Editing Hyperreal number (section)

== Development ==
The hyperreals can be developed either axiomatically or by more constructively oriented methods. The essence of the axiomatic approach is to assert (1) the existence of at least one infinitesimal number, and (2) the validity of the transfer principle. In the following subsection we give a detailed outline of a more constructive approach. This method allows one to construct the hyperreals if given a set-theoretic object called an [[Ultrafilter (set theory)|ultrafilter]], but the ultrafilter itself cannot be explicitly constructed.

=== From Leibniz to Robinson ===
When [[Isaac Newton|Newton]] and (more explicitly) [[Gottfried Leibniz|Leibniz]] introduced differentials, they used infinitesimals and these were still regarded as useful by later mathematicians such as [[Leonhard Euler|Euler]] and [[Augustin Louis Cauchy|Cauchy]]. Nonetheless these concepts were from the beginning seen as suspect, notably by [[George Berkeley]].  Berkeley's criticism centered on a perceived shift in hypothesis in the definition of the derivative in terms of infinitesimals (or fluxions), where ''dx'' is assumed to be nonzero at the beginning of the calculation, and to vanish at its conclusion (see [[Ghosts of departed quantities]] for details).  When in the 1800s [[calculus]] was put on a firm footing through the development of the [[(ε, δ)-definition of limit]] by [[Bernard Bolzano|Bolzano]], Cauchy, [[Karl Weierstrass|Weierstrass]], and others, infinitesimals were largely abandoned, though research in [[non-Archimedean field]]s continued (Ehrlich 2006).

However, in the 1960s [[Abraham Robinson]] showed how infinitely large and infinitesimal numbers can be rigorously defined and used to develop the field of [[nonstandard analysis]].<ref name=robinson>{{Citation | last1=Robinson | first1=Abraham | author1-link=Abraham Robinson | title=Non-standard analysis | publisher=[[Princeton University Press]] | isbn=978-0-691-04490-3 | year=1996}}. The classic introduction to nonstandard analysis.</ref> Robinson developed his theory [[Constructive_proof#Non-constructive_proofs|nonconstructively]], using [[model theory]]; however it is possible to proceed using only [[algebra]] and [[topology]], and proving the transfer principle as a consequence of the definitions. In other words hyperreal numbers ''per se'', aside from their use in nonstandard analysis, have no necessary relationship to model theory or first order logic, although they were discovered by the application of model theoretic techniques from logic.  Hyper-real fields were in fact originally introduced by [[Edwin Hewitt|Hewitt]] (1948) by purely algebraic techniques, using an ultrapower construction.

=== 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]].

=== An intuitive approach to the ultrapower construction ===

The following is an intuitive way of understanding the hyperreal numbers. The approach taken here is very close to the one in the book by [[Robert Goldblatt|Goldblatt]].<ref>{{Citation | last1=Goldblatt | first1=Robert | title=Lectures on the hyperreals: an introduction to nonstandard analysis | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-98464-3 | year=1998}}</ref> Recall that the sequences converging to zero are sometimes called infinitely small. These are almost the infinitesimals in a sense; the true infinitesimals include certain classes of sequences that contain a sequence converging to zero.

Let us see where these classes come from. Consider first the sequences of real numbers. They form a [[ring (abstract algebra)|ring]], that is, one can multiply, add and subtract them, but not necessarily divide by a non-zero element. The real numbers are considered as the constant sequences, the sequence is zero if it is identically zero, that is,  ''a''<sub>''n''</sub>&nbsp;=&nbsp;0 for all ''n''.

In our ring of sequences one can get ''ab''&nbsp;=&nbsp;0 with neither ''a''&nbsp;=&nbsp;0 nor ''b''&nbsp;=&nbsp;0. Thus, if for two sequences <math>a, b</math> one has ''ab''&nbsp;=&nbsp;0, at least one of them should be declared zero. Surprisingly enough, there is a consistent way to do it. As a result, the equivalence classes of sequences that differ by some sequence declared zero will form a field, which is called a hyperreal [[field (mathematics)|field]]. It will contain the infinitesimals in addition to the ordinary real numbers, as well as infinitely large numbers (the reciprocals of infinitesimals, including those represented by sequences diverging to infinity). Also every  hyperreal that is not infinitely large will be infinitely close to an ordinary real, in other words, it will be the sum of an ordinary real and an infinitesimal.

This construction is parallel to the construction of the reals from the rationals given by [[Georg Cantor|Cantor]]. He started with the ring of the [[Cauchy sequence]]s of rationals and declared all the sequences that converge to zero to be zero. The result is the reals. To continue the construction of hyperreals, consider the zero sets of our sequences, that is, the <math>z(a)=\{i: a_i=0\}</math>, that is, <math>z(a)</math> is the set of indexes <math>i</math> for which <math>a_i=0</math>. It is clear that if <math>ab=0</math>, then the union of <math>z(a)</math> and <math>z(b)</math> is '''N''' (the set of all natural numbers), so:
# One of the sequences that vanish on two complementary sets should be declared zero.
# If <math>a</math> is declared zero, <math>ab</math> should be declared zero too, no matter what <math>b</math> is.
# If both <math>a</math> and <math>b</math> are declared zero, then <math>a+b</math> should also be declared zero.
Now the idea is to single out a bunch ''U'' of [[subset]]s ''X'' of '''N''' and to declare that  <math>a=0</math> if and only if <math>z(a)</math> belongs to ''U''.  From the above conditions one can see that:
# From two complementary sets one belongs to ''U''.
# Any set having a subset that belongs to ''U'', also belongs to ''U''.
# An intersection of any two sets belonging to ''U'' belongs to ''U''.
# Finally, we do not want the [[empty set]] to belong to ''U'' because then everything would belong to ''U'', as every set has the empty set as a subset.

Any family of sets that satisfies (2–4) is called a [[Filter (set theory)|filter]] (an example: the complements to the finite sets, it is called the [[Fréchet filter]] and it is used in the usual limit theory). If (1) also holds, U is called an [[Ultrafilter (set theory)|ultrafilter]] (because you can add no more sets to it without breaking it). The only explicitly known example of an ultrafilter is the family of sets containing a given element (in our case, say, the number 10). Such ultrafilters are called trivial, and if we use it in our construction, we come back to the ordinary real numbers. Any ultrafilter containing a finite set is trivial. It is known that any filter can be extended to an ultrafilter, but the proof uses the [[axiom of choice]]. The existence of a nontrivial ultrafilter (the [[ultrafilter lemma]]) can be added as an extra axiom, as it is weaker than the axiom of choice.

Now if we take a nontrivial ultrafilter (which is an extension of the Fréchet filter) and do our construction, we get the hyperreal numbers as a result.

If <math>f</math> is a real function of a real variable  <math>x</math> then <math>f</math> naturally extends to a hyperreal function of a hyperreal variable by composition:
: <math>f(\{x_n\})=\{f(x_n)\}</math>
where <math>\{ \dots\}</math> means "the equivalence class of the sequence <math>\dots</math> relative to our ultrafilter", two sequences being in the same class if and only if the zero set of their difference belongs to our ultrafilter.

All the arithmetical expressions and formulas make sense for hyperreals and hold true if they are true for the ordinary reals. It turns out that any finite (that is, such that <math>|x| < a</math> for some ordinary real <math>a</math>) hyperreal <math>x</math> will be of the form <math>y+d</math> where <math>y</math> is an ordinary (called standard) real and <math>d</math> is an infinitesimal. It can be proven by bisection method used in proving the Bolzano-Weierstrass theorem, the property (1) of ultrafilters turns out to be crucial.