:Members of *R are called hyperreal numbers, while members of R are real
and sometimes called standa·rd. *Q consists of hyperrationals, *Z of hyperintegers,
and *N of hypernaturals. That *Q consists precisely of quotients
mjn of hyperintegers m, n E *Z follows by transfer of the sentence
'r/x E R [x E Q +-+ :3y, z E Z (z =/= 0 ;\ x = yj z)].
It is now time to examine the basic arithmetical and algebraic structure of
xlR, particularly in its relation to the structure of R.
(Un)limited, Infinitesimal, and Appreciable
Numbers
A hyperreal number b is:
•
limited if r < b < s for some r, s E JR;
• positive unlimited if r < b for all r E R;
•
negative unlimited if b < r for all r E R;
•
unlimited if it is positive or negative unlimited;
• positive infinitesimal if 0 < b < r for all positive r E R;
•
negative infinitesimal if r < b < 0 for all negative r E R;
50 5. Hyperreals Great and Small
• infinitesimal if it is positive infinitesimal, negative infinitesimal, or 0.
• appreciable if it is limited but not infinitesimal, i.e., r < lbl < s for
some r, s E +.
Thus all real numbers, and all infinitesimals, are limited. The only infinitesimal
real is 0: all other reals are appreciable. An appreciable number is one
that is neither infinitely small nor infinitely big. Observe that b is
• limited iff lbl < n for some n EN;
• unlimited iff lbl > n for all n EN;
• infinitesimal iff lbl < for all n E N;
• appreciable iffn < lbl < n for some n EN.
We denote the set *N-N of unlimited hypernaturals by *N00nSimilarly, *Ill
•
denotes the set of positive unlimited hyperreals, and *Ill the set of negative
unlimited numbers. This notation may be adapted to an arbitary subset
X of *Ill, putting Xoo = {x EX :X is unlimited}, x+ = {x EX: X> 0},
etc.
We will also use IL for the set of all limited numbers, and TI for the set of
infinitesimals.
5.2 Arithmetic of Hyperreals
Let c, 0 be infinitesimal, b, c appreciable, and H, K unlimited. Then
• Sums:
c + 0 is infinitesimal
b + c is appreciable
b + c is limited (possibly infinitesimal)
H + c and H + b are unlimited
• Opposites:
-c is infinitesimal
-b is appreciable
-H is unlimited
• Products:
c · 8 and c · b are infinitesimal
b · c is appreciable
b ·Hnand H · K are unlimited
5.3 On the Use of "Finite" and "Infinite"
• Reciprocals:
is unlimited if c =/= 0
i is appreciable
Jr is infinitesimal
• Quotients:
c E
•
d bH are mfi 'tm es•1ma1
b' H' an
is appreciable (if c =/= 0)
*' ,and I[ are unlimited (c, b =/= 0)
• Roots:
If c > 0, y'c is infinitesimal
If b > 0, \lb is appreciable
If H> 0, f/H is unlimited
• Indeterminate Forms:
, ,c·H,H+K
It follows from these rules that the set lL of limited numbers and the set ][
of infinitesimals are each a subring of *JR. Also, the infinitesimals form an
ideal in the ring of limited numbers. What then is the associated quotient
ring IL/][? Read on to Theorem 5.6.3.
With regard to nth roots, for fixed n E N the function x y'X is defined
for all positive reals, so extends to a function defined for all positive hyperreals.
But we could also consider nth roots for unlimited n. The statement
(Vn E N) (Vx E JR+) (3y E JR) (yn = x)
asserts that any positive real has a real nth root for all n EN. Its transform
asserts that every hyperreal has a hyperreal nth root for all n E *N.
Exercise 5.2.1
For any positive hyperreal a, explain why the function x ax is defined
for all x E *JR. Use transfer to explore its properties.
5.3 On the Use of "Finite" and "Infinite"
The words "finite" and "infinite" are sometimes used for "limited" and
"unlimited"n, but this does not accord well with the philosophy of our subject.
A set is regarded as being finite if it has n elements for some n E N,
and therefore is in bijective correspondence with the set
{1,n2, ... ,n} ={kEN: kn:::; n}.
52 5. Hyperreals Great and Small
However, if N is an unlimited hypernatural, then the collection
{1, 2, ... , N} = {k E *N: k N}
is set-theoretically infinite but, by transfer, has many properties enjoyed
by finite sets. Collections of this type are called hyperfinite, and will be examined
fully later. They are fundamental to the methodology of hyperreal
analysis.
There is also potential conflict with other traditional uses of the word
"infinite" in mathematics, such as in describing a series or an integral as
being infinite when it it is divergent or undefined, or in referring to the area
or volume or some more general measure of a set as being infinite when
this has nothing to do with unlimited hyperreals.
5.4 Halos, Galaxies, and Real Comparisons
• Hyperreal b is infinitely close to hyperreal c, denoted by b c, if b-c
is infinitesimal. This defines an equivalence relation on *, and the
halo of b is the -equivalence class
hal (b) = { c E * : b c}.
• Hyperreals b, c are of limited distance apart, denoted by b rv c, if b-c
is limited. The galaxy of b is the rv-equivalence class
gal(b) = {c E *: b rv c}.
So, b is infinitesimal iff b 0, and limited iff b rv 0. Thus hal(O) = II, the
set of infinitesimals, while gal(O) = L, the set of limited hyperreals.
Abraham Robinson called hal(b) the "monad" of bnand used the notation
J.L(b), which is quite common in the literature. The more evocative name
"halo" has been popularised by a French school of nonstandard analysis,
founded by George Reeb, which is also responsible for "shadow" (see Section
5.6). The work of this school is described in the book listed as item 10
in the bibliography of Chapter 20.
5.5 Exercises on Halos and Galaxies
(1) Verify that and rv are equivalence relations.
(2) If b"' x y c with bnand cnreal, show that b c. What if bnand/or
c are not real?
(3) hal(b) = {b + c: c E hal(O)}.
5.6 Shadows
(4) gal(b) = {b +c: c E gal(O)}.
(5) If x y and b is limited, prove that b · xo"' b · y. Show that the result
can fail for unlimited b.
(6) Show that any galaxy contains members of *Z, of *Qn-*Z, and of
*JRn-*Q.
Real Comparisons
Exercise (2) above embodies an important general principle, which we will
often use, about comparing the sizes of two real numbers b, c. If b > c,
then the halos of the two numbers are disjoint, with everything in hal(b)
greater than everything in hal(c). Thus to show that b :::; cit is enough to
show that something in hal(b) is less than or equal to something in hal(c).
In particular, this will hold if there is some x with either b x :::; c or
b ::S X'"'"' C.
5.6 Shadows
Theorem 5.6.1 Every limited hyperreal b is infinitely close to exactly one
real number, called the shadow of b, denoted by sh(b).
Proof Let A = {r E 1R : r < b}. Since b is limited, there exist real r, s
with r < b < s, so A is nonempty and bounded above in 1R by s. By the
completeness of .!R, it follows that A has a least upper bound c E JR.
To show b c, take any positive real EE JR. Since cis an upper bound of
A, we cannot have c + E E A; hence bo:::; c +E. Also, if bn:::; c-E, then c-c
would be an upper bound of A, contrary to the fact that c is the smallest
such upper bound. Hence b ic-E. Altogether then, c-c < bo:::; c + E, so
lb-cl :::; E. Since this holds for all positive real E, b is infinitely close to c.
Finally, for uniqueness, if b c' E JR, then as b c, we get c ,......, c', and
so c = c', since both are real. D
Theorem 5.6.2 If b and c are limited and n EN, then
(1) sh(b ±c)U= sh(b) ± sh(c),
(2) sh(b ·c)U= sh(b) · sh(c),
(3) sh(b/c) = sh(b)/sh(c) if sh(c) =1-0 (i.e., if c is appreciable},
(4) sh(bn) = sh(b)nn,
(5) sh(lbl) = ish(b)nl,
(6) sh( vb) = y!Sii(b) if b 0,
54 5. Hyperreals Great and Small
(7) if b c then sh(b) sh(c).
Proof Exercise. D
We see from these last facts that the shadow map sh : b ---+ sh(b) is an
order-preserving homomorphism from the ring lL of limited numbers onto
"-J
IR. The kernel of this homomorphism is the set {b E lL : sh(b) 0} of
infinitesimals, and the cosets of the kernel are the halos hal(b) for limited
b (cf. Exercise 5.5(3)). Thus we have an answer to our question about the
quotient of lL by IT:
Theorem 5.6.3 The quotient ring 1LIIT is isomorphic to the real number
field IR by the correspondence hal(b) ---+ sh(b). Hence ][ is a maximal ideal
of the ring lL. D
The shadow sh(b) is often called the standard part of b.
5.7 Exercises on Infinite Closeness
(1) Show that if b, cnare limited and b"" b', c c', then b ± c b' ± c',
b · c b' · c', and blc b' I c' if c ':f:. 0. Show that the last result can
fail when c 0.
(2) If c is infinitesimal, show that
sine
cos c "-J 1,
tanc 0,
rv
sinclc 1,
(cos c -1) 1 c o
(use transfer of standard properties of trigonometric functions).
(3) Show that every hyperreal is infinitely close to some hyperrational
number.
(4) Show that IRis isomorphic to the ring of limited hyperrationals *QnlL
factored by its ideal *Q n ][ of hyperrational infinitesimals.
5.8 Shadows and Completeness
We saw in the proof of Theorem 5.6.1 that the existence of shadows of
limited numbers follows from the Dedekind completeness of IR. In fact,
their existence turns out to be an alternative formulation of completeness,
as the next result shows.
5.9 Exercise on Dedekind Completeness 55
Theorem 5.8.1 The assertion "every limited hyperreal is infinitely close to a real number" implies the completeness of JR.
Proof Letnsx: N IR be a Cauchy sequence. Recall that this means that
its terms get arbitrarily close to each other as we move along the sequence.
In particular, there exists a k E N such that all terms of s beyond Sk are
within a distance of 1 of each other, i.e.,
Vm,n E N(m,n k xIsmx-snl < 1)
is true. Hence the *-transform of this sentence is also true, and applies to
the extended hypersequence (Sn : n E *N) as defined in Section 3.n13. In
particular, if we takenN to be an unlimited member of *N, then k, N k,
so
lskx-sNI < 1,
and therefore sN is limited. By the assertion quoted in the statement of
the theorem, it follows that sN L for some L E JR. We will show that the
original sequencens convergs to the real number L.
If cis any positive real number, then again, since sis Cauchy, there exists
f E N such that beyond SJe all terms are within c of each other:
Vm,n E N(m,n ic: Ism-snl