Editing Hyperreal number (section)

=== 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.