Internal and Hyperfinite Sets

The technical core that makes nonstandard analysis (as opposed to just a bigger number field) work. This page of the NSA spine develops the hypernaturals ${}^{\ast }\mathbb{N}$, the all-important internal vs. external distinction (transfer-principle.md §4), the hyperfinite sets and sums that power integration, and the overspill/underspill and saturation principles used throughout.

References: Goldblatt, Lectures on the Hyperreals, ch. 11–15; Robinson, Non-standard Analysis, ch. 3; Keisler, Foundations of Infinitesimal Calculus.

1. The hypernaturals ${}^{\ast }\mathbb{N}$

${}^{\ast }\mathbb{N}={}^{\ast }(\mathbb{N})$ is the hyperextension of $\mathbb{N}$: in the ultrapower, $[a_n]\in {}^{\ast }\mathbb{N}$ iff $a_n\in \mathbb{N}$ almost everywhere. It splits into

$${}^{\ast }\mathbb{N}=\underset{\text{standard/finite}}{\underbrace{\mathbb{N}}}\cup \underset{\text{infinite hypernaturals}}{\underbrace{\{\text{unlimited }H\}}},$$

with every unlimited $H$ larger than every ordinary $n\in \mathbb{N}$. By transfer of "every natural has an immediate successor and (except $0$) predecessor" (transfer-principle.md §3), ${}^{\ast }\mathbb{N}$ is discretely ordered: after the standard block $0,1,2,\dots$ come $\mathbb{Z}$-like blocks $\dots ,H-1,H,H+1,\dots$ of unlimited integers. Transfer also gives hyperfinite induction: any internal subset of ${}^{\ast }\mathbb{N}$ containing $0$ and closed under successor is all of ${}^{\ast }\mathbb{N}$ — the word internal is essential (§2).

2. Internal vs. external sets

An element of the nonstandard superstructure is internal if it is a member of some ${}^{\ast }A$ (equivalently, in the ultrapower, a class $[A_n]$ of a sequence of standard sets $A_n$); otherwise it is external. The working characterizations:

• Hyperextensions ${}^{\ast }A$ are internal, as are ${}^{\ast }f$, and any set defined by a first-order formula with internal parameters (the internal definition principle, §3).
• Transfer applies to internal sets only. Internal sets are precisely the ones that behave like genuine sets from the standard world's viewpoint.

The standard external sets (recurring "traps"): $$\mathbb{N},\mathbb{R},\mathrm{hal}(0)=\text{infinitesimals},>\mathcal{O}=\text{finite hyperreals},\{\text{unlimited }H\}$$ are all external. Each would break a transferred theorem if it were internal — e.g. an internal nonempty subset of ${}^{\ast }\mathbb{N}$ has a least element (by transfer of well-ordering), but $\{\text{unlimited }H\}$ has no least element (if $H$ is unlimited so is $H-1$), so it cannot be internal (transfer-principle.md §4).

3. The internal definition principle

Internal definition principle. A subset of an internal set cut out by a first-order formula whose parameters are all internal is itself internal.

This is the practical test for legitimacy. "The set of $x\in {}^{\ast }[0,1]$ with ${}^{\ast }f(x)>0$" is internal (parameters ${}^{\ast }f$, ${}^{\ast }[0,1]$ are internal), so transfer-based reasoning (e.g. it attains a max) is valid. "The set of $x$ that are infinitesimal" is not of this form — "infinitesimal" is not first-order over internal parameters (it secretly quantifies over the standard reals) — so it is external and off-limits to transfer. Learning to spot which sets are internal is learning to use NSA safely.

4. Hyperfinite sets and sums

A set is hyperfinite if it is internal and, by transfer, "finite" — i.e. it is in internal bijection with an initial segment $\{1,2,\dots ,H\}\subseteq {}^{\ast }\mathbb{N}$ for some (possibly unlimited) $H\in {}^{\ast }\mathbb{N}$. A hyperfinite set has a well-defined internal cardinality $H$ and, though it may have uncountably many external elements, transfer lets it be manipulated exactly like a finite set.

The payoff is the hyperfinite sum. For an internal function $F:\{1,\dots ,H\}\to {}^{\ast }\mathbb{R}$, the sum ${\sum }_{i=1}^{H}F(i)$ is defined by transfer of the finite-sum recursion, and obeys every finite-sum identity (linearity, reindexing, telescoping). This is the object that realizes "an infinite sum of infinitely many infinitesimal pieces":

Prototype (the integral). Partition $[a,b]$ into $H$ equal pieces of infinitesimal width $\Delta x=\frac{b-a}{H}$ ($H$ unlimited), with nodes $x_i=a+i\Delta x$. The hyperfinite Riemann sum ${\sum }_{i=1}^H{}^{\ast }>f(x_i)\Delta x$ is a single, exact hyperreal. Its standard part is the integral (nonstandard-integration.md) — Leibniz's "$\int =>\sum$ of $fdx$" taken literally, with $\int$ a genuine (hyperfinite) sum.

5. Overspill and underspill

Two principles exploit the internal/external boundary to prove things. They rest on the fact that $\mathbb{N}$ and the infinitesimals are external, so an internal set cannot coincide with them.

Overspill. If an internal set $S\subseteq {}^{\ast }\mathbb{N}$ contains arbitrarily large finite naturals (all of $\mathbb{N}$, or a cofinal part), then it contains an unlimited $H$ as well. (Otherwise $S\cap \{\text{finite}\}>=\mathbb{N}$ would be internal — impossible.)

Underspill. If an internal set of hyperreals contains arbitrarily small positive infinitesimals, it contains a positive appreciable (non-infinitesimal) element.

Typical use: a property known to hold for all standard $n\in \mathbb{N}$, if expressible as membership in an internal set, must "spill over" to some unlimited $H$ — converting a family of standard facts into a single nonstandard witness. Overspill is the nonstandard replacement for many $\epsilon$–$N$ arguments.

6. Saturation (brief)

The ultrapower ${\mathbb{R}}^{\mathbb{N}}/\mathcal{U}$ is ${\aleph }_1$-saturated: any countable, finitely-satisfiable family of internal conditions with parameters is simultaneously satisfiable. Saturation is the abstract source of overspill and of "there exists an infinitesimal finer than every member of a given sequence"–type arguments; richer nonstandard models assume higher $\kappa$-saturation (compare the types and saturation theme of model-theory.md). For this spine, ${\aleph }_1$-saturation of the ultrapower is all we need, and we invoke it only through overspill/underspill.

Where this page is used

• Hyperfinite sums are the definition of the integral in nonstandard-integration.md.
• The internal/external discipline governs every proof in infinitesimal-calculus.md and sequences-topology.md.
• Overspill/saturation power the nonstandard compactness arguments of sequences-topology.md and applications.md.

Next: Calculus via Infinitesimals — continuity and the derivative, infinitesimally.