Let \(\mathbb{N} = \{1,2,...\}\), and let \(\mathbb{R}^{\mathbb{N}}\) be the set of all sequences of real numbers. A typical member of \(\mathbb{R}^{\mathbb{N}}\) has the form \(r = \langle r_1, r_2, r_3, ... \rangle\), which may be denoted more briefly as \(\langle r_n: n \in \mathbb{N}\rangle\) or just \(\langle r_n\rangle\)·
For \(r = \langle r_n\rangle\) and \(s = \langle s_n\rangle\), put \[ \begin{align} r \oplus s &= \langle r_n + s_n: n \in\mathbb{N}\rangle \\ \\ r \odot s &= \langle r_n \cdot s_n: n \in \mathbb{N}\rangle \end{align} \] Then \(\langle \mathbb{R}^{\mathbb{N}}, \oplus, \odot\rangle\)is a commutative ring with zero \(\mathbf{0} = \langle 0, 0, 0, ... \rangle\) and unity \(\mathbf{1} = \langle 1, 1, ... \rangle\), and additive inverses given by \[ -r = \langle -r_n : n \in \mathbb{N} \rangle \] It is not, however, a field, since \[ \langle 1,0,1,0,1,...\rangle \odot \langle 0, 1, 0, 1,0,...\rangle = \mathbf{0} \] so the two sequences on the left of this equation are nonzero elements of \(\mathbb{R}^{\mathbb{N}}\) with a zero product; hence neither can have a multiplicative inverse. Indeed, no sequence that has at least one zero term can have such an inverse in \(\mathbb{R}^{\mathbb{N}}\).
Let \(\mathbb{F}\) be a fixed nonprincipal ultrafilter on the set \(\mathbb{N}\) (such exists by Corollary 2.6.2). \(\mathcal{F}\) will be used to construct a quotient ring of \(\mathbb{R}^{\mathbb{N}}\).
Define a relation \(\equiv\) on \(\mathbb{R}^{\mathbb{N}}\) by putting \[ \langle r_n \rangle \equiv \langle s_n \rangle\; iff\; \{n \in \mathbb{N}: r_n = s_n\} \in \mathcal{F} \] When this relation holds it may be said that the two sequences agree on a large set, or agree almost everywhere modulo \(\mathcal{F}\), or agree at almost all \(n\).
It is suggestive to denote the agreement set \(\{n \in \mathbb{N}: r_n = s_n\}\) by \([[ r = s ]]\), rather than \(E_{rs}\) as in Section 2.2. Thus \[ r \equiv s\; iff\; [[r = s ]] \in \mathcal{F} \] Then results like 3.3(1) and 3.3(2) can be handled by first proving properties such as those in Section 3.5 below.
The set \([[r = s]]\) may be thought of as the interpretation, or value, of the statement "\(r = s\)", or as a measure of the extent to which "\(r = s\)" is true. Normally we think of a statement as having one of two values: it is either true or false. Here, instead of assigning truth values, we take the value of a statement to be a subset of \(\mathbb{N}\). When \([[r = s]] \in \mathcal{F}\), it is sometimes said that \(r = s\) almost everywhere (modulo \(\mathcal{F}\)).
This idea can be applied to other logical assertions, such as inequalities, by defining \[ \begin{align} [[ r < s ]] &= \{n \in \mathbb{N}: r_n < s_n \},\\ [[ r > s ]] &= \{n \in \mathbb{N}: r_n > s_n \},\\ [[ r \leq s ]] &= \{n \in \mathbb{N}: r_n \leq s_n\}, \end{align} \] and so on.
The equivalence class of a sequencenr E IRN undern_ will be denoted by [r]. Thus \[ [r]={soEIRN:r s}. \] The quotient set (set of equivalence classes) of IRN by = is \[ \({}^*\mathbb{R}\) = {[r] : r E JRN}. \] Define \[ \begin{align} [r] + [s] -[rEB s] [(rn + Sn)Jo, [r] [s] [r 8 s] [(rn \end{align} \] · and \[ [r] < [s] iff [r < s] E :F iff {n EN: rn < sn} E :F. \] By 3.3(2) and 3.5(4) these notions are well-defined, which means that they are independent of the equivalence class representatives chosen to define them.
A simpler notation, which is attractive but puts some burden on the reader, is to write [rn] for the equivalence class [(rn : n EoN)] of the sequence whose nth term is rn· The definitions of addition and multiplication then read [rn] + [sn] [rn + Bn]o, [rn] · [sn] [rn · Sn]o· Theorem 3.6.1 The structure (*R, +, ·,<) is an ordered field with zero [0] and unity [1]. Proof (Sketch) As a quotient ring of JRN, \({}^*\mathbb{R}\) is readily shown to be a commutative ring with zero [OJ and unity [1], and additive inverses given by -[(r n : n E N ) ] = [(-r n : n E N ) ] , 26 3. Ultrapower Construction of the Hyperreals or more briefly, -[rn] = [-rn]· To show that it has multiplicative inverses, suppose [r] f:. [0]. Then r :/= 0, i.e., {n E N : Tn = 0} ¢. F, so as F is an ultrafilter, J = {n EN: rn f:. 0} E F. Define a sequencens by putting if n E J, otherwise. Then [r 8 s = 1] is equal to J, so [r 8 s = 1] E F, giving r 8 s 1 and hence [r] · [s] = [r 8 s] = [1] in *R. But this means that [s] is the multiplicative inverse (r]-1 of (r].
To see that the ordering < on *IR is linear, observe that N is the disjoint
union of the three sets
[r
The ring IRN is an example of what is known in algebra as a direct power of IR, a special case of the notion of direct product. An ultrapower is a quotient of a direct power that arises from the congruence relation defined by an ultrafilter.
We can identify a real number r E JR with the constant sequence r
(r, r, ... ) and hence assign to it the \({}^*\mathbb{R}\)-element
*r = [r] = [(r,or, ... )] .
It can be shown that for r, s E JR, we have
*(r + s) *r + *s,
·
*(r·s) *r *s'
*r <*s iff r
Let c = (1, , , ... ) = (-!i : n E N ) . Then
[0
The properties observed of [c] and [w] show that \({}^*\mathbb{R}\) is a proper extension
of JR, and hence a new structure. Even more directly, for any r E JR, the
set [r = w] is either 0, or equal to {r} when r E N, so cannot belong to F,
implying *r =!= [w]. Thus [w] E \({}^*\mathbb{R}\) -JR.
This argument depends crucially on the fact that F is nonprincipal. If
F were principal, then there would be some fixed J1 EN such that
F = P-={A N: '!1 E A}.
But then each sequence s E JRN would agree almost everywhere with the
sequence taking the constant value s!l, and from this it would follow that
\({}^*\mathbb{R}\) = {*r : r E JR}, and hence \({}^*\mathbb{R}\) would be isomorphic to R The details
of this are left as an exercise: the essential point is that use of a principal
ultrafilter to construct \({}^*\mathbb{R}\) does not lead to anything new.
Our discussion of c and w shows in fact that if r is any real-valued
sequence converging to zero, then [r] is an infinitesimal in *IR, while if r
diverges to oo, then [r] is unlimited in \({}^*\mathbb{R}\). Thus we have achieved the
objective proposed in Section 2.1 of building a number system with these
features.
Now that we have shown that there are infinitesimals in \({}^*\mathbb{R}\), we can begin
to apply the field operations to them to construct new numbers. What
happens for instance if we multiply or divide an infinitesimal by a positive
real number? Or by a negative real number? The general arithmetic of
hyperreals will be described in Chapter 5.
Exercise 3.8.1
Use only general properties of ordered fields to deduce from the fact that
[c] is a positive infinitesimal the conclusion that [c]-1 is greater than every
real number
A subset A of lR can be "enlarged" to a subset *A of \({}^*\mathbb{R}\): for each r E JRN,
put
[r] E *A iff {n EN: rn E A} E F.
Thus we are declaring, by the almost-all criterion, that [rn] is in *A iff rn
is in A for almost all n. Again it has to be checked that this is well-defined.
Invoking the [...] notation, put
[roEA]= {n EN: rn E A}.
Then
[r = r'] n [r E A] [r' E A],
so
r = r' & [r E A] E F implies [r' E A] E F
3.10 Exercises on Enlargement
as required. We have
[r]oE*A iff [rEA]EF.
Observe that if s E A, then [sEA] = N E F (where s = (s, s, .n.o. ) as
usual), so *s E *A. Identifying s with *s, we may regard *A as a superset of
A : A *A. Elements of *A -A may be thought of as new "nonstandard"n,
or "ideal"n, members of A that live in *R
For example, let A= N, and w = (1, 2, 3, ... ) as above. Then [woEN]=
N E F, so [w] E *N. [w] is a "nonstandard natural number".
Theorem 3.9.1 Any infinite subset of JR. has nonstandard members.
Proof Note first that this result must depend on F being nonprincipal,
because if F were principal, there would be no nonstandard elements of \({}^*\mathbb{R}\).
at all.
Now, if A JR. is infinite, then there is a sequence r of elements of A
whose terms are all distinct. Then [rEA] = N E F, so [r] E *A. But for
each s E A, { n : r n = s} is either 0 or a singleton, neither of which can
belong to F (2.5(5)), so [r] f:-*s. Hence [r] E *Ao-A. D
The converse of this theorem is also true (cf. the next exercise), so the
property of having nonstandard members exactly characterises the infinite
sets.
A function f : lR --t extends to *f : * --t *IR as follows. First, for each
sequence r E JRN, let f o r be the sequence (f( ri), f( rz), ... ) . Then put
*f([r]) = [fnor].
In other words,
or in the simplified notation,
[r = r'] [fooro= for'] ,
r -r' implies f o r -f o r',
Now, in general,
and so
ensuring that *f is well-defined. Observe that *f obeys the almost-all criterion:
*f([ r]) = [s] iff [! o r = s] E :F
iff {n EN: f(rn) = sn} E :F
iff f(r n) = Sn for almost all n.
For example, the sine function is extended to all of * by
*sin([r]n) = [(sin(r1), sin(rz), .
.. )] = [sin(rn)] .
3.8 Infinitesimals and Unlimited Numbers
3.9 Enlarging Sets
3.10 Exercises on Enlargement
3.11 Extending Functions
3.12 Exercises on Extensions
Let f A ----+ JR be a function whose domain A is a subset of JR. (e.g., f(x) = tanx). Then f extends to a functionn*!n: *An----+ \({}^*\mathbb{R}\) whose domain is the enlargement of A, i.e., dom * f = *( dom f). To define this extension, take r E JRN with [r] E *A, so that [roEA]= {n EoNn: rn E A} E F. Let Sn = { f(rn) if n E [r E A] , 0 if n [rEA] (it is enough to define Sn for almost all n). Then put *j([r]) = [s]. Essentially, we have defined as in Section 3.11, but with a modification to cater for the complication that J(rn) may not be defined for some n. The construction works because f(rn) exists for almost all n modulo F.
It is readily shown that if rEA, then */(*r) = *(f(r)), or identifying *r with r etc., we have *J(r) = f(r), so *J extends f. Therefore it would do no harm to drop the * symbol and just use f for the extension as well, and we will do so most of the time. It is a particularly natural practice for the more common mathematical functions. For instance, the function sin x is now defined for all hyperreals x E \({}^*\mathbb{R}\)..
An important case of this construction concerns sequences. A real-valued sequence is just a functionns : Nn----+ JR, and so the construction extends this to a hypersequence s : *N ----+ *R Hence the term sn is now defined even when n E *Nn-N.
Let P be a k-ary relation on JR. Thus Pis a set of k-tuples: a subset of JRk. For given sequences r1, ... , rk E JRN, define [P(r1, ... ,rk)] = {n EN: P(r, ... ,r)}. Now P can be enlarged to a k-ary relation *P on \({}^*\mathbb{R}\)., i.e., a subset of (\({}^*\mathbb{R}\).)k. For this we use the notation *P([r1], ... , [rk]) to mean that the k-tuple ([r1], ... , [rk]) belongs to * P. The definition is: *P([r1], ... , [rk]) iff [P(r\ ... , rk)] E F iff P(r, ... , r) for almost all n. As always with a definition involving equivalence classes named by particular elements, it must be shown that the notion is well-defined. In this case we can prove [r1 = s1] n · · · n [rk = sk] n [P(r\ ... , rk)] [P(s1, ... , sk)], _ so that if r1 s1 and ... and rk sk and [P(r1, ... , rk)] E F, then [P(s1, ... , sk)] _ E F. When r1, ... , rk are real numbers, P(r1, ... ,rk) iff *P(*r1, ,*rk), ..• showing that * P is an extension of P. This definition of k-ary *P encompasses the work of Sections 3.9-3.13 on extensions of sets and functions. A subset A of JR. is just a unary relation (k = 1), so the definition of *A is a special case of that of * P. When P is any of the relations =, <, >, on JR., then * P is the corresponding relation that we defined on \({}^*\mathbb{R}\)., because [r] = [s] iff [r = s] E F, [r] < [s] iff [r < s] E F, and so on. An m-ary function f : lRm --+ JR. can be identified with its ( m + 1 )-ary graph Graph f = {(r\ ... , rm, s) : J(r\ ... , rm) = s}. Then the extension of Graph f to \({}^*\mathbb{R}\). is just the graph of the extension *Jn: *!Rmn--+ *IR ofnf (Exercise 3.12(5)), i.e., *(Graph!)n= Graphn(*J). Moreover, Graph f is defined even when f is a partial function, and so that case is covered as well.
The construction of \({}^*\mathbb{R}\) as a quotient ring of JRN depends on the choice of the nonprincipal ultrafilter :F that determines the congruence =· But there are many such ultrafilters on N, as many as there are subsets of P(N)-the set of all nonprincipal ultrafilters on N is in bijective correspondence with P(P(N)).
Now, it has been shown that under a certain set-theoretic assumption called the continuum hypothesis the choice of \(\mathcal{F}\) is irrelevant: all quotients of \(\mathbb{R}^{\mathbb{N}}\) with respect to nonprincipal ultrafilters on \(\mathbb{N}\) are isomorphic as ordered fields. To explain this assumption, let us say that a set \(A\) is smaller than set \(B\), and that \(B\) is larger than \(A\), if there exists an injective function from \(A\) to \(B\), but none from \(B\) to \(A\). A famous result of Cantor is that \(\mathbb{R}\) is bigger than \(\mathbb{N}\) (and more generally that a set \(A\) is always smaller than its power set \(\mathcal{P}(A)\)). The continuum hypothesis asserts that there is no subset of \(\mathbb{R}\) that is smaller than \(\mathbb{R}\) but bigger than \(\mathbb{N}\). This implies that \(\mathbb{R}\) represents the least "infinite size" greater than the size of \(\mathbb{N}\).
The continuum hypothesis is neither provable nor disprovable from the generally accepted axioms of set theory, including the axiom of choice. Thus we can say that if we take the continuum hypothesis as an axiom, then our construction of \({}^*\mathbb{R}\) produces a unique result. Without this assumption the situation is undetermined.