Editing Hyperreal number (section)

=== Ultrapower construction ===

We are going to construct a hyperreal field via [[sequence]]s of reals.<ref>{{Citation | last1=Loeb | first1=Peter A. |authorlink = Peter A. Loeb| title=Nonstandard analysis for the working mathematician | publisher=Kluwer Acad. Publ. | location=Dordrecht | series=Math. Appl. | year=2000 | volume=510 | chapter=An introduction to nonstandard analysis | pages=1–95}}</ref> In fact we can add and multiply sequences componentwise; for example:
: <math> (a_0, a_1, a_2, \ldots) + (b_0, b_1, b_2, \ldots) = (a_0 +b_0, a_1+b_1, a_2+b_2, \ldots) </math>
and analogously for multiplication.
This turns the set of such sequences into a [[commutative ring]], which is in fact a real [[algebra over a field|algebra]] '''A'''. We have a natural embedding of '''R''' in '''A''' by identifying the real number ''r'' with the sequence (''r'', ''r'', ''r'', …) and this identification preserves the corresponding algebraic operations of the reals. The intuitive motivation is, for example, to represent an infinitesimal number using a sequence that approaches zero. The inverse of such a sequence would represent an infinite number. As we will see below, the difficulties arise because of the need to define rules for comparing such sequences in a manner that, although inevitably somewhat arbitrary, must be self-consistent and well defined. For example, we may have two sequences that differ in their first ''n'' members, but are equal after that; such sequences should clearly be considered as representing the same hyperreal number. Similarly, most sequences [[Oscillation_(mathematics)|oscillate]] [[Random_sequence|randomly]] forever, and we must find some way of taking such a sequence and interpreting it as, say, <math>7+\epsilon</math>, where <math>\epsilon</math> is a certain infinitesimal number.

Comparing sequences is thus a delicate matter. We could, for example, try to define a relation between sequences in a componentwise fashion:
: <math> (a_0, a_1, a_2, \ldots) \leq (b_0, b_1, b_2, \ldots) \iff  (a_0 \leq b_0) \wedge (a_1 \leq b_1) \wedge (a_2 \leq b_2) \ldots </math>
but here we run into trouble, since some entries of the first sequence may be bigger than the corresponding entries of the second sequence, and some others may be smaller.   It follows that the relation defined in this way is only a [[partial order]].  To get around this, we have to specify which positions matter. Since there are infinitely many indices, we don't want finite sets of indices to matter. A consistent choice of index sets that matter is given by any free [[Ultrafilter (set theory)|ultrafilter]] ''U'' on the [[natural number]]s; these can be characterized as ultrafilters that do not contain any finite sets. (The good news is that [[Zorn's lemma]] guarantees the existence of many such ''U''; the bad news is that they cannot be explicitly constructed.) We think of ''U'' as singling out those sets of indices that "matter": We write (''a''<sub>0</sub>, ''a''<sub>1</sub>, ''a''<sub>2</sub>, ...) ≤ (''b''<sub>0</sub>, ''b''<sub>1</sub>, ''b''<sub>2</sub>, ...) if and only if the set of natural numbers { ''n'' : ''a''<sub>''n''</sub> ≤ ''b''<sub>''n''</sub> } is in ''U''.

This is a [[total preorder]] and it turns into a [[total order]] if we agree not to distinguish between two sequences ''a'' and ''b'' if ''a'' ≤ ''b'' and ''b'' ≤ ''a''. With this identification, the ordered field '''*R''' of hyperreals is constructed. From an algebraic point of view, ''U'' allows us to define a corresponding [[ideal (ring theory)|maximal ideal]] '''I''' in the commutative ring '''A''' (namely, the set of the sequences that vanish in some element of ''U''), and then to define '''*R''' as '''A'''/'''I'''; as the [[Quotient ring|quotient]] of a commutative ring by a maximal ideal, '''*R''' is a field. This is also notated '''A'''/''U'', directly in terms of the free ultrafilter ''U''; the two are equivalent. The maximality of '''I''' follows from the possibility of, given a sequence ''a'', constructing a sequence ''b'' inverting the non-null elements of ''a'' and not altering its null entries. If the set on which ''a'' vanishes is not in ''U'', the product ''ab'' is identified with the number 1, and any ideal containing 1 must be ''A''. In the resulting field, these ''a'' and ''b'' are inverses.

The field '''A'''/''U'' is an [[ultraproduct|ultrapower]] of '''R'''.
Since this field contains '''R''' it has [[cardinality]] at least that of the [[cardinality of the continuum|continuum]]. Since '''A''' has cardinality
: <math>(2^{\aleph_0})^{\aleph_0} = 2^{\aleph_0^2} =2^{\aleph_0},</math>
it is also no larger than <math>2^{\aleph_0}</math>, and hence has the same cardinality as '''R'''.

One question we might ask is whether, if we had chosen a different free ultrafilter ''V'', the quotient field '''A'''/''U'' would be isomorphic as an ordered field to '''A'''/''V''. This question turns out to be equivalent to the [[continuum hypothesis]]; in [[ZFC]] with the continuum hypothesis we can prove this field is unique up to [[order isomorphism]], and in ZFC with the negation of continuum hypothesis we can prove that there are non-order-isomorphic pairs of fields that are both countably indexed ultrapowers of the reals.<ref>{{cite arXiv|last=Hamkins|first=Joel David|author-link=Joel David Hamkins|date=22 July 2024|title=How the Continuum Hypothesis Could Have Been a Fundamental Axiom|eprint=2407.02463|class=math.LO}}</ref>

For more information about this method of construction, see [[ultraproduct]].