I am not a specialist, but here is my take. One has a hierarchy of reals, hyperreals, superreals and surreals. These are ordered fields, with surreals being the largest possible ordered field. A field is a set with the four arithmetic operations that satisfy all the usual properties, in particular, commutativity of addition and multiplication. An ordered field requires that the order is compatible with the operations.
Both hyper- and surreals have infinite (greater than every sum and infinitesimal (smaller than for every natural ) numbers. Both are real closed fields, i.e., every first-order formula (a logical formula that quantifies over numbers only, not sets of numbers) using plus, times and order is true in reals iff it is true in these fields. Examples of such formulas true on reals and therefore true on hyper- and surreals include:
- every positive number is a square of another number;
- any polynomial of odd degree has at least one root;
- the polynomial has no roots.
One difference between them, according to Wikipedia, is that the field of hyperreals has "all the first-order properties of the real number system for statements involving any relations (regardless of whether those relations can be expressed using +, ×, and ≤). For example, there would have to be a sine function that was well defined for infinite inputs." Because of this, hyperreal numbers are used in non-standard analysis. In contrast, "certain transcendental functions can be carried over to the surreals, including logarithms and exponenentials, but most, e.g., the sine function, cannot."
Another way to extend reals is by adding roots of polynomials that have no solutions in real numbers. For example, complex numbers are obtained by considering a single polynomial with real coefficients that has no real root and adding a new element that is supposed to be a root. Every complex number can then be written as , , i.e., as a two-dimensional vector with real coordinates. Then the Fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. (Such field extension is called algebraic.) However, complex numbers cannot be ordered in a way that is compatible with arithmetic operations.
The Frobenius theorem states that the only finite-dimensional vector spaces over reals with multiplication and division are complex numbers (dimension 2) and quaternions (dimension 4). In fact, quaternions do not even form a field because their multiplication is not commutative.
Philosophically, there are reasons to believe that real numbers are, in fact, non-real, i.e., are not part of this universe. First, some physicists believe that our space-time may be discrete rather than continuous (Google "discrete space-time"). Second, most reals are given by an infinite decimal expansion, but once we assume that an infinite collection, though not physically realizable, is a real object, we start getting different difficulties in mathematics. For example, most real numbers are unnameable. Indeed, any description of a real number is a finite text, and there are only countably many of those, whereas the set of real numbers has a stricter greater cardinality of continuum.
Another example is the Continuum hypothesis: "There is no set whose cardinality is strictly between that of the integers and that of the real numbers." It can be neither proved nor disproved from the axioms of set theory (those axioms, in turn, have been introduce to avoid paradoxes when considering infinite objects). So, mathematicians are kind of stuck on that because, unlike physicists, they can't perform an experiment that would say whether our universe really contains sets with an intermediate cardinality.
In this respect, the provocative statement by the 19th-century mathematician Leopold Kronecker, "God made the integers; all else is the work of man" does not seem completely unreasonable. To me, however, it seems that real numbers and other infinite objects are too "natural" to be the work of man. They are not physical, but they are "out there." This reminds me a scene from the movie "The Thirteenth Floor" where a man arrived at a frontier where our "real" universe smoothly extends into a "pure design space."
I also think that natural numbers are not as special: they are just the simplest nontrivial term algebra formed formed from the constant 0 and a unary functional symbol S (successor). One can as easily consider one or more binary, ternary, etc. functional symbols.