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Hyperreal number

From Academic Kids

In mathematics, particularly in non-standard analysis and mathematical logic, hyperreal numbers 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 one that is larger than all numbers representable in the form

<math>1 + 1 + \cdots + 1.<math>

The use of the definite article the in the phrase the hyperreal numbers 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. 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 epsilon-delta definition of a 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 the use of them in nonstandard analysis, have no necessary relationship to model theory or first order logic.

Contents

The transfer principle

Historically, the concept of number has been repeatedly generalized. At each step in this process of generalization, mathematicians knew that they wished to retain as many properties as possible from the earlier concepts of numbers. However, some properties always had to be given up. In the case of the hyperreals, a long historical delay in their development was caused by uncertainty among mathematicians as to exactly which properties could be retained, and which would have to be given up. 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:

<math> \forall x \in \mathbb{R} \quad \exists y \in\mathbb{R}\quad x < y <math>

The same will then also hold for hyperreals:

<math> \forall x \in \star \mathbb{R} \quad \exists y \in\star \mathbb{R}\quad x < y <math>

Another example is the statement that if you add 1 to a number you get a bigger number:

<math> \forall x \in \mathbb{R} \quad x < x+1 <math>

which will also hold for hyperreals:

<math> \forall x \in \star \mathbb{R} \quad x < x+1 <math>

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

<math> 1

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, in the paper linked to at the end of this article, 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,

<math> (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) <math>

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, <math>7+\epsilon<math>, where <math>\epsilon<math> 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:

<math> (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 <math>

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 a 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 <math>\Bbb{R}<math>. Since this field contains R it has cardinality at least the continuum. Since A has cardinality

<math>(2^{\aleph_0})^{\aleph_0} = 2^{\aleph_0^2} =2^{\aleph_0},\,<math>

it is also no larger than <math>2^{\aleph_0}<math>, and hence has the same cardinality as R. As a real closed field with cardinality the continuum, it is isomorphic as a field to R but is not isomorphic as an ordered field to R. Thus in some sense of "larger" we do not need to go to a larger field to do nonstandard analysis.

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, check out ultraproducts and ultrapowers.

Infinitesimal and infinite numbers

A hyperreal number r is called infinitesimal if it is smaller than every positive real number and bigger 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 of F of *R form a local 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. This operation is an order-preserving homomorphism and hence well-behaved both algebraically and order theoretically. However, it is order-preserving but not isotonic, which means <math> x \le y<math> implies <math>\operatorname{st}(x) \le \operatorname{st}(y)<math>, but it is not the case that <math>x < y <math> implies <math>\operatorname{st}(x) < \operatorname{st}(y)<math>

  • We have, if both x and y are finite,
<math> \operatorname{st}(x + y) = \operatorname{st}(x) + \operatorname{st}(y) <math>
<math> \operatorname{st}(x y) = \operatorname{st}(x) \operatorname{st}(y) <math>
  • If x is finite and not infinitesimal.
<math> \operatorname{st}(1/x) = 1 / \operatorname{st}(x) <math>
  • x is real if and only if
<math> \operatorname{st}(x) = x <math>

The map st is locally constant, which entails that its derivative is identically zero and that it is continuous with respect to the order topology on the finite hyperreals.

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 have the cardinality of the continuum, in which case F is isomorphic as a field to R, but is not order isomorphic to R.

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 <math>\Bbb{R}^\kappa<math> 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. We give a particular example, commonly used in nonstandard analysis, below.

Compare with:

References

  • H. Garth Dales and W. Hugh Woodin: Super-Real Fields, Clarendon Press, 1996.
  • L. Gillman and M. Jerison: Rings of Continuous Functions, Van Nostrand, 1960.
  • Robert Goldblatt, Lectures on the hyperreals : an introduction to nonstandard analysis, Springer, 1998.
  • Abraham Robinson: Nonstandard Analysis, Princeton University Press 1966. The classic introduction to nonstandard analysis.

External links

  • Vladimir Kanovei and Saharon Shelah: A definable nonstandard model of the reals, Journal of Symbolic Logic 69 (2004) pp. 159-164, available at http://shelah.logic.at/files/825.pdf

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