It sounds very much to me like you would like mathematicians to change our notation to accommodate someone who has not put in the effort to learn mathematics. Do you wish the same from structural engineers? Programmers? Physicists? Medical doctors? Musicians?
> "In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do."
This isn't a mathematical definition of a field. This is an encyclopedia's intuitive definition. It is a good one, in my opinion, as it'll evoke the correct definition in a person who is trained at mathematics, and hopefully convey the gist of the idea to someone who is not. Very good for one sentence! But a mathematical definition it is not!
A mathematical definition of a field would be as follows:
BEGIN DEFINITION
A field is a set F together with two functions A:F×F→F and M:F×F→F and two elements z∈F and o∈F that together satisfy the following properties:
ASSOCIATIVITY OF A: A(A(a, b), c) = A(a, A(b, c)) for all a,b,c∈F.
ASSOCIATIVITY OF M: Same as above with M in place of A.
COMMUTATIVITY OF A: A(a, b) = A(b, a) for all a,b∈F.
COMMUTATIVITY OF A: Same as above with M in place of A.
NEUTRALITY OF z WRT A: A(a, z) = a for all a∈F.
NEUTRALITY OF e WRT M: M(a, o) = a for all a∈F.
INVERSE FOR A: For all a∈F, there exists an element na∈F such that A(a, na) = z.
INVERSE FOR M: For all a∈F such that a≠z, there exists an element ra∈F such that M(a, ra) = o.
M DISTRIBUTES OVER A: M(a, A(b, c)) = A(M(a, b), M(a, c)) for all a,b,c∈F.
END DEFINITION.
(This definition presupposes that one knows what a set is under a standard framework.)
Since this notation is cumbersome, it is common to write A(a,b) as a+b and M(a,b) as a·b or ab. Likewise, z if often written 0 and and o is often written 1 (or e). Similarly, na is often written -a and ra is often written 1/a, but do take care to recall that "-a" and "1/a" are just symbols. One typically compresses notation even further, and writes "a + -b" as "a - b" (not to be confused for the juxtaposition of a and -b).
Do you feel better about this definition than Wikipedia's informal one? I invite you to scribble out a verification that for example the reals form a field under this definition (with ordinary addition as A, ordinary multiplication as M, ordinary 0 as z, ordinary 1 as o).
> Because if the field had a limited number of elements and you added the last two (highest) elements together, the property which requires that the result also be present in the same set could not be met because the result would be greater than the highest element in that set...
You're reading too much into "behave like the corresponding operations on real numbers do". One does not demand that addition preserves order. Notice how there is no reference to ordering or elements being "larger" or "smaller" in the definition above. It is for example the case that in the field of two elements, 1+1=0. That's fine.
> The problem is that if you start with a highly abstracted math concept and you dig through all the links and definitions of sub-concepts, they all have huge gaps like this
Not at all. You are trying to do rigorous mathematics using informal statements (not to detract from the informal statement; someone who has seen the formal definition of a lot of mathematical structure can almost surely reconstruct the correct formal definition of a field in a second from the informal one).
> I think math definitions should be more elaborate and repetitive if necessary.
So mathematicians should make communication between ourselves more cumbersome in order to please outsiders? I'm sorry, I don't mean to sound like a gatekeeper, but this is ridiculous.
> They should not try to sound terse and clever. They should not assume that the reader can fill in the gaps.
Do you demand this of musicians and engineers and chefs and mechanics and pilots and doctors and nurses too?
> The most rational readers will not be able to fill in the gaps because rational people know the dangers of making assumptions.
The most rational readers who have studied mathematics will. To a point, of course. There is often a tradeoff to be made, but your blanket statement is just plain wrong.