No, that doesn’t make sense to me. The inverse of & would be a computable function that takes the Gödel number of a unary numeral and returns the number represented by that numeral. #[...] is syntactic sugar somewhat similar to the $NAME in e.g. Perl’s “Hello, $NAME!”.
But “#[&7]” = “7″, and if you replace the 7s with some other number, it’s still true, right?
“n >= 7” would mean an expression that contains the variable n, rather than an expression containing the unary numeral representing the numeric value of n.
Got it.
No, wait, that’s completely wrong. We need to distinguish between applying repr(.) to an argument and constructing the Gödel number of an expression in which repr(.) is applied to an argument. For example, in my earlier comment, plus(x,y) was not equal to (x+y): if x and y are the Gödel numbers of two expressions, then (x+y) would simply add those Gödel numbers (which isn’t useful), whereas plus(x,y) returns the Gödel number of an expression (namely the expression adding the expressions denoted by the Gödel numbers x and y). Do you see what I mean?
Oh, I see. Everything after that paragraph, I’m going to have to think about for a while. Edit: got it (I think).
But “#[&7]” = “7″, and if you replace the 7s with some other number, it’s still true, right?
Ah! This is correct, but I would conceptualize it differently, as the combination of two distinct phenomena. First, #[...] is sort of the inverse of ”...”, which makes more sense to me because both of these are kinds of syntactic sugar—we have both “#[e]” = e and “x + #[‘y’]” = “x + y”.
Second, “7” is the Gödel number of the unary numeral 7, and repr(7) also returns this Gödel number. In other words, “7“ = &7. Putting the two together: “#[&7]” = &7 = “7”.
But “#[&7]” = “7″, and if you replace the 7s with some other number, it’s still true, right?
Got it.
Oh, I see. Everything after that paragraph, I’m going to have to think about for a while. Edit: got it (I think).
Ah! This is correct, but I would conceptualize it differently, as the combination of two distinct phenomena. First, #[...] is sort of the inverse of ”...”, which makes more sense to me because both of these are kinds of syntactic sugar—we have both “#[e]” = e and “x + #[‘y’]” = “x + y”.
Second, “7” is the Gödel number of the unary numeral 7, and repr(7) also returns this Gödel number. In other words, “7“ = &7. Putting the two together: “#[&7]” = &7 = “7”.
:-) Thanks again for sticking with it!