The Ultrapower Construction of the Hyperreals

The construction page of the NSA spine. We build the hyperreal field ${}^{\ast }\mathbb{R}$ concretely as an ultrapower of $\mathbb{R}$ — a quotient of sequences of reals by a free ultrafilter — and prove Łoś's theorem, the single result from which everything else (the transfer principle, infinitesimals, the standard part) follows. This specializes the ultraproduct machinery of model-theory.md to the real field.

References: Goldblatt, Lectures on the Hyperreals, ch. 2–3; Robinson, Non-standard Analysis, ch. 2; Chang & Keisler, Model Theory, §4.1 (Łoś).

1. Filters and ultrafilters

A filter on a set $I$ is a family $\mathcal{F}\subseteq \mathcal{P}(I)$ of "large" subsets, closed under supersets and finite intersections, containing $I$ but not $\varnothing$:

$$I\in \mathcal{F},\varnothing \notin \mathcal{F},A,B\in \mathcal{F}\Rightarrow A\cap B\in \mathcal{F},A\in \mathcal{F}\wedge A\subseteq B\Rightarrow B\in \mathcal{F}.$$

A filter is an ultrafilter if it is maximal — equivalently, for every $A\subseteq I$ exactly one of $A$, $I\setminus A$ is in $\mathcal{F}$. Read "$A\in \mathcal{U}$" as "$A$ holds at almost every index." The ultrafilter axioms are exactly the closure properties needed to make "almost everywhere" behave like a two-valued (finitely additive $\{0,1\}$) measure.

An ultrafilter is principal if $\mathcal{U}=\{A:i_0\in A\}$ for a fixed point $i_0$ (concentrated at one index) and free (non-principal) otherwise. A free ultrafilter contains no finite set, hence contains every cofinite set (complement of a finite set); it therefore extends the Fréchet filter $\{A\subseteq I:I\setminus A\text{ finite}\}$.

2. Existence of a free ultrafilter — the choice caveat

Ultrafilter lemma. Every filter extends to an ultrafilter. In particular the Fréchet filter on $\mathbb{N}$ extends to a free ultrafilter $\mathcal{U}$.

Proof. Order the filters extending the given one by inclusion; a union of a chain of filters is a filter, so Zorn's lemma yields a maximal one, which is an ultrafilter. $\square$

This is the only non-constructive step in the whole construction, and it is worth being honest about its strength. The ultrafilter lemma is equivalent to the Boolean prime ideal theorem (BPI), which is strictly weaker than the full axiom of choice but not provable in ZF alone (see metatheory-of-first-order-logic.md for BPI's exact place). Consequently:

A free ultrafilter on $\mathbb{N}$ cannot be exhibited explicitly — its existence rests on BPI. This makes NSA non-constructive, in pointed contrast to the nilpotent-infinitesimal approach of smooth-infinitesimal.md, and connects to why full choice is not constructively innocent (choice-and-excluded-middle.md).

Fix one such $\mathcal{U}$ on $\mathbb{N}$ for the rest of the construction.

3. The ultrapower ${}^{\ast }\mathbb{R}={\mathbb{R}}^{\mathbb{N}}/\mathcal{U}$

Consider sequences of reals $⟨a_n⟩=(a_0,a_1,a_2,\dots )\in {\mathbb{R}}^{\mathbb{N}}$. Declare two sequences equivalent mod $\mathcal{U}$ if they agree at almost every index:

$$⟨a_n⟩{\sim }_{\mathcal{U}}⟨b_n⟩⟺\{n:a_n=b_n\}\in \mathcal{U}.$$

Ultrafilter closure makes ${\sim }_{\mathcal{U}}$ an equivalence relation. The hyperreals are the quotient

$${}^{\ast }\mathbb{R}:={\mathbb{R}}^{\mathbb{N}}/{\sim }_{\mathcal{U}},$$

with $[a_n]$ the class of $⟨a_n⟩$. The reals embed via constant sequences, $r\mapsto {}^{\ast }r:=[r,r,r,\dots ]$; identify $\mathbb{R}$ with its image, so $\mathbb{R}\subseteq {}^{\ast }\mathbb{R}$.

4. Order and field structure

Define operations and order coordinatewise, almost everywhere:

$$[a_n]+[b_n]=[a_n+b_n],[a_n]\cdot [b_n]=[a_n{b}_n],[a_n]<[b_n]⟺\{n:a_n<b_n\}\in \mathcal{U}.$$

These are well-defined (independent of representatives) because $\mathcal{U}$ is closed under finite intersection. The decisive point is that ${}^{\ast }\mathbb{R}$ is a field and the order is total — and both rely on $\mathcal{U}$ being an ultrafilter, not merely a filter:

• Totality of $<$. For any $[a_n],[b_n]$, the three index sets $\{a_n<b_n\}$, $\{a_n=b_n\}$, $\{a_n>b_n\}$ partition $\mathbb{N}$; an ultrafilter contains exactly one of any partition into finitely many pieces, so exactly one of $<,=,>$ holds. (A mere filter could contain none.)
• Inverses (no zero divisors). If $[a_n]\ne 0$ then $A=\{n:a_n\ne 0\}\in \mathcal{U}$. Define $b_n=1/a_n$ on $A$ and $b_n=0$ off $A$; then $[a_n][b_n]=1$ because $a_n{b}_n=1$ on $A\in \mathcal{U}$. This is where a general ultraproduct of a ring can fail to be a field but an ultraproduct of a field succeeds.

So ${}^{\ast }\mathbb{R}$ is an ordered field properly extending $\mathbb{R}$.

4.1 The first infinitesimal and the first infinite number

The class of the sequence $⟨\frac{1}{n}⟩=(1,\frac{1}{2},\frac{1}{3},\dots )$ is a positive infinitesimal: for every real $r>0$, $\{n:\frac{1}{n}<r\}$ is cofinite, hence in $\mathcal{U}$, so $0<\left[\frac{1}{n}\right]<{}^{\ast }r$. Its reciprocal $[n]=(1,2,3,\dots )$ is infinite: larger than ${}^{\ast }r$ for every real $r$. Thus ${}^{\ast }\mathbb{R}$ is non-Archimedean — the whole point — and we have produced concrete infinitesimal and infinite elements without any appeal to compactness.

The reals sitting inside ${}^{\ast }\mathbb{R}$ are exactly the classes of eventually constant sequences (and their "almost-constant" cousins); the genuinely new hyperreals come from sequences that escape to $0$ or $\infty$ or oscillate.

5. Łoś's theorem — the fundamental theorem of ultrapowers

Everything hinges on the following, which says truth in the ultrapower is decided coordinatewise, almost everywhere. Work in the first-order language of the ordered field of reals (with a symbol for every relation and function on $\mathbb{R}$ — the full structure, as in transfer-principle.md).

Łoś's theorem. For any first-order formula $\phi (x_1,\dots ,x_k)$ and hyperreals $[a_n^1],\dots ,[a_n^k]$, $${}^{\ast }\mathbb{R}\models \phi ([a_n^1],\dots ,[a_n^k])>⟺>\{n:\mathbb{R}\models \phi (a_n^1,\dots ,a_n^k)\}\in \mathcal{U}.$$

Proof (induction on $\phi$). Atomic formulas hold by the definitions of the operations and order in §4. The Boolean cases use the ultrafilter axioms: negation uses that exactly one of $A,I\setminus A$ is in $\mathcal{U}$; conjunction uses closure under intersection. The existential case is the substantive one: if ${}^{\ast }\mathbb{R}\models \exists y\psi ([a_n],y)$, a witness $[b_n]$ gives $\{n:\mathbb{R}\models \psi (a_n,b_n)\}\in \mathcal{U}$, so almost every coordinate has a witness $b_n$; conversely, from "almost every coordinate has a witness" assemble a witnessing sequence $⟨b_n⟩$ by choosing $b_n$ per coordinate (a use of choice at the level of $\mathbb{R}$) — its class witnesses the existential in ${}^{\ast }\mathbb{R}$. $\square$

Taking $\phi$ to be a sentence (no free variables), the index set is either $\mathbb{N}\in \mathcal{U}$ or $\varnothing \notin \mathcal{U}$, so:

Corollary (transfer, sentence form). A first-order sentence holds in ${}^{\ast }\mathbb{R}$ iff it holds in $\mathbb{R}$.

This corollary is the transfer principle, developed and exploited on the next page. It is why ${}^{\ast }\mathbb{R}$ is not some pathological field but an elementary extension of $\mathbb{R}$ — indistinguishable from it by any first-order property, yet containing infinitesimals.

Where this page is used

• ${}^{\ast }\mathbb{R}$ and Łoś's theorem are the foundation of the entire spine.
• The sentence-form corollary is upgraded to the full transfer principle in transfer-principle.md.
• The infinitesimal $[\frac{1}{n}]$ and infinite $[n]$ built here are organized into the algebra of hyperreals.md.
• Specializes the ultraproduct construction of model-theory.md.

Next: The Transfer Principle — from Łoś to a working dictionary between $\mathbb{R}$ and ${}^{\ast }\mathbb{R}$.