20 1. Higher order Fourier analysis

and F parameters to disappear, at the cost of converting the finitary theory

to an infinitary one. Ultralimit analysis is discussed in Section 2.1; we give

a quick review here.

We first fix a non-principal ultrafilter α∞ ∈ βN\N (see Section 2.1 for a

definition of a non-principal ultrafilter). A property Pα pertaining to a natu-

ral number α is said to hold for all α suﬃciently close to α∞ if the set of α for

which Pα holds lies in the ultrafilter α∞. Two sequences (xα)α∈N, (yα)α∈N

of objects are equivalent if one has xα = yα for all α suﬃciently close to

α∞, and we define the ultralimit limα→α∞ xα to be the equivalence class of

all sequences equivalent to (xα)α∈N, with the convention that x is identified

with its own ultralimit limα→α∞ xα. Given any sequence Xα of sets, the

ultraproduct

α→α∞

Xα is the space of all ultralimits limα→α∞ xα, where

xα ∈ Xα for all α suﬃciently close to α∞. The ultraproduct

α→α∞

X of

a single set X is the ultrapower of X and is denoted

∗X.

Ultralimits of real numbers (i.e., elements of

∗R)

will be called limit real

numbers; similarly one defines limit natural numbers, limit complex num-

bers, etc. Ordinary numbers will be called standard numbers to distinguish

them from limit numbers, thus for instance a limit real number is an ul-

tralimit of standard real numbers. All the usual arithmetic operations and

relations on standard numbers are inherited by their limit analogues; for in-

stance, a limit real number limα→α∞ xα is larger than another limα→α∞ yα

if one has xα yα for all α suﬃciently close to α∞. The axioms of a

non-principal ultrafilter ensure that these relations and operations on limit

numbers obey the same axioms as their standard

counterparts3.

Ultraproducts of sets will be called limit sets; they are roughly analogous

to “elementary sets” in measure theory. Ultraproducts of finite sets will be

called limit finite sets. Thus, for instance, if N = limα→α∞ Nα is a limit

natural number, then [N] =

α→α∞

[Nα] is a limit finite set, and can be

identified with the set of limit natural numbers between 1 and N.

Remark 1.1.22. In the language of non-standard analysis, limit numbers

and limit sets are known as non-standard numbers and internal sets, respec-

tively. We will, however, use the language of ultralimit analysis rather than

non-standard analysis in order to emphasise the fact that limit objects are

the ultralimits of standard objects; see Section 2.1 for further discussion of

this perspective.

3The formalisation of this principle is Los’s theorem, which roughly speaking asserts that

any first-order sentence which is true for standard objects, is also true for their limit counterparts.