There can’t be uncountably many propositions to which you assign probability 0, because you can only express countably many propositions.
Regarding your Pascal’s mugging argument, VNM-rational agents don’t assign infinite or negative infinite utility to anything. The variant using utility that is vast but finite in magnitude need not convince an agent that assigns the extreme outcome comparably tiny but nonzero probability. And it doesn’t work for agents with bounded utility functions, because they don’t assign such high utilities to any outcome, and thus there aren’t any outcomes that they must assign extremely tiny probabilities to in order to avoid weird behavior.
Do you agree that there are (uncountably) infinite many propositions to which we can assign a probability of 1. Then we assign a probability of 0 to the negation of those propositions.
It appears that you’re starting with some countably infinite set S of propositions, and then trying to make a proposition for each subset of S by taking the conjunction of all propositions in the subset. But all but countably many of those subsets are infinite, and you can’t take the conjunction of infinitely many propositions.
A program can’t pick out arbitrary subsets of an infinite set either. Programs can’t do uncountably many things, even if you give them an infinite amount of time to work with.
As is written, g(var) picks out one arbitrary subset of the infinite set. There are 2^Aleph_null possible subsets g(var) can produce, thus, g(var) can (not does) produce uncountably infinite many true propositions.
Ok, I see what you’re trying to do now (though the pseudocode you wrote still doesn’t do it successfully). It’s true that with randomness, there are uncountably many infinite strings that could be produced. But you still have no way of referring to each one individually, so there’s little point in calling them “propositions”, which typically refers to claims that can actually be stated.
There can’t be uncountably many propositions to which you assign probability 0, because you can only express countably many propositions.
Regarding your Pascal’s mugging argument, VNM-rational agents don’t assign infinite or negative infinite utility to anything. The variant using utility that is vast but finite in magnitude need not convince an agent that assigns the extreme outcome comparably tiny but nonzero probability. And it doesn’t work for agents with bounded utility functions, because they don’t assign such high utilities to any outcome, and thus there aren’t any outcomes that they must assign extremely tiny probabilities to in order to avoid weird behavior.
Do you agree that there are (uncountably) infinite many propositions to which we can assign a probability of 1. Then we assign a probability of 0 to the negation of those propositions.
No, of course not. As I said, there are only countably many propositions you can express at all.
I showed a method for constructing uncountably many propositions using recursion.
It appears that you’re starting with some countably infinite set S of propositions, and then trying to make a proposition for each subset of S by taking the conjunction of all propositions in the subset. But all but countably many of those subsets are infinite, and you can’t take the conjunction of infinitely many propositions.
Why can’t you take the conjunction of infinitely many propositions?
You can’t write it down in any finite amount of time.
Must I write them down? I wrote a program that could write them down.
No, you didn’t. A program can’t write down an infinite amount of information either.
Grant the program an infinite amount of time. I didn’t say the program must terminate did I.
A program can’t pick out arbitrary subsets of an infinite set either. Programs can’t do uncountably many things, even if you give them an infinite amount of time to work with.
As is written, g(var) picks out one arbitrary subset of the infinite set. There are 2^Aleph_null possible subsets g(var) can produce, thus, g(var) can (not does) produce uncountably infinite many true propositions.
Ok, I see what you’re trying to do now (though the pseudocode you wrote still doesn’t do it successfully). It’s true that with randomness, there are uncountably many infinite strings that could be produced. But you still have no way of referring to each one individually, so there’s little point in calling them “propositions”, which typically refers to claims that can actually be stated.