If (someone tells you If [I tell you] X, then X is true)
Are you trying to have the box be interchangeable with [I tell you]? Because it seems to me that usually, when “if” is followed by parentheses, those parentheses include the condition. In which case it should be “If [I tell you X]”
Also, what’s the difference between the turnstile (I think that’s what it’s called) and the box?
[I tell you] X is analogous to “box X”. Formally, the box is how we represent the statement (PA concludes X) inside the formal system S (which could also be PA). The brackets were meant to show which part corresponded with the box itself—the proper parentheses placement would be as you describe.
The turnstile means that S (or whatever system) concludes X. In other words, that X is provable in S. The box represents that (PA concludes X). So one difference is that it is for PA specifically. But more importantly, the box is a representation of that statement inside the system S. So the statement X must be encoded as a number—called the Godel nubmer of X. This is similar to how an image is encoded as a binary number on your computer. Then there is a function Bew—which is a ‘compiler’ for a number X. This is just a property of a natural number, carefully designed to represent the property that if n = encoding(X), then Bew(n) is true if PA proves the statement X. So “box X” means “Bew(encoding(X))”.
It’s like the difference between knowing you will do a certain thing—and you actually doing that thing. The box is what S thinks PA will do, and the turnstile is what it actually does.
Are you trying to have the box be interchangeable with [I tell you]? Because it seems to me that usually, when “if” is followed by parentheses, those parentheses include the condition. In which case it should be “If [I tell you X]”
Also, what’s the difference between the turnstile (I think that’s what it’s called) and the box?
[I tell you] X is analogous to “box X”. Formally, the box is how we represent the statement (PA concludes X) inside the formal system S (which could also be PA). The brackets were meant to show which part corresponded with the box itself—the proper parentheses placement would be as you describe.
The turnstile means that S (or whatever system) concludes X. In other words, that X is provable in S. The box represents that (PA concludes X). So one difference is that it is for PA specifically. But more importantly, the box is a representation of that statement inside the system S. So the statement X must be encoded as a number—called the Godel nubmer of X. This is similar to how an image is encoded as a binary number on your computer. Then there is a function Bew—which is a ‘compiler’ for a number X. This is just a property of a natural number, carefully designed to represent the property that if n = encoding(X), then Bew(n) is true if PA proves the statement X. So “box X” means “Bew(encoding(X))”.
It’s like the difference between knowing you will do a certain thing—and you actually doing that thing. The box is what S thinks PA will do, and the turnstile is what it actually does.