Table Of ContentHyperreal Calculus
MAT2000 –– Project in Mathematics
Arne Tobias Malkenes Ødegaard
Supervisor: Nikolai Bjørnestøl Hansen
Abstract
Thisprojectdealswithdoingcalculusnotbyusingepsilonsanddeltas,but
by using a number system called the hyperreal numbers. The hyperreal
numbers is an extension of the normal real numbers with both infinitely
small and infinitely large numbers added. We will first show how this
systemcanbecreated,andthenshowsomebasicpropertiesofthehyperreal
numbers. Then we will show how one can treat the topics of convergence,
continuity, limits and differentiation in this system and we will show that
the two approaches give rise to the same definitions and results.
Contents
1 Construction of the hyperreal numbers 3
1.1 Intuitive construction . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Ultrafilters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Formal construction . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Infinitely small and large numbers . . . . . . . . . . . . . . . . . 5
1.5 Enlarging sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.6 Extending functions . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 The transfer principle 6
2.1 Stating the transfer principle . . . . . . . . . . . . . . . . . . . . 6
2.2 Using the transfer principle . . . . . . . . . . . . . . . . . . . . . 7
3 Properties of the hyperreals 8
3.1 Terminology and notation . . . . . . . . . . . . . . . . . . . . . . 8
3.2 Arithmetic of hyperreals . . . . . . . . . . . . . . . . . . . . . . . 9
3.3 Halos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.4 Shadows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4 Convergence 11
4.1 Convergence in hyperreal calculus. . . . . . . . . . . . . . . . . . 11
4.2 Monotone convergence . . . . . . . . . . . . . . . . . . . . . . . . 12
5 Continuity 13
5.1 Continuity in hyperreal calculus. . . . . . . . . . . . . . . . . . . 13
5.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5.3 Theorems about continuity . . . . . . . . . . . . . . . . . . . . . 15
5.4 Uniform continuity . . . . . . . . . . . . . . . . . . . . . . . . . . 16
6 Limits and derivatives 17
6.1 Limits in hyperreal calculus . . . . . . . . . . . . . . . . . . . . . 17
6.2 Differentiation in hyperreal calculus . . . . . . . . . . . . . . . . 18
6.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
6.4 Increments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
6.5 Theorems about derivatives . . . . . . . . . . . . . . . . . . . . . 19
1
1 Construction of the hyperreal numbers
1.1 Intuitive construction
We want to construct the hyperreal numbers as sequences of real numbers
hr i = hr ,r ,...i, and the idea is to let sequences where lim r = 0
n 1 2 n→∞ n
represent infinitely small numbers, or infinitesimals, and let sequences where
lim r =∞ represent infinitely large numbers.
n→∞ n
However, if we simply let each hyperreal number be defined as a sequence
of real numbers, and let addition and multiplication be defined as elementwise
additionandmultiplicationofsequences,wehavetheproblemthatthisstructure
is not a field, since
h1,0,1,0,...i(cid:12)h0,1,0,1,...i=h0,0,0,0,...i.
The way we solve this is by introducing an equivalence relation on the set of
real-valued sequences. We want to identify two sequences if the set of indices for
which the sequences agree is a large subset of N, for a certain technical meaning
of large. Let us first discuss some properties we should expect this concept of
largeness to have.
• N itself must be large, since a sequence must be equivalent with itself.
• If a set contains a large set, it should be large itself.
• The empty set ∅ should not be large.
• We want our relation to be transitive, so if the sequences r and s agree on
a large set, and s and t agree on a large set, we want r and t to agree on a
large set.
1.2 Ultrafilters
Our model of a large set is a mathematical structure called an ultrafilter.
Definition 1.1 (Ultrafilters). We define an ultrafilter on N, F, to be a set of
subsets of N such that:
• If X ∈ F and X ⊆ Y ⊆ N, then Y ∈ F. That is, F is closed under
supersets.
• If X ∈F and Y ∈F, then X∩Y ∈F. F is closed under intersections.
• N∈F, but ∅6∈F.
• For any subset A of N, F contains exactly one of A and N\A.
We say that an ultrafilter is free if it contains no finite subsets of N. Note that a
free ultrafilter will contain all cofinite subsets of N (sets with finite complement)
due to the last property of an ultrafilter.
Theorem 1.2. There exists a free ultrafilter on N.
Proof. See [Kei76, p. 49]. (cid:4)
2
1.3 Formal construction
Let F be a fixed free ultrafilter on N. We define a relation ≡ on the set of
real-valued sequences RN by letting
hr i≡hs i ⇐⇒ {n∈N|r =s }∈F.
n n n n
Proposition 1.3 (Equivalence). The relation ≡ is an equivalence relation on
RN.
Proof. We check all needed properties of an equivalence relation.
Reflexivity Since the set {n∈N|r =r }=N, and N∈F, ≡ is reflexive.
n n
Symmetry The sets {n∈N|r =s } and {n∈N|s =r } are the same, so
n n n n
if one belongs to F, so does the other.
Transitivity Assume that hr i ≡ hs i and hs i ≡ ht i. Then both {n ∈ N |
n n n n
r =s }∈F and {n∈N|s =t }∈F. Since {n∈N|r =s }∩{n∈
n n n n n n
N|s =t }⊆{n∈N|r =t }, and F is closed under intersections and
n n n n
supersets, {n∈N|r =t }∈F, and so hr i≡ht i, as desired.
n n n n
(cid:4)
Since ≡ is an equivalence relation, we can define the set of hyperreal numbers
∗R as the set of real-valued sequences modulo the equivalence relation ≡. In
symbols,
∗R={[r]|r ∈RN}=RN/≡.
We define addition and multiplication of elements in ∗R by doing elementwise
addition and multiplication in the related sequences, more formally as
[r]+[s]=[hr i]+[hs i]=[hr +s i]
n n n n
[r]·[s]=[hr i]·[hs i]=[hr ·s i].
n n n n
We define the ordering relation < by letting
[r]<[s] ⇐⇒ {n∈N|r <s }∈F.
n n
At this point, let us introduce some notation to make our arguments easier
to read. For two sequences hr i and hs i, we denote the agreement set {n∈N|
n n
r =s } by r =s . We can apply the same notation to other relations, so for
n n
example we h(cid:74)ave r(cid:75)<s ={n∈N|r <s }.
n n
(cid:74) (cid:75)
Proposition 1.4. Theoperations+and·arewell-defined, andsoistherelation
<.
Proof. We first show that + is well-defined. If we have that hr i ≡ hr0i and
n n
hs i ≡ hs0 i, then r = r0 ∈ F and s = s0 ∈ F, which means that r =
n n
r0 ∩ s = s0 ∈ F.(cid:74)What w(cid:75)e now need(cid:74) to sho(cid:75)w is that r+s = r0+s0 (cid:74)∈ F.
If,(cid:75)for(cid:74)some(cid:75)k ∈ N, both r = r0 and s = s0, then r +(cid:74) s = r0 +s0,(cid:75)hence
k k k k k k k k
if k ∈ r = r0 ∩ s = s0 , then k ∈ r + s = r0 + s0 , which shows that
r = r0(cid:74)∩ s =(cid:75)s0 (cid:74)⊆ r+(cid:75)s = r0 +s0 .(cid:74)Since r = r0 ∩(cid:75) s = s0 ∈ F, so is
(cid:74)r+s=(cid:75) r0(cid:74)+s0 . S(cid:75)o if(cid:74)r ≡r0 and s≡s(cid:75)0, r+s≡(cid:74)r0+s0(cid:75), wh(cid:74)ich sho(cid:75)ws that the
(cid:74)operation is wel(cid:75)l-defined. Showing that · is well-defined is similar.
3
We will now show that < is well-defined, which means that we need to show
that if hr i ≡ hr0i and hs i ≡ hs0 i, then if r < s ∈ F, then r0 < s0 ∈ F.
n n n n
Firstly, assumethat r =r0 ∈F andthat s=(cid:74) s0 ∈(cid:75)F. Then, we(cid:74)needto(cid:75)prove
that if r <s ∈F t(cid:74)hen r(cid:75)0 <s0 ∈F. So(cid:74)let us(cid:75)assume that r <s ∈F, and
then pr(cid:74)ove th(cid:75)at r0 <s0 (cid:74)∈F. (cid:75) (cid:74) (cid:75)
By our assum(cid:74)ptions(cid:75), we have that r = r0 ∩ s = s0 ∩ r < s ∈ F. If
k ∈ r = r0 ∩ s = s0 ∩ r < s , then(cid:74) r =(cid:75)r0,(cid:74)s = s(cid:75)0 an(cid:74)d r <(cid:75) s , and
k k k k k k
there(cid:74)fore r0 (cid:75)<s(cid:74)0, so k(cid:75)∈ r(cid:74)0 <s0 (cid:75). So, r =r0 ∩ s=s0 ∩ r <s ⊆ r0 <s0 ,
k k
and since F is closed und(cid:74)er supe(cid:75)rsets,(cid:74)we con(cid:75)clu(cid:74)de that(cid:75) r(cid:74)0 < s0(cid:75)∈ F(cid:74), which(cid:75)
shows that < is well-defined. (cid:74) (cid:75) (cid:4)
1.4 Infinitely small and large numbers
One of the main reasons for constructing the hyperreals is that we want to have
access to infinitely large and infinitely small numbers, and now we can prove
their existence.
Theorem 1.5. There exists a number ε ∈ ∗R such that 0 < ε < r for any
positive real number r, and there exists a number ω ∈ ∗R such that ω > r for
any real number r.
Proof. First, we need to talk about real numbers in ∗R. The way to do this is
that given a real number r ∈R, we can identify this with a hyperreal number
∗r ∈ ∗R as ∗r = hr,r,...i. We will generally omit the ∗-decoration, and simply
refer to this number as r.
Now, let us turn to the actual proof. Let ε=(cid:2)h1,1,...i(cid:3)=(cid:2)h1i(cid:3). For any
2 n
positive real number r, the set {n ∈ N | 1 > r} must be finite, and therefore
n
{n∈N| 1 <r}iscofinite,andhencebelongstoourfreeultrafilterF. Therefore,
n
we can conclude that ε<r. Also, since {n∈N|0< 1}=N∈F, it must be
n
the case that 0 < ε. So the number ε is a hyperreal number which is greater
than 0, but smaller than any positive real number.
Letω =[h1,2,...i]=[hni]. Foranyrealnumberr, theset{n∈N|r ≥n}is
finite, andhence{n∈N|r <n}iscofinite, andbelongstoF, whichmeansthat
ω >r. This proves that ω is a hyperreal number greater than any real number.
(cid:4)
1.5 Enlarging sets
For a given subset A of R we can define an “enlarged” subset ∗A of ∗R by saying
that a hyperreal number r is an element in ∗A if and only if the set of n such
that r is an element in A is large. Formally this can be defined as
n
[r]∈∗A ⇐⇒ {n∈N|r ∈A}∈F.
n
Again, we need to check that this is well-defined. Using the ... notation,
let r ∈A ={n∈N|r ∈A}. We have that (cid:74) (cid:75)
n
(cid:74) (cid:75)
r =r0 ∩ r ∈A ⊆ r0 ∈A ,
(cid:74) (cid:75) (cid:74) (cid:75) (cid:74) (cid:75)
so if r ≡r0 and r ∈A ∈F, then r0 ∈A ∈F, which shows that enlargements
are well-defined.(cid:74) (cid:75) (cid:74) (cid:75)
4
An example of this is if A=N and ω =h1,2,3,...i. Then ω ∈N =N∈F,
soω ∈∗N. Wewillrefertotheset∗Nasthehypernaturals. Simi(cid:74)larly,if(cid:75)A=(0,1)
and r =h0.9,0.99,0.999,...i. Then r ∈N =N∈F, so r ∈∗(0,1).
(cid:74) (cid:75)
1.6 Extending functions
An important tool in non-standard analysis is to take a function f: R → R
and extend it to a function ∗f: ∗R→∗R. This is done by applying the function
to each element in the sequence representing the given hyperreal number. We
define the extension as follows:
∗f([hr ,r ,...i])=[hf(r ),f(r ),...i].
1 2 1 2
Again, we need to prove that this is well-defined. First, let f ◦r denote
hf(r ),f(r ),...i. In general, r =r0 ⊆ f ◦r =f ◦r0 , and so if r ≡r0, then
1 2
∗f(r)=f ◦r ≡f ◦r0 =∗f(r0).(cid:74)Hence(cid:75)the(cid:74)function is we(cid:75)ll-defined.
A function f: A→R defined on some subset A of R can also be extended to
a function ∗f: ∗A→∗R, but not in exactly the same way as above. Since r can
be in ∗A without all elements of r being in A, there can be indices i for which
f(r ) is not defined. In order to get around this, we let f(r ) = 0 whenever
i i
r 6∈A. More formally, let
i
(
f(r ) if r ∈A
s = n n
n 0 otherwise
and define
∗f([hr i])=[hs i].
n n
Sincewehavethat∗f(r)=f(r)wheneverr ∈A, ∗f extendsf. Thereforewewill
often simply drop the ∗-decoration, and simply refer to the extended function as
f as well.
An important subject related to this construction is sequences. A sequence
hs ,s ,...i is simply a function s: N→R, and so by this construction can be
1 2
extendedtoahypersequence s: ∗N→∗R,whichmeansthattheterms isdefined
n
even when n∈∗N\N.
2 The transfer principle
2.1 Stating the transfer principle
Oneofthemostimportanttoolsofnon-standardanalysisisthetransferprinciple,
a way to show that a certain type of statement is true when talking about the
realnumbersifandonlyifacertainrelatedstatementistruewhentalkingabout
the hyperreal numbers. 1
First, we introduce the set of sentences which the transfer principle applies
to. This set is basically the set of all sentences (formulas with no free variables)
in a language of first-order logic which consists of a constant for each real
number, a function symbol for each real function, and a relation symbol for each
1This is a rather cursory introduction to the tranfer principle. For a more in-depth
explanation,see[Gol98,pp.35-47].
5
relationonthereals. However,insteadofusingthequantifiers(∀x)and(∃y),our
sentences use quantifiers of the form (∀x∈A) and (∃y ∈B) where A and B are
subsets of R. Some examples of such sentences are (∀n∈N)(∃m∈N)(m>n),
(∃x ∈ R)(∀y ∈ R)(x+y = y) which state respectively that there is no biggest
natural number and there is an additive identity for the reals. Let us call such a
sentence an L-sentence.
Now, we define the ∗-transform of an L-sentence. We take a sentence ϕ, and
create a related sentence ∗ϕ. An L-sentence ϕ contains symbols P, f, and r for
relations, functions, and constants on R. To create ∗ϕ, we replace P by ∗P for
all relations P, replace f by ∗f for all functions f, and replace r by ∗r for all
constants r. Some examples of this are:
• The ∗-transform of the sentence (∀n ∈ N)(∃m ∈ N)(m > n) is (∀n ∈
∗N)(∃m∈∗N)(m∗>n).
• The ∗-transform of (∀x∈R)(sin(x)<2) is (∀x∈∗R)(∗sin(x)∗<∗2).
Wewillgenerallyfollowtheconventionsthatweomitthe∗forconstants,most
functions, and simple equalities and inequalities. With these conventions, the
above sentences become (∀n∈∗N)(∃m∈∗N)(m>n) and (∀x∈∗R)(sin(x)<2).
Now we state the transfer principle, which we will take as true without proof.
Theorem 2.1 (Transfer principle). An L-sentence ϕ is true if and only if its
∗-transform ∗ϕ is true.
Some remarks are in order. It is worth pointing out that one can go in both
directions, that is one can go from R to ∗R, and from ∗R to R. If one decides to
go in this last direction, it is important that the statement is the ∗-transform of
an L-sentence, so for example it can contain no hyperreal constants. A way to
get around this is by replacing the constant with a variable x, and adding the
quantifier (∃x∈∗A) for some A⊆R in front, which is a technique we will use.
In many cases, we will not explicitly write down the full sentence, but rather
state things like “since s<n for all natural n, by transfer it also also true for
any hypernatural n”.
2.2 Using the transfer principle
Theorem 2.2. The structure h∗R,+,·,<i is an ordered field with zero and unity.
Proof. The way we prove this is by using the transfer principle. We take the
fact that R is an ordered field as true. This can be stated by a number of logical
sentences. The fact that addition is commutative in R can be expressed as the
sentence (∀x,y ∈ R)(x+y = y+x), and so by the transfer principle, we can
conclude that (∀x,y ∈ ∗R)(x+y = y+x), and so addition is commutative in
∗R. We leave out the full details, but this procedure can then be done for all
the axioms for ordered fields (since they are all first-order axioms), and so we
conclude that h∗R,+,·,<i is also an ordered field. (cid:4)
Remark. One important property of the standard real numbers is that they are
complete, that is any subset of R which is non-empty and bounded above has
a least upper bound. The reason for why this cannot be proven to hold for ∗R
is that this can only be expressed using second-order logic, since you need to
6
talk about subsets of R, not just elements of R. In fact, ∗R is not complete. An
example of this is that the open interval of real numbers (0,1) does not have a
least upper bound in ∗R.
Proposition 2.3. For any two subsets A and B of R, we have that
• ∗(A∪B)=∗A∪∗B
• ∗(A∩B)=∗A∩∗B
• ∗(A\B)=∗A\∗B.
Proof. We prove the statement about unions, but the other two statements can
be proven similarly. The statement (∀x ∈ R)(x ∈ (A∪B) ↔ x ∈ A∨x ∈ B)
is true for any two subsets A and B of R, basically by the definition of unions.
Using the transfer principle, the statement (∀x ∈ ∗R)(x ∈ ∗(A∪B) ↔ x ∈
∗A∨x∈∗B) is also true. We also have that for any two subsets X and Y of ∗R,
(∀x∈∗R)(x∈(X∪Y)↔x∈X∨x∈Y). Combining these last two statements,
lettingX =∗AandY =∗B,wegetthat(∀x∈∗R)(x∈∗(A∪B)↔x∈(∗A∪∗B)),
which shows that ∗(A∪B)=∗A∪∗B. (cid:4)
Remark. It is worth noting that ∗(cid:0)S A (cid:1) does not need to be equal to
n∈N n
(cid:0)S ∗A (cid:1). If A ={n}, then ∗(cid:0)S A (cid:1)=∗N, but (cid:0)S ∗A (cid:1)=N.
n∈N n n n∈N n n∈N n
3 Properties of the hyperreals
3.1 Terminology and notation
At this point we introduce some terminology and notation for talking about
hyperreal numbers. We say that a hyperreal number b is:
• limited if r <b<s for some r,s∈R,
• positive unlimited if r <b for all r ∈R,
• negative unlimited if b<r for all r ∈R,
• unlimited if it is positive or negative unlimited,
• positive infinitesimal if 0<b<r for all positive r ∈R,
• negative infinitesimal if r <b<0 for all negative r ∈R,
• infinitesimal if it is positive infinitesimal, negative infinitesimal or 0,
• appreciable if it is limited but not infinitesimal.
Wewillusethetermslimitedandunlimited,ratherthanfiniteandinfinite,when
referring to individual numbers. Finite and infinite are terms we use for sets
only.
For any subset X of ∗R, we define X = {x ∈ X | x is unlimited}, X+ =
∞
{x ∈ X | x > 0}, and X− = {x ∈ X | x < 0}. These notations can also be
combined, and so X+ denotes all positive unlimited members of X.
∞
7
3.2 Arithmetic of hyperreals
Whenreasoningabouthyperreals, itisusefultohavecertainrulesforcomputing
them, for example that the sum of two infinitesimals is itself infinitesimal. Here
are some such rules for computing with hyperreal numbers. If ε and δ are
infinitesimals, b and c are appreciable, and H and K are unlimited, then:
• ε+δ is infinitesimal,
• b+ε is appreciable,
• H +ε and H +b are unlimited,
• b+c is limited,
• −ε is infinitesimal,
• −b is appreciable,
• −H is unlimited,
• ε·δ and ε·b are infinitesimal,
• b·c is appreciable,
• b·H and H ·K are unlimited,
• 1 is unlimited if ε6=0,
ε
• 1 is appreciable,
b
• 1 is infinitesimal,
H
• ε, ε and b are infinitesimal,
b H H
• b is appreciable,
c
• b, H and H are unlimited if ε6=0.
ε ε b
We do not give a proof for any of these rules, but they can be proven by using
the transfer principle, or by reasoning about sequences of reals.
The following expressions do not have such a rule, and can all take on
infinitesimal, appreciable, and unlimited values: ε, H, ε·H, H +K.
δ K
3.3 Halos
A hyperreal b is said to be infinitely close to a hyperreal c if b−c is infinitesimal,
and this is denoted by b’c. This defines an equivalence relation on ∗R, and we
define the halo of b to be the ’-equivalence class
hal(b)={c∈∗R|b’c}.
Said differently, the halo of b is the set of all hyperreals which are infinitely close
to b.
Proposition 3.1. If two real numbers b and c are infinitely close, that is if
b’c, then b=c.
Proof. Suppose that b ’ c with b and c real, but that b 6= c. Then there is a
non-zero real number r such that b−c=r. But this contradicts the assumption
that b’c, since r is not an infinitesimal. (cid:4)
8
Proposition 3.2. Suppose that b and c are limited, and that b’b0 and c’c0.
Then b±c’b0±c0 and b·c’b0·c0. Furthermore, if c6’0, then b/c’b0/c0.
Proof. From our assumptions, we have that b−b0 =ε and c−c0 =ε , with ε
b c b
and ε being infinitesimal. It is also the case that both b0 and c0 are limited.
c
We want to show that b±c ’ b0 ±c0, and this is done by showing that
(b±c)−(b0±c0) is infinitesimal. We have that
(b±c)−(b0±c0)=(b−b0)±(c−c0)=ε ±ε .
b c
Since both the sum of and the difference between two infinitesimals is itself
infinitesimal by Section 3.2, we have that (b±c)−(b0±c0) is infinitesimal, and
hence that b±c’b0±c0.
The case b·c’b0·c0 is proven similarly. We have that
b·c−b0·c0 =b·c−b·c0+b·c0−b0·c0
=b·(c−c0)+(b−b0)·c0
=b·ε +ε ·c0
c b
whichisinfinitesimalsincetheproductofalimitednumberwithaninfinitesimalis
infinitesimalandthesumoftwoinfinitesimalsisinfinitesimal. Henceb·c’b0·c0.
For the last case we have that
b b0 b·c0−b0·c
− =
c c0 c·c0
b·c0−b·c+b·c−b0·c
=
c·c0
b·(c−c0)+c·(b−b0)
=
c·c0
c·ε −b·ε
= b c.
c·c0
Now, ifc6’0, thedenominatoristheproductoftwoappreciablenumbers, which
is also appreciable. Since the numerator is infinitesimal by a similar argument to
thecaseofproducts,thequotientisitselfinfintesimal,andhenceb/c’b0/c0. (cid:4)
Remark. The first part of the prosition, namely that b±c’b0±c0, holds also
for unlimited b and c, but the other parts do not. To show this, let H be some
positive unlimited number, and let b0, c and c0 equal H, and let b equal H + 1.
H
Then b’b0 and c’c0, but
(cid:18) 1 (cid:19)
b·c−b0·c0 = H + ·H −H ·H =H2+1−H2 =1,
H
which is not infinitesimal, and so b·c 6’ b0·c0. A similar counterexample can
also be produced for b/c.
3.4 Shadows
Theorem 3.3 (Existence of shadows). Every limited hyperreal b is infinitely
close to one and only one real number s. This real number is called the shadow
of b, which is denoted by sh(b).
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