Hyperreal numbers and nonstandard analysis are intimately connected. First, the nonstandard analysis consists of an extension of the axioms for the real numbers, which introduces a new predicate $\text{st}(x)$ (read: "$x$ is a standard real number"). This predicate is not defined but obeys three new axioms:
- Transfer principle. For any standard formula $F(x)$, one has: $\forall^{st}x:F(x) \Leftrightarrow \forall x:F(x)$ (a standard formula is a formula not involving in any way the predicate $\text{st}(\cdot)$, and $\forall^{st}x$ is an abbreviation for $[\forall x:\text{st}(x) \Rightarrow \dots]$, similarly for $\exists^{st}$)
- Idealization principle. For any standard statement $B(x,y)$, one has $[\forall^{st}Y: Y\ {finite} \Rightarrow \exists x \forall y\in Y:B(x,y)]\Leftrightarrow [\exists x\forall^{st}y B(x,y)]$.
- Standardization principle. For any formula $F(x)$, standard or non-standard, one has: $\forall^{st}E\exists^{st}S_F \forall^{st} x [x\in S_F \Leftrightarrow x\in E \land F(x)]$.
These three principles, together with the usual axioms of the real numbers, allow to construct a theory which admits a model containing a set isomorphic to $\mathbb{R}$, but in addition the model contain also "nonstandard" objects which can be differently interpreted as "infinitesimal numbers", "infinitely large numbers" and so on. The hyperreals numbers are a non-archimedean field which is a model for these axioms.
What I find enigmatic is the following statement by Diener & Diener (1995): one important consequence of Idealization principle is there exists a finite set $\mathcal{F}$ that contains all standard objects. I am no sure about this, but it seems a parallel case to that of the Skolem's paradox: Skolem's paradox is that every countable axiomatisation of set theory in first-order logic, if it is consistent, has a model that is countable. (This appears contradictory because it is possible to prove in ZFC a sentence that intuitively says that there exist sets that are not countable). We have something of that sort here, namely, that any model for the axioms of non-standard analysis will model the standard objects as a finite subset. I don't know if this interpretation is correct.