The system of hyperreal numbers represents a rigorous method of treating the ideas about infinite and infinitesimal numbers that had been used casually by mathematicians, scientists, and engineers ever since the invention of calculus by Newton and Leibniz. The hyperreals, or nonstandard reals (usually denoted as *R), denote an ordered field which is a proper extension of the ordered field of real numbers R and which satisfies the transfer principle. This principle allows true first order statements about R to be reinterpreted as true first order statements about *R.
An important property of *R is that it has infinitely large as well as infinitesimal numbers, where an infinitely large number is a number that is larger than all numbers representable in the form
The condition of being a hyperreal field is a stronger one than that of being a real closed field strictly containing R. It is also stronger than that of being a superreal field in the sense of Dales and Woodin.
However, in the 1960s Abraham Robinson showed how infinitely large and infinitesimal numbers can be rigorously defined and used to develop the field of nonstandard analysis. Robinson developed his theory nonconstructively, using model theory; however it is possible to proceed using only algebra and topology, and proving the transfer principle as a consequence of the definitions. In other words hyperreal numbers per se, aside from their use in nonstandard analysis, have no necessary relationship to model theory or first order logic.
The correct general statement that formulates these equivalences is called the transfer principle. Note that in many formulas in analysis quantification is over higher order objects such as functions and sets which makes the transfer principle somewhat more subtle than the above examples suggest.
The transfer principle however doesn't mean that R and *R have identical behavior. For instance, in *R there exists an element w such that
The hyperreals *R form an ordered field containing the reals R as a subfield. Unlike the reals, the hyperreals do not form a standard metric space, but by virtue of their order they carry an order topology.
The hyperreals can be developed either axiomatically or by more constructively oriented methods. The essence of the axiomatic approach is to assert (1) the existence of at least one infinitesimal number, and (2) the validity of the transfer principle. In the following subsection we give a detailed outline of a more constructive approach. This method allows one to construct the hyperreals if given a set-theoretic object called an ultrafilter, but the ultrafilter itself cannot be explicitly constructed. (Kanovei and Shelah have found a method that gives an explicit construction, at the cost of a significantly more complicated treatment.)
We are going to construct a hyperreal field via sequences of reals. In fact we can add and multiply sequences componentwise; for example,
Comparing sequences is thus a delicate matter. We could, for example, try to define a relation between sequences in a componentwise fashion:
This is a total preorder and it turns into a total order if we agree not to distinguish between two sequences a and b if a≤b and b≤a. With this identification, the ordered field *R of hyperreals is constructed. From an algebraic point of view, U allows us to define a corresponding maximal ideal I in the commutative ring A, and then to define *R as A/I; as the quotient of a commutative ring by a maximal ideal, *R is a field. This is also notated A/U, directly in terms of the free ultrafilter U; the two are equivalent.
The field A/U is an ultrapower of R. Since this field contains R it has cardinality at least the continuum. Since A has cardinality
One question we might ask is whether, if we had chosen a different free ultrafilter V, the quotient field A/U would be isomorphic as an ordered field to A/V. This question turns out to be equivalent to the continuum hypothesis; in ZFC with the continuum hypothesis we can prove this field is unique up to order isomorphism, and in ZFC with the continuum hypothesis false we can prove that there are non-order-isomorphic pairs of fields which are both countably indexed ultrapowers of the reals.
For more information about this method of construction, see ultraproduct.
The following is an intuitive way of understanding the hyperreal numbers. The approach taken here is very close to the one in the book by Goldblatt. Recall that the sequences converging to zero are sometimes called infinitely small. These are almost the infinitesimals in a sense, the true infinitesimals are the classes of sequences that contain a sequence converging to zero. Let us see where these classes come from. Consider first the sequences of real numbers. They form a ring, that is, one can multiply add and subtract them, but not always divide by non-zero. The real numbers are considered as the constant sequences, the sequence is zero if it is identically zero, that is, for all .
In our ring of sequences one can get with neither nor . Thus, if for two sequences one has , at least one of them should be declared zero. Surprisingly enough, there is a consistent way to do it. As a result, the classes of sequences that differ by some sequence declared zero will form a field which is called a hyperreal field. It will contain the infinitesimals in addition to the ordinary real numbers, as well as infinitely large numbers (the reciprocals of infinitesimals, they will be represented by the sequences converging to infinity). Also every hyperreal which is not infinitely large will be infinitely close to an ordinary real, in other words, it will be an ordinary real + an infinitesimal.
This construction is parallel to the construction of the reals from the rationals given by Cantor. He started with the ring of the Cauchy sequences of rationals and declared all the sequences that converge to zero to be zero. The result is the reals. To continue the construction of hyperreals, let us consider the zero sets of our sequences, that is, the , that is, is the set of indexes for which . It is clear that if , then the union of and is N (the set of all natural numbers), so:
Any family of sets that satisfies (ii)-(iv) is called a filter (an example: the complements to the finite sets, it is called the Fréchet filter and it is used in the usual limit theory). If (i) also holds, U is called an ultrafilter (because you can add no more sets to it without breaking it). The only explicitly known example of an ultrafilter is the family of sets containing a given element (in our case, say, the number 10). Such ultrafilters are called trivial, and if we use it in our construction, we come back to the ordinary real numbers (exercise). Any ultrafilter containing a finite set is trivial (exercise). It is known that any filter can be extended to an ultrafilter, but the proof uses the axiom of choice. The existence of a nontrivial ultrafilter (the ultrafilter lemma) can be added as an extra axiom, it's weaker than the axiom of choice (that says that for any bunch of nonempty sets there is a function f that picks an element from any of them, f(X) is an element of X).
Now if we take a nontrivial ultrafilter (which is an extension of the Fréchet filter, exercise) and do our construction, we get the hyperreal numbers as a result. The infinitesimals can be represented by the non-vanishing sequences converging to zero in the usual sense, that is with respect to the Fréchet filter (exercise).
If is a real function of a real variable then naturally extends to a hyperreal function of a hyperreal variable by composition:
All the arithmetical expressions and formulas make sense for hyperreals and hold true if they are true for the ordinary reals. One can prove that any finite (that is, such that for some ordinary real ) hyperreal will be of the form where is an ordinary (called standard) real and is an infinitesimal.
It is parallel to the proof of the Bolzano-Weierstrass lemma that says that one can pick a convergent subsequence from any bounded sequence, done by bisection, the property (i) of the ultrafilters is again crucial.
Now one can see that is continuous means that is infinitely small whenever is, and is differentiable means that
A hyperreal number r is called infinitesimal if it is less than every positive real number and greater than every negative real number. Zero is an infinitesimal, but non-zero infinitesimals also exist: take for instance the class of the sequence (1, 1/2, 1/3, 1/4, 1/5, 1/6, ...) (this works because the ultrafilter U contains all index sets whose complement is finite).
A hyperreal number x is called finite (or limited by some authors) if there exists a natural number n such that -n < x < n; otherwise, x is called infinite (or illimited). Infinite numbers exist; take for instance the class of the sequence (1, 2, 3, 4, 5, ...). A non-zero number x is infinite if and only if 1/x is infinitesimal.
The finite elements F of *R form a local ring, and in fact a valuation ring, with the unique maximal ideal S being the infinitesimals; the quotient F/S is isomorphic to the reals. Hence we have a homomorphic mapping, st(x), from F to R whose kernel consists of the infinitesimals and which sends every element x of F to a unique real number whose difference from x is in S; which is to say, is infinitesimal. Put another way, every finite nonstandard real number is "very close" to a unique real number, in the sense that if x is a finite nonstandard real, then there exists one and only one real number st(x) such that x – st(x) is infinitesimal. This number st(x) is called the standard part of x, conceptually the same as x to the nearest real number. This operation is an order-preserving homomorphism and hence is well-behaved both algebraically and order theoretically. It is order-preserving though not isotonic, i.e. implies , but does not imply .
An important special case is where the topology on X is the discrete topology; in this case X can be identified with a cardinal number κ and C(X) with the real algebra of functions from κ to R. The hyperreal fields we obtain in this case are called ultrapowers of R and are identical to the ultrapowers constructed via free ultrafilters in model theory.