>Again, thanks for making me think and showing me the limits of my understanding.
Yes this was a fun discussion, thanks.
Your objection stands if you have (and know you have) at least one instance of every value for the quantity. So suppose that we are given a countably infinite set of variables and told that each integer is denoted by at least one of these variables, and then further given a function over pairs of variables f(x,y), such that f(x,y) = 1 if x and y differ by less than 3 and = 0 otherwise. Then, yes, we can figure out which variables are exactly identical to which others.
However, I would regard this as irrelevant scenario in the sense that we could never know, via observation, that we had obtained such a set of variables (even if we allow the possibility of making a countably infinite number of observations). Suppose that we make an infinite series of observations and end up with at least one variable denoting each member of the following set (with the ellipses counting up/down to +/-infinity):
...,0,2,3,4,5,6,7,9,...
In other words, we have variables with every integer value except 1 and 8. Then for any variable x with the value 4 and variable y with the value 5, f(x,z) = f(y,z) for all variables z. In other words, there'll be no way to distinguish 4-valued variables from 5-valued variables. It's only in the case where some oracle tells us that we have a variable for every integer value that we can figure out which variables have exactly the same values as which others.
Yes this was a fun discussion, thanks.
Your objection stands if you have (and know you have) at least one instance of every value for the quantity. So suppose that we are given a countably infinite set of variables and told that each integer is denoted by at least one of these variables, and then further given a function over pairs of variables f(x,y), such that f(x,y) = 1 if x and y differ by less than 3 and = 0 otherwise. Then, yes, we can figure out which variables are exactly identical to which others.
However, I would regard this as irrelevant scenario in the sense that we could never know, via observation, that we had obtained such a set of variables (even if we allow the possibility of making a countably infinite number of observations). Suppose that we make an infinite series of observations and end up with at least one variable denoting each member of the following set (with the ellipses counting up/down to +/-infinity):
In other words, we have variables with every integer value except 1 and 8. Then for any variable x with the value 4 and variable y with the value 5, f(x,z) = f(y,z) for all variables z. In other words, there'll be no way to distinguish 4-valued variables from 5-valued variables. It's only in the case where some oracle tells us that we have a variable for every integer value that we can figure out which variables have exactly the same values as which others.