Hyperreal Number

The hyperreal numbers, denoted by $\ast \mathbb{R}$, are a fascinating extension of the real numbers that elegantly incorporate both infinite and infinitesimal quantities. Think of them as the real numbers, but with a few extra, rather opinionated, guests at the party. A hyperreal number $x$ is considered “finite” if its absolute value $|x|$ is less than some integer $n$. If $|x|$ is smaller than $1/n$ for every positive integer $n$, then $x$ is deemed “infinitesimal.” The term “hyper-real” itself was coined by Edwin Hewitt back in 1948, a man who clearly appreciated a certain flair for the dramatic in mathematics.

What truly sets the hyperreals apart is their adherence to the transfer principle . This isn’t just some vague notion; it’s a rigorous formalization of what Gottfried Wilhelm Leibniz mused about as the law of continuity . In essence, any true statement expressible in first-order logic about the real numbers also holds true for the hyperreal numbers. Take the commutative law of addition, $x + y = y + x$. It’s true for reals, and thanks to the transfer principle, it’s equally true for hyperreals. The hyperreal numbers, much like the reals, form a real closed field . This principle extends even to more complex statements. For instance, if $\sin(\pi n) = 0$ for all integers $n$, then it also holds that $\sin(\pi H) = 0$ for all hyperintegers $H$. The transfer principle, particularly as applied through ultrapowers , is a direct consequence of Łoś’s theorem from 1955.

The suspicion surrounding arguments involving infinitesimals dates back to antiquity. Even Archimedes found it prudent to replace such proofs with more conventional methods, like the method of exhaustion . For centuries, these infinitesimals were treated with a mix of utility and distrust. However, in the 1960s, Abraham Robinson provided a monumental service by proving that the logical consistency of hyperreal numbers was intrinsically tied to that of the real numbers. If the reals were sound, then so were the hyperreals, and any proof employing infinitesimals, when manipulated according to Robinson’s meticulously outlined logical rules, was likewise sound. This effectively banished the lingering fear that proofs involving these “infinitely small” quantities were inherently flawed.

The application of hyperreal numbers, and especially the transfer principle, to the study of analysis is known as nonstandard analysis . One of its most immediate and powerful benefits is the ability to define fundamental concepts like the derivative and integral in a remarkably direct manner, sidestepping the convoluted logical apparatus of multiple quantifiers often required in standard analysis. For example, the derivative of a function $f(x)$ can be expressed as:

$f’(x) = \text{st}\left(\frac{f(x + \Delta x) - f(x)}{\Delta x}\right)$

Here, $\Delta x$ is an infinitesimal, and $\text{st}(\cdot)$ denotes the standard part function , which essentially “rounds off” a finite hyperreal number to its nearest real counterpart. Similarly, the integral can be defined as the standard part of a suitably constructed infinite sum . It’s a way of making the intuitive leaps of calculus concrete and rigorous.

Transfer principle

The core idea behind the hyperreal system is to create an extension of the real numbers, $\mathbb{R}$, which we’ll call $\ast\mathbb{R}$, that embraces infinitesimal and infinite numbers without disrupting the fundamental algebraic axioms of the reals. This means any statement of the form “for any number $x$…” that holds true for the reals must also hold true for the hyperreals. For instance, the axiom stating “for any number $x$, $x + 0 = x$” remains inviolable. This principle extends to statements involving multiple numbers, like “for any numbers $x$ and $y$, $xy = yx$.” This ability to seamlessly carry over statements from the reals to the hyperreals is precisely what the transfer principle is all about. However, it’s crucial to note that this carry-over doesn’t apply to statements involving quantification over sets or other higher-level structures typically built from sets. Only those logical sentences classified as being in first-order logic are guaranteed to obey this restriction. Each real set, function, and relation finds its natural hyperreal extension, preserving the same first-order properties.

Despite this principle, it’s a mistake to assume that $\mathbb{R}$ and $\ast\mathbb{R}$ behave identically. Consider this: in $\ast\mathbb{R}$, there exists an element $\omega$ such that $1 < \omega$, $1+1 < \omega$, $1+1+1 < \omega$, and so on, ad infinitum. There is absolutely no such number in $\mathbb{R}$ – $\ast\mathbb{R}$ is not Archimedean . This is perfectly permissible because the non-existence of such a number $\omega$ cannot be expressed as a first-order statement. The transfer principle operates on what is true, not on what isn’t.

Use in analysis

The historical notations for non-real quantities in calculus, such as $dx$ for infinitesimals and $\infty$ for limits of integration in improper integrals , find their rigorous footing in hyperreal numbers.

Let’s revisit the transfer principle with an example. The statement “for any nonzero number $x$, $2x \neq x$” is unequivocally true for real numbers. Since this is a first-order statement, the transfer principle dictates it must also be true for hyperreal numbers. This implies that a generic symbol like $\infty$ cannot be used to represent all infinite quantities in the hyperreal system. Infinite quantities, much like finite ones, possess distinct magnitudes relative to each other, and similarly for infinitesimals.

Furthermore, the casual assertion $1/0 = \infty$ is fundamentally invalid. The transfer principle protects the statement that zero has no multiplicative inverse. The correct hyperreal counterpart to such a calculation would be: if $\varepsilon$ is a non-zero infinitesimal, then $1/\varepsilon$ is an infinite hyperreal number. The magnitude of $1/\varepsilon$ will depend on the magnitude of $\varepsilon$.

For any finite hyperreal number $x$, its standard part , $\text{st}(x)$, is defined as the unique real number closest to $x$. The difference between $x$ and $\text{st}(x)$ is, by definition, infinitesimal. The standard part function can be extended to infinite hyperreal numbers: if $x$ is a positive infinite hyperreal, $\text{st}(x) = +\infty$; if $x$ is negative infinite, $\text{st}(x) = -\infty$. This captures the idea that an infinite hyperreal number is “infinitely close” to infinity, yet still distinct from any real number.

Differentiation

One of the most compelling applications of the hyperreal number system is providing a precise definition for the differential operator $d$, which Leibniz famously employed to define the derivative and the integral.

For any real-valued function $f$, its differential $df$ is defined as a mapping. For any pair $(x, dx)$, where $x$ is real and $dx$ is a nonzero infinitesimal, the differential $df(x, dx)$ is defined as an infinitesimal:

$df(x,dx) := \text{st}\left(\frac{f(x+dx) - f(x)}{dx}\right) \ dx.$

It’s worth noting that the very notation “$dx$” used for an infinitesimal is consistent with this definition. If we interpret $x$ as the function $f(x) = x$, then for any $(x, dx)$, the differential $d(x)$ will indeed equal the infinitesimal $dx$.

A real-valued function $f$ is considered differentiable at a point $x$ if the quotient

$\frac{df(x,dx)}{dx} = \text{st}\left(\frac{f(x+dx) - f(x)}{dx}\right)$

yields the same value for all nonzero infinitesimals $dx$. If this condition holds, the quotient is defined as the derivative of $f$ at $x$.

Let’s take the function $f(x) = x^2$ as an example. Let $dx$ be a non-zero infinitesimal. Then:

$$ \frac{df(x,dx)}{dx} = \text{st}\left(\frac{f(x+dx) - f(x)}{dx}\right) $$

$$ = \text{st}\left(\frac{(x+dx)^2 - x^2}{dx}\right) $$

$$ = \text{st}\left(\frac{x^2 + 2x \cdot dx + (dx)^2 - x^2}{dx}\right) $$

$$ = \text{st}\left(\frac{2x \cdot dx + (dx)^2}{dx}\right) $$

$$ = \text{st}\left(\frac{2x \cdot dx}{dx} + \frac{(dx)^2}{dx}\right) $$

$$ = \text{st}(2x + dx) $$

$$ = 2x $$

The use of the standard part function here offers a rigorous alternative to the historical practice of simply discarding the $(dx)^2$ term. While $(dx)^2$ is not zero in the hyperreal system (since $dx \neq 0$), it is infinitesimally small compared to $dx$. This reveals a hierarchy of infinitesimal quantities within the hyperreal system. Dual numbers are a number system that arose from this very idea of handling such terms.

Using hyperreal numbers for differentiation offers a more algebraically tractable approach. In standard calculus, partial and higher-order differentials are not always easily manipulated algebraically. The hyperreal system, however, provides a framework for such manipulations, albeit with a slightly modified notation.

Integration

The hyperreal number system also provides a precise interpretation for Leibniz’s integral sign, $\int$, enabling a rigorous definition of the definite integral.

For any infinitesimal function $\varepsilon(x)$, one can define the integral $\int \varepsilon$ as a mapping. For any ordered triple $(a, b, dx)$, where $a$ and $b$ are real numbers and $dx$ is an infinitesimal with the same sign as $b-a$, the integral is assigned the value:

$$ \int_{a}^{b}(\varepsilon, dx) := \text{st}\left(\sum_{n=0}^{N} \varepsilon(a + n \ dx)\right) $$

Here, $N$ is any hyperinteger such that $\text{st}(N \ dx) = b-a$.

A real-valued function $f$ is then defined as integrable over a closed interval $[a, b]$ if, for any non-zero infinitesimal $dx$, the integral

$$ \int_{a}^{b}(f \ dx, dx) $$

remains constant, independent of the choice of $dx$. If this condition holds, the integral is termed the definite integral (or antiderivative) of $f$ on $[a, b]$. This demonstrates how, within the hyperreal framework, Leibniz’s notation for the definite integral transforms into a meaningful algebraic expression, much like the derivative.

Properties

The hyperreals, $\ast\mathbb{R}$, constitute an ordered field that properly contains the reals, $\mathbb{R}$, as a subfield . Unlike the reals, $\ast\mathbb{R}$ doesn’t adhere to the standard definition of a metric space . However, its inherent order allows it to possess an order topology .

The use of the definite article “the” in “the hyperreal numbers” can be slightly misleading, as there isn’t a single, unique ordered field that universally fits the description in most treatments. Nevertheless, a significant development occurred in 2003 with a paper by Vladimir Kanovei and Saharon Shelah . They demonstrated the existence of a definable, countably saturated (meaning $\omega$-saturated but not countable ) elementary extension of the reals. This finding gives strong justification for referring to this specific construction as “the” hyperreal numbers. Furthermore, if one assumes the continuum hypothesis , the field obtained through the ultrapower construction from the space of all real sequences becomes unique up to isomorphism.

The condition of being a hyperreal field is more stringent than being a real closed field that strictly contains $\mathbb{R}$. It also imposes stronger conditions than being a superreal field as defined by Dales and W. Hugh Woodin .

Development

The hyperreals can be approached through axiomatic methods or through more constructive techniques. The axiomatic route essentially posits two key principles: (1) the existence of at least one infinitesimal number, and (2) the unwavering validity of the transfer principle. For a more constructive perspective, one often relies on a set-theoretic construct known as an ultrafilter . While this method allows for the construction of the hyperreals, the ultrafilter itself remains elusive – it cannot be explicitly constructed.

From Leibniz to Robinson

When Isaac Newton and, more explicitly, Gottfried Wilhelm Leibniz introduced differentials, they embraced infinitesimals. These concepts remained useful for mathematicians like Leonhard Euler and Augustin-Louis Cauchy . However, they were also met with skepticism from the outset, most notably from George Berkeley . Berkeley’s critique famously targeted the perceived logical inconsistency in the definition of the derivative, where $dx$ was treated as both non-zero (at the beginning of a calculation) and zero (at its conclusion). This is elaborated in his work, “The Analyst” (see Ghosts of departed quantities ). The advent of the (ε, δ)-definition of limit in the 19th century by mathematicians like Bernard Bolzano , Cauchy, and Karl Weierstrass provided a rigorous foundation for calculus, largely displacing infinitesimals. Yet, research into non-Archimedean fields persisted.

The breakthrough came in the 1960s when Abraham Robinson provided a rigorous framework for dealing with infinitely large and infinitesimal numbers, thereby establishing the field of nonstandard analysis . Robinson’s initial work was nonconstructive, relying on model theory . However, it is possible to develop hyperreals using purely algebraic and topological methods, proving the transfer principle as a direct consequence of the definitions. This means that hyperreal numbers, in their essence, are not intrinsically tied to model theory or first-order logic, even though their discovery was facilitated by model-theoretic techniques. In fact, Edwin Hewitt had introduced hyperreal fields in 1948 using purely algebraic methods, specifically through an ultrapower construction.

Ultrapower construction

A common and constructive way to build the hyperreal field involves sequences of real numbers. We can define addition and multiplication of 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 similarly for multiplication. This process transforms the set of all such sequences into a commutative ring , which is also a real algebra . We can embed the real numbers $\mathbb{R}$ into this ring by identifying each real number $r$ with the constant sequence $(r, r, r, \ldots)$. This embedding preserves the algebraic operations of the reals. The underlying intuition is to represent infinitesimals by sequences that approach zero and infinities by sequences that diverge. However, the devil is in the details of how we compare these sequences. For instance, two sequences might differ in their initial terms but become identical later on; these should intuitively represent the same hyperreal number. The challenge lies in establishing consistent rules for comparing sequences, especially those that seem to “oscillate randomly” forever, ensuring they can be interpreted meaningfully, perhaps as $7 + \epsilon$, where $\epsilon$ is a specific infinitesimal.

To address this, we need a robust method for comparing sequences. A naive componentwise comparison, $(a_0, a_1, a_2, \ldots) \leq (b_0, b_1, b_2, \ldots) \iff (a_0 \leq b_0) \wedge (a_1 \leq b_1) \wedge (a_2 \leq b_2) \ldots$, falters because some components of the first sequence might be larger than their counterparts in the second, while others are smaller. This results in only a partial order . To overcome this, we must define which sets of indices “matter.” Since there are infinitely many indices, finite sets shouldn’t dictate the comparison. This is where free ultrafilters on the natural numbers come into play. An ultrafilter $U$ is a collection of subsets of $\mathbb{N}$ such that:

1. Any set containing a set from $U$ is also in $U$.
2. The intersection of any two sets in $U$ is also in $U$.
3. The empty set is not in $U$.
4. Either a set or its complement must be in $U$. (This is the “ultra” property that distinguishes ultrafilters from mere filters).

We define $(a_0, a_1, a_2, \ldots) \leq (b_0, b_1, b_2, \ldots)$ if and only if the set of natural numbers ${n : a_n \leq b_n}$ belongs to the ultrafilter $U$. This establishes a total preorder . By identifying sequences $a$ and $b$ if $a \leq b$ and $b \leq a$, we obtain a total order and construct the ordered field $\ast\mathbb{R}$.

Algebraically, the ultrafilter $U$ corresponds to a maximal ideal $I$ in the ring of sequences. The hyperreal field $\ast\mathbb{R}$ is then constructed as the quotient ring $A/I$. The maximality of $I$ ensures that $A/I$ is indeed a field. This construction is often denoted as $A/U$, directly referencing the ultrafilter. The existence of non-trivial ultrafilters (those not containing any finite sets) is guaranteed by Zorn’s lemma , though they remain unconstructible.

The field $A/U$ is an ultrapower of $\mathbb{R}$. Its cardinality is at least that of the continuum . Since the ring $A$ has cardinality $(2^{\aleph_0})^{\aleph_0} = 2^{\aleph_0}$, the hyperreal field also has the same cardinality as $\mathbb{R}$.

A crucial question arises: if we choose a different free ultrafilter $V$, is the resulting quotient field $A/U$ isomorphic as an ordered field to $A/V$? This question is intimately linked to the continuum hypothesis . Under ZFC with the continuum hypothesis, we can prove this field is unique up to order isomorphism . Conversely, without the continuum hypothesis, there exist pairs of non-order-isomorphic fields constructed this way.

For a deeper dive into this construction, consult the concept of ultraproduct .

An intuitive approach to the ultrapower construction

To grasp the hyperreal numbers more intuitively, consider sequences of real numbers that converge to zero. These are often termed “infinitely small.” They are precursors to the true infinitesimals, which are classes of sequences containing such converging sequences.

Imagine our ring of sequences. We can add, subtract, and multiply them. Division is trickier. Real numbers are represented by constant sequences. The issue is that $ab = 0$ can occur even if neither $a$ nor $b$ is the zero sequence. To address this, we need a consistent way to declare certain sequences “zero.” When this is done correctly, the resulting equivalence classes of sequences form a field – the hyperreal field. This field contains infinitesimals, infinitely large numbers (reciprocals of infinitesimals), and importantly, any finite hyperreal number can be expressed as the sum of an ordinary real number and an infinitesimal.

This construction mirrors Cantor’s method of constructing the reals from Cauchy sequences of rationals. He declared sequences converging to zero as the “zero” element. To build hyperreals, we focus on the “zero set” of a sequence $a$, denoted $z(a) = {i : a_i = 0}$, the set of indices where the sequence is zero. If $ab = 0$, then the union of $z(a)$ and $z(b)$ must be the set of all natural numbers $\mathbb{N}$. This leads to the following conditions for declaring a sequence “zero”:

• If two complementary sets cover $\mathbb{N}$, one of them must contain a zero set.
• If $a$ is declared zero, then $ab$ must also be declared zero for any $b$.
• If both $a$ and $b$ are declared zero, then $a+b$ must be declared zero.

We achieve this by choosing a collection $U$ of subsets of $\mathbb{N}$. A sequence $a$ is declared zero if and only if $z(a) \in U$. For this to work, $U$ must satisfy properties analogous to filters and ultrafilters:

• $U$ must contain all supersets of its elements.
• The intersection of any two sets in $U$ must be in $U$.
• $U$ cannot contain the empty set .
• For any set $S \subseteq \mathbb{N}$, either $S \in U$ or its complement $\mathbb{N} \setminus S \in U$.

A family of sets satisfying these properties is an ultrafilter . If $U$ is a non-trivial ultrafilter (meaning it doesn’t contain any finite sets), and we apply our construction, we obtain the hyperreal numbers. The existence of such non-trivial ultrafilters, while not explicitly constructible, is guaranteed by the axiom of choice .

When a real function $f(x)$ is extended to hyperreals, it is done via composition: $f({x_n}) = {f(x_n)}$, where ${\dots}$ denotes the equivalence class of the sequence relative to the chosen ultrafilter. All arithmetic expressions involving hyperreals hold true if they hold true for the reals. Crucially, any finite hyperreal $x$ can be expressed as $y+d$, where $y$ is a standard real number and $d$ is an infinitesimal. The property that for any set $S$, either $S \in U$ or its complement is in $U$ is fundamental to proving this.

Properties of infinitesimal and infinite numbers

The finite elements within $\ast\mathbb{R}$ form a local ring , specifically a valuation ring . The unique maximal ideal within this ring consists of the infinitesimals. The quotient of this ring by the ideal of infinitesimals is isomorphic to the real numbers. This gives us a homomorphic mapping, $\text{st}(x)$, from the finite elements to the reals. The kernel of this map is precisely the set of infinitesimals. This function $\text{st}(x)$ assigns to each finite hyperreal $x$ a unique real number such that their difference, $x - \text{st}(x)$, is infinitesimal. In essence, $\text{st}(x)$ is the closest real number to $x$. This standard part function is order-preserving, although not strictly isotonic: $x \leq y$ implies $\text{st}(x) \leq \text{st}(y)$, but $x < y$ does not necessarily imply $\text{st}(x) < \text{st}(y)$.

Key properties of the standard part function for finite $x$ and $y$:

• $\text{st}(x+y) = \text{st}(x) + \text{st}(y)$
• $\text{st}(xy) = \text{st}(x)\text{st}(y)$

If $x$ is finite and non-infinitesimal:

• $\text{st}(1/x) = 1/\text{st}(x)$

Furthermore, $x$ is a real number if and only if $\text{st}(x) = x$. The standard part map is continuous with respect to the order topology on the finite hyperreals; it’s even locally constant .

Hyperreal fields

Consider a Tychonoff space $X$, and let $C(X)$ be the algebra of continuous real-valued functions on $X$. If $M$ is a maximal ideal in $C(X)$, then the factor algebra $A = C(X)/M$ forms a totally ordered field $F$ containing $\mathbb{R}$. If $F$ strictly contains $\mathbb{R}$, then $M$ is termed a hyperreal ideal (a term introduced by Edwin Hewitt in 1948), and $F$ is called a hyperreal field. Notably, this definition doesn’t impose any constraints on the cardinality of $F$; it can, in fact, be the same as that of $\mathbb{R}$.

A particularly significant instance arises when $X$ has the discrete topology . In this scenario, $X$ can be identified with a cardinal number $\kappa$, and $C(X)$ becomes equivalent to $\mathbb{R}^{\kappa}$, the algebra of functions from $\kappa$ to $\mathbb{R}$. The hyperreal fields obtained in this specific context are known as ultrapowers of $\mathbb{R}$ and are identical to those constructed via free ultrafilters in model theory.

This rewrite aims to expand upon the original Wikipedia article, injecting a certain analytical tone and clarifying the concepts with additional descriptive language. The internal links have been preserved precisely as requested, maintaining the integrity of the original structure while elaborating on the mathematical ideas.