Nonstandard Analysis

Leibniz and Newton built the calculus on quantities infinitely small yet nonzero. Newton's $o$ could be multiplied by any finite number and stay negligible, but had to be nonzero so one could divide by it; Leibniz's $\mathbf{d}x$ was smaller than any assignable quantity, again nonzero. The idea drew attacks through the eighteenth century, and the nineteenth century replaced it with the $\epsilon$-$\delta$ treatment of limits now standard. In 1961 Abraham Robinson recovered infinitesimals rigorously by working inside a nonstandard model of the theory of the reals. The tools are compactness, to build the model, and elementary equivalence, to transfer truths into it.

Building the hyperreal field

The construction uses a deliberately enormous first-order language: one that can name every feature of $\mathbb{R}$. Its parameters are

• $\forall$, ranging over the reals;
• an $n$-place predicate symbol $P_R$ for every $n$-ary relation $R$ on $\mathbb{R}$;
• a constant symbol $c_r$ for every real $r$;
• an $n$-place function symbol $f_F$ for every $n$-ary operation $F$ on $\mathbb{R}$.

The standard structure $\mathfrak{R}$ interprets each symbol by the object it names. A nonstandard structure comes from compactness. Let

where $c_r{P}_{<}{v}_1$ formalizes $r$ is less than $v_1$. Any finite subset of $\Gamma$ names finitely many reals below $v_1$, and is satisfied in $\mathfrak{R}$ by assigning $v_1$ a large enough real. By compactness $\Gamma$ has a model $\mathfrak{A}$ with an element $a$ satisfying every $c_r{P}_{<}{v}_1$. Since $\mathfrak{A}\models \text{Th }\mathfrak{R}$, we have $\mathfrak{A}\equiv \mathfrak{R}$.

The map $h(r)=c_r^{\mathfrak{A}}$ embeds $\mathbb{R}$ into $\mathfrak{A}$. It is injective and preserves every relation and operation, because each such preservation is a sentence true in $\mathfrak{R}$ and hence in the elementarily equivalent $\mathfrak{A}$. Replacing the image points by the reals themselves yields a structure ${}^{\ast }\mathbb{R}$ containing $\mathbb{R}$ as a substructure, with an element $b$ larger than every real.¹

Notation follows the reals with an asterisk: ${}^{\ast }R$ is the relation, and ${}^{\ast }F$ the operation, that ${}^{\ast }\mathbb{R}$ assigns to $P_R$ and $f_F$. Each standard relation is the restriction of its starred version to $\mathbb{R}$.

The transfer principle

The value of $\mathfrak{A}\equiv \mathfrak{R}$ is a general method for proving facts about the extension. To show that a starred relation or operation has a property, observe that (1) the standard relation or operation has it, (2) the property is expressible by a sentence of the language, and (3) $\mathfrak{R}\equiv {}^{\ast }\mathbb{R}$. The property then transfers.

Transfer proves at once that ${}^{\ast }<$ is a linear order, that ${}^{\ast }+$ is commutative, and that $({}^{\ast }\mathbb{R};0,1,{}^{\ast }+,{}^{\ast }\cdot )$ is a field, since each field axiom is a single sentence true in $\mathbb{R}$. Properties not expressible by a sentence can fail. The least-upper-bound property is one: $\mathbb{R}$ is a bounded subset of ${}^{\ast }\mathbb{R}$, bounded by any infinite $b$, yet has no least upper bound there. The least-upper-bound property quantifies over subsets, which first-order sentences cannot reach.

Finite elements and infinitesimals

The extension stratifies by magnitude. Write ${}^{\ast }|x|$ for the starred absolute value.

• Finite elements. $\mathbb{F}=\{x\in {}^{\ast }\mathbb{R}\mid {}^{\ast }|x|<y\text{ for some }y\in \mathbb{R}\}$: bounded by a standard real.
• Infinitesimals. $\mathbb{I}=\{x\in {}^{\ast }\mathbb{R}\mid {}^{\ast }|x|<y\text{ for every positive }y\in \mathbb{R}\}$: below every positive standard real.

The reciprocal of the infinite $b$ is a nonzero infinitesimal, so $\mathbb{I}$ contains more than $0$. The only standard infinitesimal is $0$ itself. Unbounded standard sets acquire infinite points: ${}^{\ast }\mathbb{N}$ contains infinite natural numbers, since for every real there is a larger member of $\mathbb{N}$ transfers.

In algebraic language $\mathbb{F}$ is a subring of ${}^{\ast }\mathbb{R}$ and $\mathbb{I}$ is an ideal of $\mathbb{F}$. The proofs are direct bounds: if ${}^{\ast }|x|<a$ and ${}^{\ast }|y|<b$ for standard $a,b$, then ${}^{\ast }|x\cdot y|<a\cdot b$, a standard real, so the product is finite; and if $x$ is infinitesimal while $z$ is finite with ${}^{\ast }|z|<b$, then for any positive standard $a$ one has ${}^{\ast }|x|<a/b$, whence ${}^{\ast }|x\cdot z|<a$.

The standard part

Infinitesimal closeness organizes $\mathbb{F}$ into a copy of $\mathbb{R}$.

By the arithmetic of finite and infinitesimal elements this is an equivalence relation, compatible with addition and (on finite elements) multiplication. Distinct standard reals are never infinitely close, since $0$ is the only standard infinitesimal. Every finite hyperreal is infinitely close to exactly one real.

So $\mathrm{st}$ is a ring homomorphism from $\mathbb{F}$ onto $\mathbb{R}$ with kernel $\mathbb{I}$, and the quotient $\mathbb{F}/\mathbb{I}$ is isomorphic to the real field.² From here the asterisks on the arithmetic operations are dropped.

Convergence and the derivative

The infinitesimal apparatus rephrases limits without $\epsilon$-$\delta$ quantifiers. Instead of variables approaching a value, variables land infinitely close to it.

This matches the classical definition. If $F$ converges to $b$ in the ordinary sense, then for each standard $\epsilon >0$ the sentence guaranteeing a $\delta$ transfers to ${}^{\ast }\mathbb{R}$, and an $x$ infinitely close to $a$ satisfies ${}^{\ast }|x-a|<\delta$, forcing ${}^{\ast }|b-{}^{\ast }F(x)|<\epsilon$ for every standard $\epsilon$, so ${}^{\ast }F(x)\simeq b$. Conversely, if the infinitesimal condition holds, then for each standard $\epsilon >0$ a suitable $\delta$ exists in ${}^{\ast }\mathbb{R}$ (take it infinitesimal), and the sentence transfers back to $\mathbb{R}$. The limit is then the standard part of a single evaluation:

The derivative becomes an ordinary quotient. Writing $\mathbf{d}F={}^{\ast }F(a+\mathbf{d}x)-F(a)$ for a nonzero infinitesimal $\mathbf{d}x$,

and the division is genuine division in the field, not a limit of ratios.

The classical theorems follow with nonstandard proofs. Differentiability at $a$ implies continuity at $a$: if $\mathbf{d}F/\mathbf{d}x\simeq F^{\prime }(a)$ is finite, multiplying by the infinitesimal $\mathbf{d}x$ shows $\mathbf{d}F$ is infinitesimal, so ${}^{\ast }F(a+\mathbf{d}x)\simeq F(a)$. The chain rule reads off as a product of quotients:

where $\mathbf{d}G={}^{\ast }G(a+\mathbf{d}x)-G(a)$, handling the case $\mathbf{d}G=0$ separately. These are classical theorems with nonstandard proofs, not analogues of them; the method has produced original results in analysis, including in the theory of Hilbert spaces.³

Three tools combine in the construction: compactness supplies the infinite element, elementary equivalence supplies the transfer principle carrying every first-order truth, and the arithmetic of infinitesimals does the rest. What separates ${}^{\ast }\mathbb{R}$ from $\mathbb{R}$ is precisely what first-order logic cannot express: the least-upper-bound property and the distinction between finite and infinite elements.