Internal set theory

The internal set theory (Engl. Internal Set Theory (IST) ) is a syntactic version of the nonstandard analysis , which in 1977 by Edward Nelson was introduced. In contrast to the model theoretical approach, infinitesimals are not constructed with the help of a non- Archimedean field extension, but are defined by an extension of set theory within the real numbers.

Language and axioms

In addition to the set-theoretical elementity , a predicate (for standard ) is introduced, which is described in the following by three axiom schemes. Formulas that do not contain are called internal formulas; those that contain are called external formulas. The following quantifiers are defined as an abbreviation: ${\displaystyle \in }$ ${\displaystyle \mathrm {st} }$ ${\displaystyle \mathrm {st} }$ ${\displaystyle \mathrm {st} }$

• ${\displaystyle \forall ^{\mathrm {st} }x:\Phi }$for (standard applies to all ...) ${\displaystyle \forall x:\operatorname {st} (x)\Rightarrow \Phi }$ ${\displaystyle x}$
• ${\displaystyle \exists ^{\mathrm {st} }x:\Phi }$for (there is (at least) one standard so that ...) ${\displaystyle \exists x:\operatorname {st} (x)\land \Phi }$ ${\displaystyle x}$
• ${\displaystyle \forall ^{\mathrm {fin} }x:\Phi }$for (for all finite sets ...) ${\displaystyle \forall x:\operatorname {fin} (x)\Rightarrow \Phi }$ ${\displaystyle x}$
• ${\displaystyle \exists ^{\mathrm {fin} }x:\Phi }$for (there is (at least) a finite set such that ...) ${\displaystyle \exists x:\operatorname {fin} (x)\land \Phi }$ ${\displaystyle x}$

As well as combinations of these abbreviations such as or , the formal definition of which can be given similarly. ${\displaystyle \forall ^{\mathrm {st\,fin} }x:\Phi }$ ${\displaystyle \forall ^{\mathrm {st} }x\in A:\Phi }$

In addition to the Zermelo-Fraenkel set theory with the axiom of choice (whereby the axiom schemes may only use formulas that do not appear), three further axiom schemes are used: ${\displaystyle \mathrm {st} }$

The transfer axiom

For every internal formula (with free variables) in which the predicate does not occur, the following applies: ${\displaystyle \Phi }$ ${\displaystyle n+1}$ ${\displaystyle \mathrm {st} }$

${\displaystyle \forall ^{\mathrm {st} }t_{1},...,t_{n}(\forall ^{\mathrm {st} }x:\Phi (x,t_{1},...,t_{n})\Leftrightarrow \forall x:\Phi (x,t_{1},...,t_{n}))}$

The reformulation

${\displaystyle \forall ^{\mathrm {st} }t_{1},...,t_{n}(\exists ^{\mathrm {st} }x:\Phi (x,t_{1},...,t_{n})\Leftrightarrow \exists x:\Phi (x,t_{1},...,t_{n}))}$

shows that any set whose existence and uniqueness can be proven in classical theory is a standard set.

The axiom of idealization

For every internal formula in which the variable is not free and the predicate does not occur, the following applies: ${\displaystyle \Phi }$ ${\displaystyle z}$ ${\displaystyle \mathrm {st} }$

${\displaystyle (\forall ^{\mathrm {st\,fin} }z\exists x\forall y\in z:\Phi )\iff (\exists x\forall ^{\mathrm {st} }y:\Phi )}$

The idealization axiom provides two important conclusions:

1. A set is standard and finite if and only if all of its elements are finite.
2. There is a finite set that contains all standard sets.

The second statement in particular takes some getting used to: There is a finite set that (according to the transfer axiom) contains all sets that can be constructed in classical set theory. However, this finite set is not standard, since otherwise it contains all elements according to the transfer axiom. However, the term "finally" itself is not standard, or as Nelson himself says: "'finally' does not mean what we always thought."

Although this definition takes getting used to, it is the key to non-standard analysis: we can infer that there are real numbers that are greater than 0 but less than any positive standard number.

The axiom of standardization

For every (internal or external) formula (in which the variable does not appear), the following applies: ${\displaystyle \Phi }$ ${\displaystyle y}$

${\displaystyle \forall ^{\mathrm {st} }x\exists ^{\mathrm {st} }y\forall ^{\mathrm {st} }z(z\in y\iff (z\in x\land \Phi (x)))}$

The axiom of standardization allows (as the only axiom) the construction of sets with the help of formulas that use the predicate . However, a set constructed in this way can contain non-standard elements that do not comply. ${\displaystyle \mathrm {st} }$ ${\displaystyle \Phi }$

Examples

Three classic examples from calculus are intended to show how various approaches can be justified in the Internal Set Theory that would not be formulated without the additional axioms. In contrast to other approaches of non-standard analysis, such arguments can be formulated without a body extension and without difficult logical preparation.

A real number is infinitely small or infinitesimal number, if for every real number standard applies: . In more recent publications one also reads the term i-small to avoid the historical but possibly misleading term "infinite". You still write when the difference is infinitesimal. ${\displaystyle x}$ ${\displaystyle r>0}$ ${\displaystyle |x|<r}$ ${\displaystyle x\approx y}$ ${\displaystyle x-y}$

continuity

Using these infinitesimal continuity for example, can be characterized: a standard feature in a point is continuous if for all applies: . The function is continuous if and only if it is continuous in all standard points and if and only if it is continuous in all points. ${\displaystyle f\colon \mathbb {R} \to \mathbb {R} }$ ${\displaystyle x\in \mathbb {R} }$ ${\displaystyle y\approx x}$ ${\displaystyle f(y)\approx f(x)}$

In contrast to the “ - definition” (with the help of limit values), this definition is a little clearer: If the argument is changed just a little bit, then the picture changes just a little bit. ${\displaystyle \epsilon }$ ${\displaystyle \delta }$

For example, the function is continuous because be standard and so is ${\displaystyle f(x)=x^{2}}$ ${\displaystyle x_{0}}$ ${\displaystyle \epsilon \approx 0,\epsilon \neq 0}$

${\displaystyle f(x_{0}+\epsilon )=x_{0}^{2}+2x_{0}\epsilon +\epsilon ^{2}\approx x_{0}^{2}=f(x)}$

However, it is not uniformly continuous because it is not continuous at about the point : ${\displaystyle f}$ ${\displaystyle \epsilon ^{-1}\approx \infty }$

${\displaystyle f(\epsilon ^{-1}+\epsilon )=\epsilon ^{-2}+2\epsilon ^{-1}\epsilon +\epsilon ^{2}\approx \epsilon ^{-2}+2\not \approx f(\epsilon ^{-1})}$

Differentiation

The derivative of a function is generally defined as usual. For standard functions, however, there is an equivalent formulation: The derivative of a (real) standard function is a standard function that assigns a standard number to each standard point (in which it is differentiable), so that applies to all : ${\displaystyle f}$ ${\displaystyle f'}$ ${\displaystyle x}$ ${\displaystyle f}$ ${\displaystyle \epsilon \approx 0,\epsilon \neq 0}$

${\displaystyle f'(x)\approx {\frac {f(x+\epsilon )-f(x)}{\epsilon }}}$

This formulation can help to find the derivative with the help of the transfer axiom. For example, what is the derivative of ? ${\displaystyle f(x)=x^{2}}$

The function is standard. Assume is some standard number. Then applies to everyone ${\displaystyle x_{0}}$ ${\displaystyle \epsilon \approx 0,\epsilon \neq 0}$

${\displaystyle f'(x_{0})\approx {\frac {(x_{0}+\epsilon )^{2}-x_{0}^{2}}{\epsilon }}={\frac {x_{0}^{2}+2x_{0}\epsilon +\epsilon ^{2}-x_{0}^{2}}{\epsilon }}=2x_{0}+\epsilon \approx 2x_{0}}$

So is for all standard values , and with the transfer axiom this must apply to all . ${\displaystyle f'(x)=2x}$ ${\displaystyle x}$

integration

If there is a standard set, an integrable standard function, and a finite set that contains all standard numbers in , then is , where is the distance from from to the next larger point from . ${\displaystyle D\subseteq \mathbb {R} }$ ${\displaystyle f\colon D\rightarrow \mathbb {R} }$ ${\displaystyle F}$ ${\displaystyle D}$ ${\displaystyle \int _{D}f(x)dx\approx \sum _{F}f(x)\Delta x}$ ${\displaystyle \Delta x}$ ${\displaystyle x}$ ${\displaystyle F}$

This allows the substitution rule for the integral to be derived quite simply and clearly : If this sum is to be replaced by (where is a suitable standard function), it must also be replaced by a suitable one. ${\displaystyle x}$ ${\displaystyle g(y)}$ ${\displaystyle g}$ ${\displaystyle \Delta x}$ ${\displaystyle \Delta y}$

But if differentiable, then (see example differentiation) ${\displaystyle g}$

${\displaystyle g'(y)\approx {\frac {g(y+\Delta y)-g(y)}{\Delta y}}={\frac {\Delta g(y)}{\Delta y}}={\frac {\Delta x}{\Delta y}}}$

and this term can - unlike the formal object - simply be transformed and used: ${\displaystyle {\frac {dx}{dy}}}$

${\displaystyle \int _{D}f(x)dx\approx \sum _{F}f(x)\Delta x\approx \sum _{F'}f(g(y))g'(y)\Delta y\approx \int _{D'}f(g(y))g'(y)dy}$

And since both , and are standard functions, the integrals must be equal. ${\displaystyle f}$ ${\displaystyle g}$

swell

Edward Nelson : Internal Set Theory: A new approach to Nonstandard Analysis . In: Bulletin of the AMS . 83, No. 6, November 1977, pp. 1165-1198.

Individual evidence

1. ↑ Internal set theory . In: Guido Walz (Ed.): Lexicon of Mathematics . 1st edition. Spectrum Academic Publishing House, Mannheim / Heidelberg 2000, ISBN 3-8274-0439-8 .
2. ↑ "Perhaps it is fair to say that 'finite' does not mean what we have always thought it to mean." In: E. Nelson, Internal Set Theory, Ch. 1, p. 9 (The text can be here [1] downloaded.).