Why does ANSI SQL define SUM(no rows) as NULL?

Question

The ANSI SQL standard defines (chapter 6.5, set function specification) the following behaviour for aggregate functions on empty result sets:

COUNT(...) = 0
AVG(...) = NULL
MIN(...) = NULL
MAX(...) = NULL
SUM(...) = NULL

Returning NULL for AVG, MIN and MAX makes perfect sense, since the average, minimum and maximum of an empty set is undefined.

The last one, however, bothers me: Mathematically, the SUM of an empty set is well-defined: 0. Using 0, the neutral element of addition, as the base case makes everything consistent:

SUM({})        = 0    = 0
SUM({5})       = 5    = 0 + 5
SUM({5, 3})    = 8    = 0 + 5 + 3
SUM({5, NULL}) = NULL = 0 + 5 + NULL

Defining SUM({}) as null basically makes "no rows" a special case that does not fit in with the others:

SUM({})     = NULL  = NULL
SUM({5})    = 5    != NULL + 5 (= NULL)
SUM({5, 3}) = 8    != NULL + 5 + 3 (= NULL)

Is there some obvious advantage of the choice that was made (SUM being NULL) that I have missed?