What's the maximum quantifier depth for doing everyday mathematics? I count groups of quantifiers of the same type as 1, so that a block of variables of the shape (all x,y,z), or the statement that there's a magic square of size 3x3 (exists a,b,c,d,e,f,g,h,i) are counted as one quantifier. So I'm really talking about Alternating Quantifier Depth. The definition of "a function f is continuous at a point x" has depth 3: (all epsilon) (exists delta) (all y) 0 < |y-x| < delta => |f(y)-f(x)| < epsilon and "f is continuous" is also depth 3: (all x,epsilon) (exists delta) (all y) etc. etc. The theorem that "the sum of continuous functions is continuous" is depth 5: (all f,g) (exists h) (all x,epsilon) (exists delta) (all y) h(x) = f(x)+g(x) And { 0 < |y-x| < delta => ... } So it looks like routine math might use a maximum depth of 8 or so. Presumably there are a few specialties that go much deeper. (?) Evaluating a game tree is an example. Rich