Section 2.8: Nonstandard Analysis

In 1961 Abraham Robinson introduced a new method for treating limits... This method combines the intuitive advantages of working with infinitely small quantities with modern standards of rigor. The basic idea is to utilize a nonstandard model of the theory of the real numbers.

Construction of *𝕽

Let the language include the following parameters.

$\forall$ .
$P_{R}$ , an $n$ -place predicate symbol, for every $n$ -ary relation $R$ on $\mathbb{R}$ .
$f_{F}$ , an $n$ -place function symbol, for every $n$ -ary operation $F$ on $\mathbb{R}$ .
$c_{r}$ , a constant symbol, for every $r\in\mathbb{R}$ .

$\mathfrak{R}$ is a structure in this language, where $|\mathfrak{R}|=\mathbb{R}$ , $P_{R}^{\mathfrak{R}}=R$ , $f_{F}^{\mathfrak{R}}=F$ , and $c_{r}^{\mathfrak{R}}=r$ .

Let $\Gamma=\mbox{Th}\mathfrak{R}\cup\{P_{<}c_{r}v_{1}\}_{r\in\mathbb{R}}$ . By the compactness theorem, there is a structure $\mathfrak{A}$ and $a\in|\mathfrak{A}|$ such that $\Gamma$ is satisfied in $\mathfrak{A}$ when $v_{1}$ is assigned $a$ .

$\mathfrak{A}\equiv\mathfrak{R}$ .
$h(r)=c_{r}^{\mathfrak{A}}$ is an isomorphism of $\mathfrak{R}$ into $\mathfrak{A}$ .

Given the isomorphism $h$ , we can find a structure $^{*}\mathfrak{R}$ such that a) $\mathfrak{R}$ is a substructure of $^{*}\mathfrak{R}$ , b) $^{*}\mathfrak{R}\cong\mathfrak{A}$ , in particular, $^{*}\mathfrak{R}\equiv\mathfrak{R}$ , and c) there is $b\in|{}^{*}\mathfrak{R}|$ such that $^{*}\mathfrak{R}$ satisfies $\Gamma$ when $v_{1}$ is assigned $b$ .

Asterisk Notation

$^{*}R=P_{R}^{^{*}\mathfrak{R}}$ , $^{*}F=f_{F}^{^{*}\mathfrak{R}}$ .

$c_{r}^{^{*}\mathfrak{R}}=r$ , so no special notation needed.
$R$ is the restriction of $^{*}R$ to $\mathbb{R}$ . $F$ is the restriction of $^{*}F$ to $\mathbb{R}$ .

Properties

If a property of $R$ or $F$ can be expressed by a sentence of the language, then $^{*}R$ or $^{*}F$ , correspondingly, has the property.

$^{*}<$ is an ordering relation.
$(^{*}\mathbb{R};0,1,^{*}+,^{*}\cdot)$ is a field.

The set $\mathcal{F}$ of finite elements of $^{*}\mathbb{R}$ is $\{x\in^{*}\mathbb{R}|^{*}|x|^{*}<y\mbox{ for some }y\in\mathbb{R}\}$ .

The set $\mathcal{I}$ of infinitesimals of $^{*}\mathbb{R}$ is $\{x\in^{*}\mathbb{R}|^{*}|x|^{*}<y\mbox{ for all positive }y\in\mathbb{R}\}$ .

$\mathcal{F}$ and $\mathcal{I}$ are closed under $^{*}+$ , $^{*}-$ , as well as multiplication $^{*}\cdot$ by a number from $\mathcal{F}$ .

$x$ is infinitely close to $y$ , $x\simeq y$ , iff $x^{*}-y$ is infinitesimal.

$\simeq$ is an equivalence relation on $^{*}\mathbb{R}$ .
If $u\simeq v$ and $x\simeq y$ , then $u^{*}\pm x\simeq v^{*}\pm y$ . If $u,v,x,y$ are finite, then $u^{*}\cdot x\simeq v^{*}\cdot y$ .
If $x<y$ and at least one of them is finite, then there is $q\in\mathbb{R}$ such that $x<q<y$ .
For every $x\in\mathcal{F}$ there is unique $y\in\mathbb{R}$ such that $x\simeq y$ . If $x,y\in\mathbb{R}$ , then $x\simeq y$ iff $x=y$ .
- For $x\in\mathcal{F}$ there is a unique decomposition $x=^{\circ}x{}^{*}+i$ into the standard part of $x$ , $\mbox{st}(x)=^{\circ}x\in\mathbb{R}$ , and an infinitesimal $i\in\mathcal{I}$ .
- $\mbox{st}$ is a homomorphism of the ring $\mathcal{F}$ onto $\mathbb{R}$ with kernel $\mathcal{I}$ , hence, $\mathcal{F}/\mathcal{I}$ is isomorphic to $\mathbb{R}$ .

Applications

Subsets and relations

$^{*}\mathbb{Q}$ is dense in $^{*}\mathbb{R}$ : for every $x\in{}^{*}\mathbb{R}$ there is $q\in{}^{*}\mathbb{Q}$ such that $q\simeq x$ .
- $^{*}\mathbb{Q}$ and $^{*}\mathbb{N}$ both have cardinality at least $2^{\aleph_{0}}$ .
$^{*}A-A\cap\mathbb{R}=\emptyset$ . $^{*}A=A$ iff $A$ is finite.
Bolzano–Weierstrass Theorem. If $A\subset\mathbb{R}$ is bounded and infinite, then there is $p\in\mathbb{R}$ that is infinitely close to, but different from a point in $^{*}A$ .
A least upper bound property does not hold for $^{*}\mathbb{R}$ .
If $F:A\rightarrow B$ , then (considered as a relation on $\mathbb{R}$ ) $^{*}F:{}^{*}A\rightarrow{}^{*}B$ .
- If $F$ is one-to-one, then for every $x\in{}^{*}A-A$ , $^{*}F(x)\notin\mathbb{R}$ .

Convergence

A function $F:\mathbb{R}\rightarrow\mathbb{R}$ converges at $a$ to $b$ iff $x\simeq a$ , $x\neq a$ implies $^{*}F(x)\simeq b$ . In particular, $\lim_{x\rightarrow a}F(x)=\mbox{st}({}^{*}F(a{}^{*}+i))$ for any $i\in\mathcal{I}-\{0\}$ . $F$ is continuous at $a$ iff $x\simeq a$ implies $^{*}F(x)=F(a)$ .

Derivative. If $F'(a)$ exists, then for every $dx\in\mathcal{I}-\{0\}$ , $F'(a)=\mbox{st}(dF{}^{*}/dx)$ .
- If $F'(a)$ exists, then $F$ is continuous at $a$ .
- If $F'(a)$ and $G'(F(a))$ exist, then $(G\circ F)'(a)$ exists and $(G\circ F)'(a)=G'(F(a))\cdot F'(a)$ .
Convergence of sequences. $S:\mathbb{N}\rightarrow\mathbb{R}$ converges to $b$ iff $^{*}S(x)\simeq b$ for every infinite $x\in{}^{*}\mathbb{N}$ .

Subpages

Section 2.8: Nonstandard Analysis

Section 2.8: Nonstandard Analysis

Construction of *𝕽

Asterisk Notation

Properties

Applications

Subsets and relations

Convergence