Hyperreal numbers

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number system
nonstandard analysis
axiom of choice
set theory
ring
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 $\{n \mid a_n = b_n\} \in U$.

Comments

(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

Links

Data properties

Entries are of type: hyperreal number