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Hyperreal number

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

1 + 1 + cdots + 1.
The use of the definite article the in the phrase the hyperreal number is somewhat misleading in that there is not a unique ordered field that is referred to in most treatments. However, a 2003 paper by Kanovei and Shelah shows that there is a definable, countably saturated (meaning ω-saturated, but not of course countable) elementary extension of the reals, which therefore has a good claim to the title of the hyperreal numbers.

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.

The application of hyperreal numbers and in particular the transfer principle to problems of analysis is called nonstandard analysis; some find it more intuitive than standard real analysis.

From Newton to Robinson

When Newton and (more explicitly) Leibniz introduced differentials, they used infinitesimals and these were still regarded as useful by later mathematicians such as Euler and Cauchy. Nonetheless these concepts were from the beginning seen as suspect, notably by Berkeley, and when in the 1800s calculus was put on a firm footing through the development of the (ε, δ)-definition of limit by Cauchy, Weierstrass, and others, they were largely abandoned.

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 transfer principle

The self-consistent development of the hyperreals turned out to be possible if every true first-order logic statement that uses basic arithmetic (the natural numbers, plus, times, comparison) and quantifies only over the real numbers was assumed to be true in a reinterpreted form if we presume that it quantifies over hyperreal numbers. For example, we can state that for every real number there is another number greater than it:
forall x in mathbb{R} quad exists y inmathbb{R}quad x < y
The same will then also hold for hyperreals:
forall x in {}^starmathbb{R} quad exists y in {}^starmathbb{R}quad x < y
Another example is the statement that if you add 1 to a number you get a bigger number:
forall x in mathbb{R} quad x < x+1
which will also hold for hyperreals:
forall x in {}^starmathbb{R} quad x < x+1

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

but there is no such number in R. This is possible because the nonexistence of this number cannot be expressed as a first order statement of the above type. A hyperreal number like w is called infinitely large; the reciprocals of the infinitely large numbers are the infinitesimals.

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.)

The ultrapower construction

We are going to construct a hyperreal field via sequences of reals. In fact we can add and multiply sequences componentwise; for example,

(a_0, a_1, a_2, ldots) + (b_0, b_1, b_2, ldots) = (a_0 +b_0, a_1+b_1, a_2+b_2, ldots)
and analogously for multiplication. This turns the set of such sequences into a commutative ring, which is in fact a real algebra A. We have a natural embedding of R in A by identifying the real number r with the sequence (r, r, r, ...) and this identification preserves the corresponding algebraic operations of the reals. The intuitive motivation is, for example, to represent an infinitesimal number using a sequence that approaches zero. The inverse of such a sequence would represent an infinite number. As we will see below, the difficulties arise because of the need to define rules for comparing such sequences in a manner that, although inevitably somewhat arbitrary, must be self-consistent and well defined. For example, we may have two sequences that differ in their first n members, but are equal after that; such sequences should clearly be considered as representing the same hyperreal number. Similarly, most sequences oscillate randomly forever, and we must find some way of taking such a sequence and interpreting it as, say, 7+epsilon, where epsilon is a certain infinitesimal number.

Comparing sequences is thus a delicate matter. We could, for example, try to define a relation between sequences in a componentwise fashion:

(a_0, a_1, a_2, ldots) leq (b_0, b_1, b_2, ldots) iff a_0 leq b_0 wedge a_1 leq b_1 wedge a_2 leq b_2 ldots
but here we run into trouble, since some entries of the first sequence may be bigger than the corresponding entries of the second sequence, and some others may be smaller. It follows that the relation defined in this way is only a partial order. To get around this, we have to specify which positions matter. Since there are infinitely many indices, we don't want finite sets of indices to matter. A consistent choice of index sets that matter is given by any free ultrafilter U on the natural numbers; these can be characterized as ultrafilters which do not contain any finite sets. (The good news is that the axiom of choice guarantees the existence of many such U, and it turns out that it doesn't matter which one we take; the bad news is that they cannot be explicitly constructed.) We think of U as singling out those sets of indices that "matter": We write (a0, a1, a2, ...) ≤ (b0, b1, b2, ...) if and only if the set of natural numbers { n : anbn } is in U.

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 ab and ba. 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

(2^{aleph_0})^{aleph_0} = 2^{aleph_0^2} =2^{aleph_0},,
it is also no larger than 2^{aleph_0}, and hence has the same cardinality as R.

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.

An intuitive approach to the ultrapower construction

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, a_n=0quad for all nquad.

In our ring of sequences one can get ab=0quad with neither a=0quad nor b=0quad. Thus, if for two sequences a, bquad one has ab=0quad, 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 z(a)={i: a_i=0}quad, that is, z(a)quad is the set of indexes iquad for which a_i=0quad. It is clear that if ab=0quad, then the union of z(a)quad and z(b)quad is N (the set of all natural numbers), so:

(i) one of the sequences that vanish on 2 complementary sets should be declared zero
(ii) if aquad is declared zero, abquad should be declared zero too, no matter what bquad is.
(iii) if both aquad and bquad are declared zero, then a^2+b^2quad should also be declared zero.
Now the idea is to single out a bunch U of subsets X of N and to declare that a=0quad if and only if z(a)quad belongs to U. From the conditions (i), (ii) and (iii) one can see that
(i) From 2 complementary sets one belongs to U
(ii) Any set containing a set that belongs to U, also belongs to U.
(iii) An intersection of any 2 sets belonging to U belongs to U.
(iv) we don't want an empty set to belong to U
because then everything becomes zero because every set contains an empty set.

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 fquad is a real function of a real variable xquad then fquad naturally extends to a hyperreal function of a hyperreal variable by composition:

where { dots} means "the equivalence class of the sequence dots relative to our ultrafilter", two sequences being in the same class if and only if the zero set of their difference belongs to our ultrafilter.

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 |x| < aquad for some ordinary real aquad) hyperreal xquad will be of the form y+dquad where yquad is an ordinary (called standard) real and dquad 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 fquad is continuous means that f(a)-f(x)quad is infinitely small whenever x-aquad is, and fquad is differentiable means that

is infinitely small whenever x-aquad is. Remarkably, if one allows aquad to be hyperreal, the derivative will be automatically continuous (because, fquad being differentiable at xquad,
is infinitely small when x-aquad is, therefore f'(x)-f'(a)quad is also infinitely small when x-aquad is).

Infinitesimal and infinite numbers

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. x le y implies operatorname{st}(x) le operatorname{st}(y), but x < y does not imply operatorname{st}(x) < operatorname{st}(y).

  • We have, if both x and y are finite,

operatorname{st}(x + y) = operatorname{st}(x) + operatorname{st}(y)
operatorname{st}(x y) = operatorname{st}(x) operatorname{st}(y)

  • If x is finite and not infinitesimal.

operatorname{st}(1/x) = 1 / operatorname{st}(x)

  • x is real if and only if

operatorname{st}(x) = x
The map st is continuous with respect to the order topology on the finite hyperreals, in fact it is locally constant.

Hyperreal fields

Suppose X is a Tychonoff space, also called a T3.5 space, and C(X) is the algebra of continuous real-valued functions on X. Suppose M is a maximal ideal in C(X). Then the factor algebra A = C(X)/M is a totally ordered field F containing the reals. If F strictly contains R then M is called a hyperreal ideal and F a hyperreal field. Note that no assumption is being made that the cardinality of F is greater than R; it can in fact have the same cardinality.

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 Bbb{R}^kappa 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.

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