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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:

  1. 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 $\exists^{st}$)
  2. Idealization principle. For any standard statement $B(x,y)$, one has $[\forall^{st}Y: Y\ {finite} \Rightarrow \exists x \forall y\in Y:B(x,y)]\Leftrightarrow [\exists x\forall^{st}y B(x,y)]$.
  3. 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.

Davius
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  • I believe the discussion here may be related. – Noah Schweber Dec 20 '22 at 18:35
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    Two things: (1) These are not the axioms of nonstandard analysis, these appear to be the axioms of Nelson's IST. There are other inequivalent ways of axiomatizing the concept of "nonstandard analysis". The theorem you're asking about is definitely false in some of those. (2) "...will model the standard objects as a finite subset." This is incorrect; the claim is that standard objects are contained in a finite subset, not that they form a finite subset. These two properties are distinct in an internal theory like IST. – Nicholas Todoroff Dec 20 '22 at 21:39
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    I should add, what IST calls "finite" corresponds to hyperfinite in Robinson's superstructure approach; this might give an idea of what's going on here. – Nicholas Todoroff Dec 20 '22 at 21:41

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