Hyperreal numbers

(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".