Non-Standard Analysis

A self-contained development of non-standard analysis (NSA) — Robinson's rigorous theory of infinitesimals. Starting from an explicit construction of the hyperreal field ${}^{\ast }\mathbb{R}$, it establishes the transfer principle and then rebuilds one-variable calculus (continuity, the derivative, the integral, limits, compactness) with infinitely small and infinitely large numbers, proving at each step that the infinitesimal definitions agree with the classical $\epsilon$–$\delta$ ones of the analysis spine.

These pages are the rigorous backing for the remark Why the Physicist's Calculus Is Legitimate §4.1: they make "$dx$ is a real tiny thing" literally true, and show that "$\int fdx$ = sum of infinitesimal slices" and "$\frac{dy}{dx}$ is a fraction" are theorems, not heuristics. NSA sits at the intersection of logic and analysis: it specializes the ultraproduct machinery of model-theory.md and re-derives the calculus of the analysis spine.

Contents

A. Foundations and construction

1. Motivation and History — Leibniz/Euler infinitesimals, the Weierstrass exorcism, Berkeley's paradox, and Robinson's 1966 resolution; the ultrapower vs. IST routes.
2. The Ultrapower Construction — filters and ultrafilters, the free-ultrafilter (BPI) caveat, ${}^{\ast }\mathbb{R}={\mathbb{R}}^{\mathbb{N}}/\mathcal{U}$, and Łoś's theorem.
3. The Transfer Principle — the $\ast$-map, transfer, and the crucial internal vs. external boundary.

B. The hyperreal number system

4. The Hyperreal Numbers — infinitesimal / finite / infinite, halos and galaxies, and the standard part $\mathrm{st}$.
5. Internal and Hyperfinite Sets — hypernaturals, the internal definition principle, hyperfinite sums, overspill, saturation.

C. Calculus via infinitesimals

6. Calculus via Infinitesimals — microcontinuity and the derivative as a standard part, with the classical equivalences.
7. Non-Standard Integration — the integral as the standard part of a hyperfinite Riemann sum; equivalence with the Riemann integral; the FTC.
8. Sequences, Limits, and Topology — convergence via unlimited indices, Bolzano–Weierstrass, and nonstandard compactness.

D. Applications and contrast

9. Applications — the transfer/standard-part method, Peano existence, Loeb measure, and the physics payoff.
10. Smooth Infinitesimal Analysis — the contrasting nilpotent, intuitionistic route (SDG) and when to prefer each.

Dependency graph

graph TD
  HIST[1 motivation-history] --> ULT[2 ultrapower]
  ULT --> TP[3 transfer-principle]
  TP --> HR[4 hyperreals]
  HR --> INT[5 internal-sets]
  HR --> CALC[6 infinitesimal-calculus]
  INT --> CALC
  CALC --> INTEG[7 nonstandard-integration]
  CALC --> TOP[8 sequences-topology]
  INTEG --> APP[9 applications]
  TOP --> APP
  APP --> SDG[10 smooth-infinitesimal]
  MT[../../logic/model-theory.md] -.reuse.-> ULT
  FOL[../../logic/first-order-logic.md] -.reuse.-> TP
  INTEG -.recovers.-> ANA[../analysis/05-riemann-integral.md]
  APP --> PHYS[../remarks/01-physics-use-of-calculus.md]

Reading order

Read 1 → 10 in order: the construction (1–3) is a prerequisite for everything, the number system (4–5) for all the calculus, and the calculus pages (6–8) before the synthesis (9–10). Readers who only want the calculus can skim the construction (2) for the statement of transfer (3) and the standard part (4), then jump to 6–8. The contrast page (10) is independent and can be read any time after 4.

Scope

• In scope: the ultrapower construction, transfer, the hyperreals and standard part, hyperfinite sums, and the infinitesimal reconstruction of one-variable calculus and elementary topology.
• Out of scope (pointers only): Nelson's Internal Set Theory (mentioned in 1); full Loeb measure / nonstandard measure theory and stochastic analysis (pointer in 7 & 9); nonstandard functional analysis. A future measure-theory.md could take up the Loeb construction.
• Relation to the analysis spine: NSA is a parallel route, not a replacement. It cites the classical statements of analysis and re-proves them infinitesimally; the arrows run NSA → analysis and NSA → logic, never the reverse.