Abstract
The expressive power of minIST comes from the fact that it allows for the notions of finite sets with unlimited cardinality, and finite subsets of the reals whose distance is at most an infinitesimal from every point in some non-empty open interval.
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Notes
- 1.
Such as the system minIST  −  discussed in Appendix A.
- 2.
Proving this could be a useful exercise for students. If \(P\left \{A\right \} \simeq 1\), then clearly a.s. A (simply choose \(N = \Omega \setminus \{ A\}\)). Conversely, if a.s. A, then the set
$$M = \left \{n \in \mathbf{N}\ :\ \exists N \subseteq \Omega \left (P(N) \leq 1/n\ \&\ \Omega \setminus \{ A\} \subseteq N\right )\right \}$$contains all standard elements of \(\mathbf{N}\). Since there is no set which consists of all standard natural numbers (see Remark 1.1), M must contain some nonstandard \({n}_{0} \in \mathbf{N}\), too. But then, \(P\left (\Omega \setminus \{ A\}\right ) \leq 1/{n}_{0} \simeq 0\), so \(P\{A\} \simeq 1\).
- 3.
In Robinsonian nonstandard analysis, this Wiener walk is known as Anderson’s [4] construction of the Wiener process.
- 4.
In Robinsonian nonstandard analysis, this Poisson walk is known as Loeb’s [51] construction of the Poisson process.
- 5.
One should note that this infinitesimal version of Stirling’s formula can also be proved in radically elementary probability theory, cf. van den Berg [11, last paragraph on p. 172].
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Herzberg, F.S. (2013). Radically Elementary Probability Theory. In: Stochastic Calculus with Infinitesimals. Lecture Notes in Mathematics, vol 2067. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33149-7_2
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