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$\newcommand\SThing[1]{\Sigma\text-\mathrm{#1}}\newcommand\SType{\SThing{Type}}\newcommand\SFun{\SThing{Fun}}\newcommand\SRel{\SThing{Rel}}$I am about to begin my doctorate in philosophy, and my thesis project involves studying non-standard analysis. I understand the intuition: we want a “bigger” set of real numbers, that contain infinite natural numbers and infinitesimal real numbers.

There is a way to construct such an extension via model theory, which is what Abraham used in his first treatment of the subject. The problem is, I don't understand Abraham's work in the slightest. The book is kind of poorly written and he doesn't define things very well.

The first place where I found a precise treatment of higher-order model theory was in Peter Johnstone's Elephant, where he defines what's a higher-order signature and a higher-order structure. I've modified the definition so that we work in a setting where there is only one sort of terms.

A higher-order signature $\Sigma$ is defined by the following datum.

  • A set $\SType$ of types, defined recursively as follows:
    • there is a distinct type, $A \in \SType$, which is our only sort, and act as the “universe” where all our elements are taken from;
    • there is a distinct type $1 \in \SType$, which is the empty product, and for every $B, C \in \SType$, we have the product type $B \times C \in \SType$;
    • for every $B, C \in \SType$, we have the function type $[B \to C] \in \SType$;
    • for every $B \in \SType$, we have the list and the part types, $LB, PB \in \SType$ respectively, which represent the “lists of elements of $B$” and the “subsets of $B$”.
  • A set $\SFun$ of function symbols. We write $f \colon B \to C$ to represent that $f$ is a function symbol between the types $B$ and $C$;
  • A set $\SRel$ of relation symbols. We write $R \rightarrowtail B$ to represent that $R$ is a relation symbol of type $B$.

A $\Sigma$-structure $M$ is then a choice of set $A^M$ and an interpretation of the other types in the natural way: we define $1^M = 1 = \{\varnothing\}$, $(B \times C)^M = B^M \times C^M$, $[B \to C]^M = \operatorname{Hom}_{\operatorname{Set}}(B^M, C^M)$, $LB^M = \cup_{n \geq 0} (B^M)^n$ and $PB^M = \mathcal{P}(B^M)$. We also ask that function symbols and relation symbols are interpreted as functions and subsets in the obvious way.

What lies in the heart of my question is: how would one go about constructing a non-standard model of the real numbers using this setting? How does this relate to the notion of higher-order signature defined in Abraham's book?

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    $\begingroup$ "Abraham" here is presumably Abraham Robinson. It would be clearer to refer to him by his surname Robinson, or by his full name. $\endgroup$ Commented yesterday
  • $\begingroup$ Out of curiosity, by "doctorate in philosophy", do you mean PhD in logic/math or actually philosophy? $\endgroup$ Commented yesterday
  • $\begingroup$ Sorry, I forgot that doctorate in philosophy is the meaning of PhD, I'm enroling in a PhD in actual philosophy. It is mainly math but it is in the philosophy department $\endgroup$ Commented yesterday

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There can be no nonstandard model as ordinarily understood in the context of second-order logic, since standardness is definable in second-order logic.

The setting you describe is using in effect full second-order logic, since for example you use the full power set and the full function spaces. But by Dedekind's categoricity theorem for the natural numbers, in second-order logic we can characterize the standard model of arithmetic up to isomorphism.

Therefore, we cannot have a nonstandard model of the real numbers with transfer and so forth in second-order logic. The standard model knows it is standard, but this assertion cannot transfer to the nonstandard model.

Meanwhile, one can do a version of nonstandard models in second-order logic with the Henkin semantics, rather than full second-order logic. Perhaps the easiest way to do this is to take an ultrapower of the whole set theoretic universe $V^{\mathbb{N}}/\mu$ by a nonprincipal ultrafilter $\mu$ on $\mathbb{N}$. From this, you get an elementary embedding $x\mapsto x^*$ of every standard object $x$ to its nonstandard counterpart $x^*$. So one gets the nonstandard reals $\mathbb{R}^*$ in the setting for second order logic provided by $V^*$, which is not full second order logic, since $V^*$ does not have all the subsets of the objects it has, but it is nevertheless a very robust setting to interpret second and all higher-order logics.

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  • $\begingroup$ But isn't by constructing such a model that Robinson first introduces the theory? Is his notion of signature weaker as to allow it? $\endgroup$ Commented yesterday
  • $\begingroup$ I'm not familiar with how Robinson had done it back then, but rather only with how we generally think about it now. The main point is that you can't interpret the second-order and higher-order logic in the full manner, if you want the transfer principle. But if you use the Henkin semantics, then you can achieve transfer. $\endgroup$ Commented yesterday
  • $\begingroup$ Is there any bibliography that defines Henkin semantics precisely, and that also constructs such a model? $\endgroup$ Commented yesterday
  • $\begingroup$ I'm asking mainly because my thesis will involve heavily the ideas proposed by Robinson to extend the notions of enlargement to study other subjects in a non-standard manner, like lie groups, complex analysis and funcional analysis, and I dont how if the ultrapower construction can achieve this. If it can, than great! $\endgroup$ Commented yesterday
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    $\begingroup$ Perhaps another way for me to have answered your question was to say: form the second-order structure as you describe, and then take the ultrapower of it. This will be the desired nonstandard model. The ultrapower structure will not be using full semantics, but only the Henkin semantics. $\endgroup$ Commented yesterday
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This is really just a footnote to Joel's answer, and the comment thread underneath it.

There's nothing special about the real line here. The ultrapower construction works on any (infinite) structure $\mathfrak{S}$, to yield a structure $\mathfrak{S}'$ together with a non-surjective elementary embedding $i:\mathfrak{S}\rightarrow\mathfrak{S}'$. (If we try to do this to a finite structure, nothing goes wrong per se, it's just that we get $\mathfrak{S}=\mathfrak{S}'$ and $i$ is the identity.)

Now we can try to shoehorn higher structure into this. For example, given any $\mathfrak{S}$ we can let $\mathfrak{A}$ be the "powerstructure" of $\mathfrak{S}$ — $\mathfrak{A}$ has two "sorts," one consisting of elements of $\mathfrak{S}$ and one consisting of subsets of $\mathfrak{S}$, and we have the structure of $\mathfrak{S}$ on the elements-sort and the relation $\in$ connecting the two sorts. Applying the ultrapower construction to $\mathfrak{A}$ works perfectly well with an important caveat: the resulting two-sorted structure $\mathfrak{A}'$ is (just like $\mathfrak{A}$) two-sorted, and the first sort of $\mathfrak{A}'$ is isomorphic to an ultrapower $\mathfrak{S}'$ of $\mathfrak{S}$. The second sort of $\mathfrak{A}'$, however, is going to be strictly smaller than the full powerset of $\mathfrak{S}'$.

This is where Henkin structures come in. The family of subsets of (and relations/functions on) $\mathfrak{S}'$ which show up in $\mathfrak{A}'$ is reasonably large and closed under several basic operations. Similarly, we can try to get nonstandard extensions of a given structure $\mathfrak{S}$ taking into account richer-than-first-order logics by applying the ultrapower construction to some appropriate expansion $\mathfrak{A}$ of $\mathfrak{S}$, with the idea that the more-than-first-order behavior of the original $\mathfrak{S}$ is reflected by the first-order behavior of the more-than-original behavior of $\mathfrak{A}$. But we probably don't get the same connection between $\mathfrak{S}'$ and $\mathfrak{A}'$ that we have between $\mathfrak{S}$ and $\mathfrak{A}$.

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  • $\begingroup$ Do you have any book recommendations on where to learn about ALL this? I'm very new to model theory, I was a group theorist until 2 seconds ago $\endgroup$ Commented yesterday
  • $\begingroup$ @LucasGiraldi Chang/Keisler "Model theory" is a classic. For ultraproducts specifically, Bell/Slomson's "Models and ultraproducts" is good but rather technical. $\endgroup$ Commented yesterday
  • $\begingroup$ @LucasGiraldi, re, I hope you are now group theorist + other, not other − group theorist! $\endgroup$ Commented yesterday
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    $\begingroup$ @LucasGiraldi, gleaning from your various comments, I think you might find Goldblatt's Lectures on the Hyperreals to be a clear introduction to nonstandard analysis, with low-cost treatment of the model theory, but also working up to treating other structures besides the hyperreals specifically (in its Part IV: Nonstandard Frameworks). $\endgroup$ Commented yesterday
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There are two main approaches to nonstandard analysis: (I) model-theoretic, and (II) axiomatic (or syntactic). Since the existing answers focus on approach (I), I will complement them by some basics on approach (II).

The axiomatic approach was developed about a decade after Robinson's contribution. It was developed independently by Karel Hrbacek and Edward Nelson in the mid-70s.

In approach (II), instead of extending the number system (such as $\mathbb N$ and $\mathbb R$), one enriches the language of set theory by incorporating a distinction that goes back to Leibniz, between assignable and inassignable numbers (nonzero infinitesimals are necessarily inassignable). In modern terminology, one speaks rather of standard and nonstandard numbers (or sets).

The underlying set theory is different from classical set theory, which is based on a single (binary) relation of set membership, namely the relation $\in$. By contrast, the background set theory of approach (II) is based on the membership relation $\in$ and a one-place predicate "standard". For example, elements of $\mathbb N$ come in two flavors: standard and nonstandard. Every standard natural number is smaller than every nonstandard natural number.

Then a positive infinitesimal $\epsilon\in\mathbb R$ is defined as a number smaller than the inverse of every standard natural number.

Approach (II) has a number of advantages, which I will list now.

(1) To practice axiomatic nonstandard analysis, one does not need to learn model theory.

(2) For a meaningful fragment of the axiomatic approach that can handle both calculus and classical analysis (more generally, "ordinary mathematics"), there is no need for either the axiom of choice or nonprincipal ultrafilters, unlike approach (I), thereby eliminating a frequently raised objection to nonstandard analysis.

(3) This approach enables a reverse-mathematical analysis of the role of the axiom of choice in nonstandard analysis, something that is not possible with the model-theoretic approach, since fairly strong forms of the axiom of choice are already required to build up the basic framework.

(4) The $\mathbb R$ of the axiomatic approach retains all the categorical properties that are often mentioned as important advantages of $\mathbb R$, thus eliminating yet another commonly heard objection to nonstandard analysis (that the hyperreals lack such categoricity, etc.).

If your thesis is in philosophy, you may also be interested in a philosophical aspect of the debate over nonstandard analysis. Recently, the logician Haim Gaifman and his protégé R. Erhardt have claimed in print that Robinson was a finitist. We have argued against the Gaifman-Erhardt characterisation, and in support of the thesis that Robinson was a Formalist, in

Katz, M.; Kuhlemann, K.; Sanders, S.; Sherry, D. "Formalism 25." Journal for General Philosophy of Science (2025). https://doi.org/10.1007/s10838-025-09726-8, https://arxiv.org/pdf/2502.14811

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