Abstract
The most important feature of Nelson’s [60] radically elementary analysis is the discretization of the continuum. The crucial step herein is the consistent use of infinitely large (“nonstandard”) numbers and infinitesimals, in a manner which was first proposed by Nelson through the axiom system of Internal Set Theory [59], motivated by the groundbreaking work of Abraham Robinson [66, 67]. One decade on, Nelson [60] introduced an even more elementary, yet still very powerful, formal system, which we shall review presently.
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Notes
- 1.
For some fascinating insights into—and some polemical comments on—the history of infinitesimals, cf. e.g., Błasczcyk et al. [19].
- 2.
Formally, let μ be a map which assigns each set of natural numbers either 0 or 1 and is such that whenever I, J are disjoint (i.e. \(I \cap J\,=\,\varnothing \)), \(\mu (I \cup J) = \mu (I) + \mu (J)\) and such that there is no natural number k such that μ(I) = 1 if and only if k ∈ I for all I. Two infinite sequences of real numbers (a n ) n , (b n ) n are called μ-equivalent, denoted \({({a}_{n})}_{n} {\sim }_{\mu }{({b}_{n})}_{n}\) if and only if \(\mu \left (\left \{n\ :\ {a}_{n} = {b}_{n}\right \}\right ) = 1\). It is not difficult to show that ∼ μ is indeed an equivalence relation. The new numbers are then just μ-equivalence classes of infinite sequences of real numbers.
- 3.
It is also not difficult to verify that the following relation and operations are well-defined. For all sequences of real numbers \({({a}_{n})}_{n}, {({b}_{n})}_{n}\),
$$\begin{array}{rcl}{ \left [{({a}_{n})}_{n}\right ]}_{{\sim }_{\mu }} >{ \left [{({b}_{n})}_{n}\right ]}_{{\sim }_{\mu }}& :\Leftrightarrow & \mu \left (\left \{n : {a}_{n} > {b}_{n}\right \}\right ) = 1 \\ {\left [{({a}_{n})}_{n}\right ]}_{{\sim }_{\mu }} +{ \left [{({b}_{n})}_{n}\right ]}_{{\sim }_{\mu }}& :=&{ \left [{({a}_{n} + {b}_{n})}_{n}\right ]}_{{\sim }_{\mu }} \\ {\left [{({a}_{n})}_{n}\right ]}_{{\sim }_{\mu }} -{\left [{({b}_{n})}_{n}\right ]}_{{\sim }_{\mu }}& :=&{ \left [{({a}_{n} - {b}_{n})}_{n}\right ]}_{{\sim }_{\mu }} \\ { \left [{({a}_{n})}_{n}\right ]}_{{\sim }_{\mu }}{\left [{({b}_{n})}_{n}\right ]}_{{\sim }_{\mu }}& :=&{ \left [{({a}_{n}{b}_{n})}_{n}\right ]}_{{\sim }_{\mu }} \\ { \left [{({a}_{n})}_{n}\right ]}_{{\sim }_{\mu }}^{-1}& :=&{ \left [{(1/{a}_{n})}_{n}\right ]}_{{\sim }_{ \mu }}\text{ if }{a}_{n}\neq 0\text{ for all }n \\ & & \\ \end{array}$$ - 4.
We will hardly ever have the need to refer to standard real numbers; we will, however, often refer to limited (standard or nonstandard) real numbers (see below).
- 5.
The name is derived from Nelson’s [59] Internal Set Theory (IST), of which even Minimal Internal Set Theory combined with the Sequence Principle (see footnote 5 on p. 6) is only a small subsystem, see Sect. A.1 of Appendix A. Although Nelson [62] did not state this explicitly, an axiom system such as minIST is most probably what he had in mind when suggesting the use of “minimal nonstandard analysis” [62, p. 30].
In his 1987 monograph on Radically elementary probability theory [60], Nelson proposes an axiom system which enlarges minIST by the following axiom scheme:
-
(Sequence Principle) If \(A\left ({v}_{0},{v}_{1}\right )\) is any formula (which may involve the predicate “standard”) with the property that for all standard natural numbers n there exists some x with \(A\left (n,x\right )\), then there exists a sequence \({({x}_{n})}_{n\in \mathbf{N}}\) such that \(A\left (n,{x}_{n}\right )\) holds for all standard n.
However, Nelson [60] only uses the Sequence Principle occasionally and conveniently marks those results which are proved through the Sequence Principle by an asterisk; the greater part of radically elementary probability theory—and in particular, all results from radically elementary probability theory which we use in this book—can be developed in minIST. Again, none of the results of the present work depend on the Sequence Principle.
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- 6.
Equivalently one could write: “All axioms …”.
- 7.
This principle is known as the Transfer Principle of nonstandard analysis. It is beyond the scope of this book to give a rigorous justification. We only point out that the extended, nonstandard real number system was devised to preserve very simple mathematical propositions (such as “x 2 ≥ 0 for all real numbers x”) and that by a beautiful theorem due to Łoś [52] this preservation property can be shown to hold for complex mathematical propositions as well.
- 8.
Thus, the formula A(v) may involve the predicate “standard”!
- 9.
In Appendix A, we shall consider an even weaker system than minIST, denoted by minIST − , which still allows for much of radically elementary mathematics to be developed and also admits a simple relative consistency proof.
- 10.
This could be an instructive exercise for students interested in the foundational aspects of minIST. By External Induction in k, one can prove for all standard k ∈ N that if n ∈ N and n ≤ k, then n is standard:
-
For the base step of the External Induction, note that the only n ∈ N with n ≤ 0 is 0, hence standard by an axiom of minIST.
-
For the induction step of the External Induction note that whenever n ∈ N with n ≤ k + 1, one has
-
(1)
either n ≤ k, in which case n is standard by induction hypothesis of the External Induction,
-
(2)
or \(n = k + 1\), whence n again is standard (as k is standard and successors of standard natural numbers are standard by another axiom of minIST).
-
(1)
Thus, there can be no pair of a nonstandard n ∈ N and a standard k such that one would have \(n \leq k\). Hence nonstandard natural numbers are always greater than all standard natural numbers.
-
References
Błasczcyk, P., Katz, M., Sherry, D.: Ten misconceptions from the history of analysis and their debunking. Found. Sci. (forthcoming) (2012). doi: 10.1007/s10699-012-9285-8
Łoś, J.: Quelques remarques, théorèmes et problèmes sur les classes définissables d’algèbres. In: Skolem, T., Hasenjaeger, G., Kreisel, G., Robinson, A., Wang, H., Henkin, L., Łoś, J. (eds.) Mathematical Interpretation of Formal Systems. Studies in Logic and the Foundations of Mathematics, vol. 16, pp. 98–113. North-Holland, Amsterdam (1955)
Nelson, E.: Internal set theory: A new approach to nonstandard analysis. Bull. Am. Math. Soc. 83(6), 1165–1198 (1977)
Nelson, E.: Radically elementary probability theory. Annals of Mathematics Studies, vol. 117. Princeton University Press, Princeton, NJ (1987)
Nelson, E.: The virtue of simplicity. In: Berg, I.v.d., Neves, V. (eds.) The Strength of Nonstandard Analysis, pp. 27–32. Springer, Vienna (2007)
Robinson, A.: Non-standard analysis. Nederlandse Akademie van Wetenschappen. Proceedings. Series A. Indagat. Math. 64, 432–440 (1961)
Robinson, A.: Non-standard Analysis. North-Holland, Amsterdam (1966)
Schmieden, C., Laugwitz, D.: Eine Erweiterung der Infinitesimalrechnung. Math. Z. 69, 1–39 (1958)
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Herzberg, F.S. (2013). Infinitesimal Calculus, Consistently and Accessibly. In: Stochastic Calculus with Infinitesimals. Lecture Notes in Mathematics, vol 2067. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33149-7_1
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