Emeritus
PF Gold
P: 16,091
 Quote by mr_homm BTW, could you give me a reference on the hypernaturals indexing the hyperreals?
It's a direct application of the transfer principle.

In the standard model, the decimal expansion of a real number s is nothing more than a function f:Z->{0, ..., 9} satisfying

$$s = \sum_{n \in \mathbb{Z}} f(n) 10^n$$
$$\lim_{n \rightarrow +\infty} f(n) = 0$$

and we have a theorem that every real number has a decimal expansion.

Applying the transfer principle tells us that in the nonstandard model, the hyperdecimal expansion of a hyperreal number s is nothing more than an (internal) function f:*Z->{0, ..., 9} satisfying

$$s = \sum_{n \in {}^\star \mathbb{Z}} f(n) 10^n$$
$$\lim_{n \rightarrow +\infty} f(n) = 0$$

and we have a theorem that every hyperreal number has a hyperdecimal expansion.