Do programming languages have consistent interpretations of {1,...,n} when n = 0?

Question

On math.SE, a question about math notation enerated a discussion of how programming languages interpret the set {1,...,n} when n=0

The question asked for a mathematical notation to represent the R code 1:n

According to the comments, the mathematical interpretation of {1,...,n} when n=0 is that this is an empty set. A subsequent comment suggested that C is consistent with this interpretation, because for (int i = 1; i < n; i++) returns a empty set because it iterates 0 times.

It is not clear to me what the equivalent statement in R is, but 1:0 returns the vector [1,0]

Thus, for (i in 1:0) print(i) iterates over 1 and 0 (I interpret as analogous to the C code above)

Is this because {1,...,n} is not the correct notation for 1:n?

Does this mean R violates a universal rule?

Is there a consistent interpretation for this set among programming languages?

Answer 1

Each mathematical formalism has its own notation. To suggest that there is a "universal notation" is very "un-mathematical". Look at the notation associated with tensors or groups if you want examples of mathematical domains where multiple notational systems exist.

In R the code x <- 1:0 returns the ordered vector c(1,0). Just as the code x <- 2:-2 returns c(2,1,0,-1,-2). The code x <- seq(1, length=0) returns a sequence of length 0 which is printed in console sessions as integer(0). R is not really designed to mimic set notation but it does have some set functions and it also has packages that more fully implement set notation.