Hyperreal numbers
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Numbers
$1$:
(n: 1 for n in NN)
$\varepsilon$:
(n: 1/n for n in NN)
$1/\varepsilon$:
(n: n for n in NN)
$S$:
(n: (-1)^n * n for n in NN)
Definition
Given a free ultrafilter $U$ on the natural numbers $\mathbb{N}$, the field of hyperreal numbers $^*\mathbb{R}$ [2] can be defined as the ultrapower $\mathbb{R}^\mathbb{N}/U$ [3], which is the set of all sequences of real numbers modulo the equivalence relation $(a_n)_n \sim (b_n)_n$ if and only if $\{n \mid a_n = b_n\} \in U$.
(1)
Our current definition of $^*\mathbb{R}$ is not well-defined as it depends on the choice of the free ultrafilter $U$. This is against a basic principle of NumberDB according to which every number in this database should be well-defined. This could possibly be resolved using the construction of Kanovei and Shelah [1].
(2)
$^*\mathbb{R}$ is an extension field of the real numbers $\mathbb{R}$: An embedding $\mathbb{R} \to {}^*\mathbb{R}$ is given by $r \mapsto (r)_n$.
(3)
The hyperreal numbers become a field with respect to element-wise operations. (Except that for division, division by $0$ may happen at some indices $n$, in which case once chooses an arbitrary real number as the result.)
(4)
The hyperreal number $\varepsilon = (1/n)_n$ is an infinitesimal: $\varepsilon > 0$ and $\varepsilon < r$ for any positive real number $r$.
(5)
Similarly, $1/\varepsilon$ is an infinite hyperreal number: $1/\varepsilon > r$ for any real number $r$.
(6)
The hyperreal $S = ((-1)^n n)_n$ is either $1/\varepsilon$ or its negative, depending on the choice of $U$. We might call $S$ "Schrödinger's infinity".
References
[1]
Kanovei, Vladimir; Shelah, Saharon, "A definable nonstandard model of the reals", Journal of Symbolic Logic, 69: 159–164, (2004). (arXiv) (doi)