not know that the sandwich is there. Furthermore, let's say that I go to bed without eating the sandwich (just to explicitly rule out the possibility that I discovered it and ate it before bed).
Was there a choice in case 1a? Case 1b? Case 2? I say yes, yes, and no respectively.
By in the moment I mean that the choice-making is the process of actually selecting the subset C.
Sure.
And that is an act of free will as I have defined it, since without the process of the choice-making the future will not be known.
You're begging the question. Suppose I'm in a highly precise MRI machine while I "make the choice". You're claiming that it would be impossible for a person looking at the MRI machine to know which choice I'm going to make before it is made.
And I'm claiming you have exactly no legitimate reason from an epistemic standpoint to make this claim. Furthermore, there's reason not to make this claim; even without an MRI machine, I can predict what thing I choose, and what thing friends of mine would choose, to a surprising degree of accuracy for certain classes of choice presentations. This sort of prediction is the basis for folk concepts such as "preference", "habit", etc.
The future becomes determined in part by the choices we make, so even though reality is deterministic it's generally unknown until the outcomes of the choices are made manifest.
Since choice is a behavior with duration, and since we so happen to be flexible in this way, we can change the choice even during the initiation of the action. I could start my march to the fridge and then suddenly stop, change my mind, and just go to bed.
So just as it's not necessarily unpredictable which choice I will make, it's not necessarily predictable what would happen even after having made a choice. And it's entirely possible that for some choices whether I stick to it or not is less predictable than for other choices whether or not I select a particular option.
The predictability aspect is a red herring that unfortunately too many fall for.
Growing your hair is not a conscious choice unless you are some sage who can regulate his or her bodily functions to that degree.
Be it known that hair grows anyway, and the conscious choice is about whether or not you want to exploit that to make your hair longer as opposed to getting haircuts or shaving your head. "Grow my hair out" doesn't mean to
make my hair grow; it means to
let it grow long.
So now that I clarified this for you, what about the question? Do you think you can point to the action in the moment when I made the choice to grow my hair out?
The future is unknown in the total sense, even for the universe, because of hypothesis 2 which says that reality is computationally irreducible.
You cannot derive a fact from a hypothesis. Furthermore, you're committing a fallacy of division here anyway; even if I grant that reality is computationally irreducible, that does not imply that choices are.
The principle of computational irreducibility says that ...
Just so you know, I'm not going to accept Wolfram as a valid authority in this area.