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 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 holds lies in the ultrafilter α∞. Two sequences (xα)α∈N, (yα)α∈N
of objects are equivalent if one has = for all α sufficiently close to
α∞, and we define the ultralimit limα→α∞ 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 of sets, the
ultraproduct
α→α∞
is the space of all ultralimits limα→α∞ xα, where
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∞ is larger than another limα→α∞
if one has 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∞ 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.
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