What Are Numbers?

Introduction

When doing mathematics,  we usually take for granted what natural numbers, integers, and rationals are. They are pretty intuitive.   Going from rational numbers to reals is more complicated.   The easiest way at the start is probably infinite decimals.  Dedekind Cuts can be used to get a bit more fancy.  A Dedekind cut is a partition of the rational numbers into two sets, A and B, such that all elements of A are less than all elements of B, and A contains no largest rational number.  It corresponds to a point on the natural number line.  A is all the rationales to the left, and B is all the rationales to the right, including the point if it is rational.

But little niggles remain.

Cracks In Mathematics

Everyone has probably seen why x = .9999999….. = 1.  It’s simple.  10*.99999999999…. = 9.9999999…. 10*x – x = 9*x = 9 or x=1.   Everything sweet.   As an exercise, go to youtube, and you will see many videos on why it is true or not true.   The reason the debate rages is exactly what do we mean by .99999999…..?  Formally it is a decimal point followed by an infinite number of 9’s.  Exactly why is the algebra that was done valid? If .99999….. means .9 + .09 + .009 ….., using the idea of limit, then indeed .99999999…. = 1 because it can be made arbitrarily close to 1.   But it is just as reasonable to think of it as a sequence .9 .99 .999 .9999 ………   Under that interpretation, as we will see, it is not one but is infinitesimally close to 1.   Infinitesimally close?   What does that mean?   Read on, and all will be revealed.

Here is another.  1 – 1 + 1 – 1 ……..  = 1/2.   But when you add it up, you get 0, 1, 0, 1. ……..   How could it ever be 1/2?   The answer is the same – what exactly do we mean by 1 – 1 + 1 -1 ………?  Using reasoning similar to .99999….., S = 1 – 1 + 1 – 1 ……   S = 1-S.  2*S = 1.  S=1/2.  Under one interpretation, it is 1/2; under another, it has no answer.   What is going on is explained by complex analysis and will not be examined here.   The point is unless care is taken, problems can occur.

The Attempted Fix

One way to prevent cracks like this is to be careful with your reasoning.   This trend started in the 19th century with the development of a subject called Analysis.   As researchers went deeper and deeper into the foundations of mathematics, it looked for a while like Nirvana had been reached.   Frege was about to publish a book that he thought put mathematics on a firm foundation – The Foundations of Arithmetic.   But just before it was published, disaster struck.   Bertrand Russell, a young mathematician working on the logical foundations of mathematics, came up with Russell’s Paradox.   He is usually thought a philosopher.   But he was more than that – having sat for the Tripos, where he was the commendable seventh wrangler.  But his interests were in mathematical logic and gradually morphed more into philosophy.

It goes like this.   Let X be the set of all sets that do not have themselves as members.   Is X ∈ X?  Then X contains itself as a member.  Contradiction.  Is X ∉ X?  But that means it is a set that not does have itself as a member, so X ∈ X.  Contradiction.  ‘Trouble, Right here in River City. With a capital “T” that rhymes with “P” and stands for pool. We’ve surely got trouble. Right here in River City. Gotta figure out a way to keep the young ones moral after school’.   Maybe best not to teach them set theory?

Russell To The Rescue

Russell was worried.  He worked for years with his doctoral advisor, Whitehead, and finally came up with his magnum opus – Principia Mathematica.  Some idea of the scope and comprehensiveness of the Principia can be gleaned from the fact that it takes over 360 pages to prove definitively that 1 + 1 = 2.  Today, it is considered one of the most important and seminal works in logic since Aristotle’s “Organon”.  It seemed remarkably successful and resilient in its ambitious aims and soon gained world fame for Russell and Whitehead.  Indeed, only Gödel’s 1931 incompleteness theorem finally showed that the Principia could not be consistent and complete.

Despite Godel’s fatal blow against its aim, it showed that set theory, developed by Cantor, needed more rules.  The so-called “axiom of infinity” guarantees the existence of at least one infinite set, namely the set of all natural numbers.   The “axiom of choice” ensures that, given any sets containing at least one object, it is possible to select exactly one thing from each set, even if there are infinitely many sets, and no rule is required to choose them.   Plus, the “axiom of reducibility”, which I will not explain because today is largely considered ‘hocus-pocus’, and even Russell came to believe it was not needed.

Zermelo-Fraenkel Set Theory

Russell had to add extra axioms to Cantor’s intuitive set theory.   It has been found others were also needed.   The final version that most mathematicians use today is Zermelo-Fraenkel set theory.   I will give two links explaining it.   Each has advantages and disadvantages compared to the other.   Please at least glance at both.   As we will see, natural numbers can be defined surprisingly simply from these axioms.

https://brilliant.org/wiki/zfc/

https://www.cantorsparadise.com/zfc-why-what-and-how-e1498aec321b

Without Loss of Generality

I will use a notation the reader may not have seen – WOLOG.  It stands for without loss of generality.  It is used to indicate the assumption that follows is chosen arbitrarily, narrowing the proof to a particular case, but does not affect the validity of the proof in general.  The other issues are sufficiently similar to those presented, that proving them follows similar logic.  As a result, once the proof is given for the particular case, it is trivial to adapt it to prove the conclusion in all other cases.

Natural Numbers

The following link is an interesting read:

https://www.revistaminerva.pt/on-the-nature-of-natural-numbers/

The main idea needed is the definition of Natural Numbers (due to von Neumann):

We define 0 = ∅ and define n+1 = n∪{n}.

Not while the link I gave is an interesting read, the definition it gives using n-1 is wrong because n-1 is not defined yet.

The hypothesis of infinity guarantees such a set exists – indeed, it is the axiom of infinity.   The following definitions are standard:

0 = {},  1 = {0}, 2 = {o,1}, 3 = {0,1,2}………..

This is an example of an inductive definition.  n+1 by definition is {0,1,……,n}.   If something is defined for n=0, and n+1 is defined for n; then it is defined for all n.  It leads to a principle of reasoning called the principle of induction.  If something is true for n=0 and if true for n implies it is true for n+1, then it is true for all n.

Operations On Natural Numbers

All the following operations are defined using the principle of induction.

Addition: m + 0 = m.  m + (n+1) = (m+n) + 1

Multiplication: m*0 = 0,  m*(n+1) = m*n + m

Because the integers have not been defined yet, n-1 is defined as 0-1 = 0.  (n+1)-1 = n.

Total Order Of The Natural Numbers

A total order on the natural numbers is defined by letting a ≤ b if and only if another natural number c exists where a + c = b.  This is the standard definition.   But because of how the natural numbers have been constructed, we can say a < b if a ∈ b.

Division

In general, dividing natural numbers and getting a natural number is impossible. As a result, the division procedure with a remainder or Euclidean division is available as a substitute.  For any two natural numbers a and b with b ≠ 0, there are natural numbers q and r such that

a = b*q + r and r < b

The number q is the quotient, and r is the remainder of the division. The numbers q and r are uniquely determined by a and b.   The proof will be left as an exercise.  Hint: use the well-ordering principle on {q: a – b*q > 1)

Concluding Remarks About Natural Numbers

The above is only a partial list of all the properties of natural numbers.   Deriving them all is not the purpose of this article; instead, it explains what numbers are.   Interested readers can search the internet for a complete list of all their essential properties.   If desired, they can even attempt to prove them.   Next, we will move on to integers.

Integers

The starting point is to take the ordered pairs (a, b) of natural numbers.  Intuitively, we will think of this ordered pair as representing the integer a − b.  Thus (7, 4) will represent the number 7 − 4 = 3, while (4, 7) will represent the number 4 − 7 = −3.

It is easily seen that (5, 3) and (8, 6) both represent the number 2 and so we will say that two ordered pairs (a, b) and (c, d) are equal if a + d = c + b.  Note that if a + d = c + b, then a − b = c − d, which is what we want.  Typically this equality would define an equivalence class, and the integers are all the equivalence classes described this way.   I, however, prefer the idea of Urelement.  It’s simple and has already been encountered in the natural numbers where 3 = {0,1,2}.   It assigns a single object to groups, pairs, or whatever you like that are defined as equal.  Urelements are single objects.   They can be elements of a set but are not sets themselves.  One can still speak of the set, sequence, pair etc., defining the Urelement, but it is still a single object.   For example, 1 ∈ 3 where 1 is the Urelement of set {0}.   Here each integer is represented by a Urelement, e.g. -3.   It is the single object of all the pairs (0,3), (1,4), (2,5)…..    It can be viewed as a single object in the set of all pairs with the definition of equality (a,b) = (c,d) if a + d = c + b is defined as the same object.   Usually, the object representing (a,0) and all pairs equal to it would be a; the object representing (o, a) and all pairs identical would be -a.  The set of all such single things is called the integers.   For more detail on Urelements, there is a Wikipedia article.

Operations On Integers.

The addition of two integers is defined as (a, b) + (c, d) = (a + c, b + d). Hence, for example, (1, 3) + (5, 2) = (6, 5). This will correspond to −2 + 3 = 1.

Multiplication.  The multiplication rule is a little trickier. To see what the correct definition should be, we recall that (a, b) and (c, d) can be thought of as a − b and c − d, so expanding the product (a − b)*(c − d), we obtain ac + bd − ad − bc = ac + bd − (ad + bc).  Hence we define the multiplication of the ordered pairs by (a, b)*(c, d) = (ac + bd, ad + bc).  Thus, for example, (1, 3)*(5, 2) = (5 + 6, 2 + 15) = (11, 17). This corresponds to −2 × 3 = −6. The commutative law for multiplication holds since (a, b)*(c, d) = (ac + bd, ad + bc) = (ca + db, cb + da) = (c, d)*(a, b).  The second equality uses commutative laws to add and multiply whole numbers.

Next, we define the negative of an ordered pair (a, b) by −(a, b) = (b, a).  So, −(a, 0) = (0, a), corresponding to −(a) = −a, and −(2, 6) = (6, 2), which can be thought of as −(−4)= 4.

Subtraction can now be defined as (a, b) − (c, d) = (a, b) + [−(c, d)] = (a, b) + (d, c) = (a + d, b + c). Thus (1, 3) − (5, 2)= (3, 8), which corresponds to −2 − 3= −5.

Suppose we have the ordered pairs (a,0) and (b,0).   These correspond to the positive integers a and b.  (a,0)*(b,0) = (a*b,0)

Suppose we have the ordered pairs (a,0) and (0,b).  These ordered pairs correspond to a and −b. When we take their product, we obtain (a, 0)*(0, b) = (0, ab), and the product corresponds to −ab. This is equivalent to the rule that a positive times a negative is a negative. For example (4, 0)*(0, 3) = (0, 12) corresponding to 4 × (−3) = −12.   Similarly (0,a)*(b,0) = (0,a*b)

We can now prove the result corresponding to the rule that the product of two negative numbers is a positive number. Suppose we have the ordered pairs (0, a) and (0, b).  These ordered pairs correspond to negative integers −a and −b.   For their product, we obtain (0, a)*(0, b) = (ab, 0), and the product corresponds to the positive integer ab.

An important question has been skipped over.  Since various ordered pairs can be equal, how do we know that the addition and multiplication of identical ordered pairs will always produce identical integers? For example, (6, 3) + (5, 7) = (11, 10). Now (6, 3) is equal to (3, 0) and (5, 7) is equal to (0, 2), and (3, 0) + (0, 2)= (3, 2).  But note that (11, 10) and (3, 2) are equal.  Similarly, (6, 3)*(5, 7) = (51, 57) and (3, 0)*(0, 2) = (0, 6), and (51, 57) is equal to (0, 6). In mathematical language, we say that the definitions of addition and multiplication must be well-defined.  Suppose (a1, b1) is equivalent to (a2, b2) and (c1, d1) is equivalent to (c2, d2).  If (a1, b1) + (c1, d1) = (a2, b2) + (c2, d2), then addition is well-defined.  Similarly, we can show multiplication is well-defined.   The details are left as an exercise.

Ordering Of The Integers

(a, b) < (c, d) if a + d < b + c. For example, (5, 3) < (6, 2), since 7 < 9. This corresponds to 2 < 4 in the integers. Also, (3, 7) < (2, 3) since 6 < 9. This corresponds to −4 < −1 in the integers.

Similarly, (a, b) > (c, d) if a + d > b + c. For example, (9, 3) > (6, 2), since 11 > 9. This corresponds in the integers to 6 > 4.

Because all the integers > zero are represented by (a,0) and those < zero by (0, a) where a are natural numbers, then since the natural numbers are well ordered, the integers are well ordered.

Just like the natural numbers, the integers have many more properties.   Again these can be found from an internet search.  The reader can prove these if desired, but as mentioned before, this article’s purpose is not to establish all the properties of the integers.   Instead, it is to show how they are defined.   We now move on to rationals.

Rationals

A rational number can be expressed as p/q, where p is an integer, and q is a non-zero whole number.

Nowadays, rational numbers are defined in terms of ratios; the term rational is not a derivation of a ratio. On the contrary, the ratio is derived from rational.  The first use of its modern meaning was in England about 1660, while the use of rationals as numbers appeared almost a century earlier, in 1570.  This meaning of rational came from the mathematical definition of irrational numbers (to be defined later, but the reader has undoubtedly come across the term before), which was first used in 1551 and used in “translations of Euclid”.   This is because of the Pythagorean Theorem.  Considering a right-angled triangle with sides of unit lengths, the hypotenuse length is √2.   √2 is not rational.   The proof will be given later.

This unique history originated when ancient Greeks “avoided heresy by forbidding themselves from thinking of those irrational lengths as numbers”.  Such lengths were irrational, in the logical sense, that is, “not to be spoken about” sense.  This etymology is similar to that of imaginary and real numbers.   Although these days, imaginary numbers can be given the straightforward interpretation of the rotation of a number by 90%, similar to -1 rotates a number by 180%.

Rational numbers are formally defined as pairs of integers (p, q) with p an integer and q is an integer greater than zero.   (p, q) is also written as p/q.  Rationals p1/q1 and p2/q2 are equal if  p1*q2 = q1*p2.    Here they are not represented by the same Urelement but by p1/q1 and p2/q2, even though they are equal.   If q < 0 then p/q is the rational -p/-q.

Every rational number a/b may be expressed uniquely as an irreducible fraction. This may be obtained by dividing a and b by their greatest common divisor and, if b < 0, changing the sign of the resulting numerator and denominator.

Operations On Rationals

Addition.  a/b + c/d = (a*d + c*d)/b*d.

Subtraction a/b – c/d = (a*d – c*d)/b*d.

Multiplication (a/b)*(c/d) = (a*c)/(b*d)

Division (a/b)/(c/d) = (a/b)*(d/c) where if c is negative, d is changed to negative.

a/b < c/d is defined as ad < cb.   Similarly a/b > c/d is defined as ad > cb.

As said for the naturals and integers, the rationals have many more properties.   Again these can be found from an internet search.  The reader can prove these if desired, but as mentioned before, this article’s purpose is not to establish all the properties of the integers.   Instead, it is to show how they are defined.   We now move on to Reals.

Hyperrational Sequences

A hyperrational sequence is a sequence whose terms A1, A2, A3, A4, ……An,……. are all rational numbers.  Two hyperrational sequences, A and B are equal if An = Bn except for a finite number of terms.  Unless specifically referred to as sequences, identical hyperrational sequences are considered a single object, i.e. Urelements.  A < B is defined as An < Bn except for a finite number of terms.  Similarly, A > B if An > Bn except for a finite number of terms.  If Q is a rational number, the sequence Q Q Q Q…….. is the hyperrational sequence of the rational Q.   Of course, if A = Q as a hyperrational sequence, then A is also the rational Q.  When referring to a rational Q, if it is the number Q or the sequence of Q will be clear from the context.

A + B  is defined as An + Bn.  A – B = An – Bn.  A*B = An*Bn.  A/B = An/Bn.   In the definition of division, if Bn = 0 for a finite number of terms, the term An/Bn is set to zero.  Of course, B ≠ 0.

A rational sequence X = Xn is finite if a positive rational Q exists |Xn| < Q for all n.

If a finite rational sequence X > zero, X is called positive and X = |X|.  If X < zero, then X is called negative.  -X is then positive and |X| = -X.   If X = zero, then |X| = X.

Infinitesimals

Let Q be any positive rational number.  Let B be the hyperrational sequence Bn = 1/n.  Then an N can be found such that 1/n < Q for any n > N.  Hence, by the definition of < in hyperrational sequences, B < Q for any positive rational number Q.  Such sequences are called infinitesimal.  A sequence, B, is infinitesimal if |B| < Q for any positive rational Q.  Normally zero is the only number with that property.   In the algebra of hyperrational sequences, we have hyperrational sequences |x| > 0, called infinitesimals, less than any positive rational number.  If x > 0 then x is a positive infinitesimal, if x < 0 it is a negative infinitesimal. Also, we have infinitesimals smaller than other infinitesimals, e.g. 1/n^2 < 1/n, except when n = 1.

Positive and Negative Unlimited Sequences

Sequences can also be larger than any rational number, called positively unlimited.  Let A be the sequence An=n.  If X is any rational number, there is an N such for all n > N, then An > X.  Again; we have positively unlimited numbers greater than other positively unlimited numbers because except for n = 1, n^2 > n. Even 1 + n > n for all n.

Similarly, sequences such as A = An = -n are less than any rational number, i.e. are negatively unlimited.

Pathological Sequences

Some hyperrational sequences are pathological, e.g. the sequence 1 0 1 0 1 0……   It is neither >, < 0r = 1.

It can be rectified by using what is called an ultrafilter.   However, this is a complication I would like to avoid.   So here I have used what Terry Tao calls a cheap version of infinitesimal and unlimited sequences based on what is called the Fréchet Filter:

https://en.wikipedia.org/wiki/Fr%C3%A9chet_filter

Terry examines the issue in detail here:

https://terrytao.wordpress.com/2012/04/02/a-cheap-version-of-nonstandard-analysis/

Stay calm if it is a bit advanced.   Just skim it and get the gist for now.   Come back to it at the end of the article.

The Fréchet Filter has been used so far but will be refined further using the concept of limit.

Convergence of Sequences

We will now look at infinitesimals in another way.  If a sequence xn converges to zero, has a limit of zero, or xn → 0; intuitively, as n is made larger, |xn| can be made arbitrarily small.  Formally, rather than intuitively, it is defined as for all ε > 0; an N exists for all n>N, then |xn| < ε.  Suppose x = xn is infinitesimal, then using our definition of infinitesimal, |x| < X for any positive rational X, or |xn| < X except for a finite number of xn.   Since it is valid except for a finite number of terms, an N can be found such that for every n > N, then |xn| < X.   We will write this out formally.  Given any X > 0, an N exists such that for all n > N, then |xn| < X.    This is precisely the same as the formal definition of converging to zero with X replacing ε.

For hyperrational sequences, x=xn is infinitesimal if and only if xn converges to zero.

This then allows the definition of convergence to a rational.   A = An converges to a rational Q, has a limit of Q, or An → Q if A – Q is infinitesimal.   Formally An converges to Q if for all ε > 0 an N exists for all n>N, then |An – Q| < ε.

If An → A an N exists for all n > N, then |An – A| < 1.  |An| < 1 + |A|.   Let M = max (|An|) for all n ≤ N.  |An| ≤ max (M, 1 + |A|).  The hyperrational A=An is finite if An has a limit.

Algebra of Limits

If An → A, Bn → B then C*An → C*A for any constant C, An + Bn → A + B, An – Bn → A – B, An*Bn → A*B, An/Bn → A/B where B ≠ zero.

C*An.  Fix ε>0.  |C*An – C*A| = |C||An – A|.  A N exists for all n > N |An – A| < ε/|C|. |C*An – C*A| < ε.

An + Bn.   Fix ε>0.   A N1 exists for any n > N1 |An – A| < ε/2.  A N2 exists for any n > N2 |Bn – B| < ε/2.  Let N = max (N1, N2).  |An + Bn – (A + B)| = |(An – A) + (Bn – A)| ≤ |An – A| + |Bn – B| < ε/2 + ε/2 = ε for any n > N.

Similar to subtraction.

It is now easy to show that limits are unique.  Suppose An → B.   An – An → A – B = 0, i.e. A = B.

An*Bn.   Fix ε>0. |An*Bn – A*B| = |An*Bn – A*Bn + (A*Bn – A*B)| ≤ |An*Bn – A*Bn| + |A*Bn – A*B)| = |Bn|*|An – A| + |A|*|Bn – B| ≤ M*|An – A| + |A|*|Bn – B| where M is a bound of the sequence Bn.  If A = 0 there exists a N for all n > N |An – A| = |An| < ε/M.  |An*Bn – A*B| = |An*Bn| < M*ε/M = ε.  If A ≠ 0 then a N1 exists for all n>N1 |Bn – B| < ε/2*|A|.  Also a N2 exists for all n > N2 |An – A| < ε/2*M.  Let N = max(N1,N2).  For all n > N |An*Bn – A*B| ≤ M*|An – A| + |A|*|Bn – B| < M*ε/2*M + |A|*ε/2*|A| = ε/2 + ε/2 = ε.

An/Bn.  Fix ε>0  An/Bn = An*(1/Bn).  If 1/Bn → 1/B then An/Bn → A/B.  Suppose B ≠ 0.  |1/Bn – 1/B| = |(B – Bn)/B*Bn| = |B – Bn|/|B*Bn| = |B – Bn|/|B|*|Bn|.  |B| – |Bn| ≤ ||B| – |Bn|| ≤ |B – Bn|.  A N1 exists for all n > N1 |Bn – B| < |B|/2.  |B| – |Bn| < |B|/2.  |Bn| – |B| > -|B|/2.  |Bn| > |B| – |B|/2 = |B|/2.  |B – Bn|/|B|*|Bn.| < |B – Bn|/(|B|*(|B|/2)).  Let C = |B|*|B|/2.  A N2 exists for all n > N2 |Bn – B| < ε*C.  Let N = max (N1,N2).  For all n > N |1/Bn – 1/B| < ε*C/C = ε

Please take a look at the pattern here.   First, we fix ε>0.   Then we do some manipulations at the end of which something is < ε.  Since ε>0 is arbitrary, it is valid for all ε>0.   This standard trick my analysis professor taught me many years ago makes such proofs a lot easier.  The reader will encounter this several times in this article.

Real Numbers

The Need for Real Numbers

Let √2 = p/q, which is taken to be irreducible.  2 = p^2/q^2.  p^2 = 2*q^2.  If p is not even then there is a remainder 0f 1 when divided by 2 ie p = 2*p’ + 1.  p^2 = (2*p’ + 1)^2 = 4*p’^2 + 4*p’ + 1 which is not even.  Hence p is even, i.e. p = 2*p’.  p^2 = 4*p’.  4*p’^2 = 2*q^2.  2*p’^2 = q^2.  Hence q is even.   Both p and q have a common divisor of 2.   Contradiction.  Hence √2 is not rational.  This means equations such as x^2 = 2 can’t be solved.

Define .999999…… as the sequence An = .9 .99 .999 .9999……  Each term is less than 1.   Thus .999999…. < 1   1 – An = .1 .01 .001 .0001 ………   Given any positive rational Q < 1, an N can be found such that for all n > N, then An > Q.  1 – An is a positive infinitesimal.   .999999….. is not equal to 1 but infinitesimally close to it.  .9999999….. is not one, but is usually taken as 1.

We will define real numbers to resolve this, what √2 is, and other issues.

Hyperrational Cauchy Sequences

Intuitively a Cauchy sequence is a sequence such that as n gets larger, the terms get closer and closer to each other until eventually they are so close the difference can be neglected, i.e. the sequence is convergent. Formally a sequence An is Cauchy; if for all ε>0, an N exists, for all m,n > N, |Am – An| < ε.

If An is a Cauchy sequence, an N can be found for all m,n > N then |An – Am| < 1.   For all m ≥ N+1 then |Am| < 1 + |Ac| where c = N+1.  Let M = max |An| if n ≤ N.  Hence |An| < max(M,1 + |Ac|).   Cauchy sequences of rational numbers are finite hyperrational sequences.

Hyperrational Cauchy Sequences Do Not Always Converge to a Rational

For rational Cauchy sequences, sometimes they converge to a rational, in which case there are no problems.   But sometimes, it converges to something that is not rational.  For example, let X1=2, Xn+1 = Xn/2 + 1/Xn be the recursively defined rational sequence Xn.  Calculate the first few terms.  Even the fourth term is close to √2. Indeed let εn’ = Xn – √2. Define εn = εn’/√2.  Xn = √2*(1+εn).  We have seen εn is small after a few terms.  Xn+1 = ((1/√2)*(1+εn)) + (1/√2)*(1/(1+εn)) = 1/√2*((1+εn) + 1/(1+εn)).  If S = 1 + x + x^2 +x^3 ….  S – S*x = 1.  S = 1/1-x = 1 + x + x^2 + x^3……  If x is small to a good approximation, 1/1-x = 1+x or 1/1+x = 1 – x.  We call this true to the first order of smallness because we neglected terms of higher powers than 1.  Hence Xn+1 = (1/√2)*((1+εn) + (1-εn)) = √2 to the first order of smallness in en.  The sequence quickly converges to √2, which, as shown, is not rational.  As an aside for those that know it, the sequence was constructed using Newton’s method, which generally converges quickly.

Real Numbers Defined

All hyperrational Cauchy sequences are defined as the set of real numbers.   Two real numbers are equal if and only if they are infinitesimally close.  They are all assigned the same Urelement, with Q assigned to the hyperrational sequences infinitesimally close to the rational Q.  This means the rational numbers are a proper subset of the real numbers.  If A and B are real numbers, then A>B if, A≠B and A’>B’ where A’ is any hyperrational sequence represented by the real A, and B’ is any hyperrational sequence represented by the real B.  Similarly, for A<B.

.999999….. is the same real number since it is infinitesimally close to 1.

If A is a hyperrational Cauchy sequence represented by the real number R, then trivially An converges to A.  Since, as real numbers, all the hyperrational sequences represented by R are equal to A, An converges to all the hyperrational sequences represented by R.   This means all hyperrational Cauchy sequences converge to some real R.  Conversely given any real number R we can find a hyperrational Cauchy sequence that converges to a hyperrational sequence represented by R.  However there are real numbers, such as √2, that can never equal a hyperrational sequence, but are only ever infinitesimally close to it.

Algebra of Real Numbers

As yet R1 + R2, R1 – R2, R1* R2, R1/R2 have yet to be defined.   Summation, subtraction, multiplication and division are defined the same as for hyperrational sequences, except each real number represents many hyperrationals.  If R1n = R1, R2n = R2 and R’1n = R1, R’2n = R2.  R2n = R1n + r1n and R’2n = R’1n + r’1n where r1n and r’1n are infinitesimal.  R’1n + R’2n =  (R1n + R1n) + (r1n r’1n).  r1n + r2n → 0. R’1n + R’2n are infinitesimally close to R1n + R2n.  Addition is well-defined.  Similar to subtraction.  R’1n*R’2n =  (R1n + r1n)*(R2n + r2n) = R1n*R2n + (R1n*r2n + R2n*r1n + x1n*x2n).   As r1n and r2n are infinitesimal, R1n and R2n finite, they converge to zero.  R1n*R2n is infinitesimally close to R’1n*R’2n, so define the same R1*R2.  Hence multiplication is well-defined.  Note that 1/R2 (R2 ≠ 0) is a finite rational sequence so R1/R2 being well defined follows from R1/R2 = R1*(1/R2).

Hyperrationals Defined

As the concept of a real number has been defined; the hyperrationals can now be defined.   First we need to determine when a hyperrational sequence is >, <, or = to a real number R.   A hyperrational sequence An is > R if An > R except for a finite number of terms.   Similarly, for < or =.   Note that a hyperrational sequence can only be = to a real number if that real number is rational.

All the positive unlimited hyperrational sequences are greater than every real number.   Similarly all the negative unlimited hyperrational numbers are less than every real number.   Only finite hyperrational sequences may be pathological and not >, <, or = to a real number.

The hyperrationals are all hyperrational sequences that are either >, < or = to all real numbers.

As will be seen later, all finite hyperrationals can be expressed as R + r, where R is a real number and r is an infinitesimal that is >, <, or = to zero.   Of course, r can only be zero if R is rational.   Pathological sequences are not part of the finite hyperrationals.  The definition of <, >, and = is an ordering of the finite hyperrationals.  For those familiar with abstract algebra, the hyperrationals are an ordered field.  However, since irrational numbers are only ever infinitesimally close to a hyperrational, there are ‘holes’ in the sense that no irrational number is equal to a hyperrational.  In a sense to be made concrete the rational numbers are not complete, ie have ‘holes’.  Even hyperrationals do not resolve this issue.  This is why we need the hyperreals to be defined later.

Completeness

First, we note the same limit algebra and their proofs apply for real sequences as for rational sequences.

Let Xn be a Cauchy sequence of real numbers.   Then a hyperrational Cauchy sequence Q=Qm exists such that Qm converges to Xn, i.e. a Qm exists such that Qm is arbitrarily close to Xn.  Given an Xn, a Qn exists |Xn – Qn| < 1/n.  Xn – Qn converges to zero.  Qn = Xn + (Qn – Xn).   Since Xn and Qn – Xn are Cauchy, Qn is Cauchy, hence converges to an actual number R.  As n → ∞ Xn = (Xn – Qn) + Qn → R.  Hence any Cauchy sequence of real numbers converges to an actual number.

Without going into the detail of ordered fields, the set of all rationals and the set of all reals are ordered lots.  An ordered field is called complete if all the Cauchy sequences converge to an element of the ordered field.  Otherwise, it is called incomplete.

The rationals are incomplete.  However, all real Cauchy sequences converge to a real number.  The reals are complete.  It turns out the real numbers are the only complete ordered field.

The Rationals Are Dense in the Reals

If A is a proper subset of B, then A is dense in B if given b ∈ B, then a ∈ A can be found that is arbitrarily close to b.  The integers, for example, are not dense in the rationals because two rationals can always be found with no integers between them.   Perhaps surprisingly, the rationals are dense in the reals.

Suppose a < b where a and b are real numbers, then an N exists, N > 2/b-a.  Nb – Na > 2.   Let M be the set of all integers > Na.  Since the integers are well ordered, M has a minimum element, m.  Since the distance between integers is 1, Nb > m > Na, b> m/N > a.   A rational can be found between two different real numbers.

There is a rational Q between R and R+ε, R < Q < R+ε where ε>0 is arbitrary.  A rational can be found arbitrarily close to any real number.   That a rational can be found between any two real numbers is sometimes the alternative definition of the rationals are dense in the reals.

The Least Upper Bound Property

I will now prove an essential property of the reals.  Given a set S of reals, an upper bound is a real number ≥ than every s ∈ S.  The Least Upper Bound property says that if S has an upper bound, S has an upper bound smaller than any other S upper bound, called the Least Upper Bound (LUB) of S.

If S has exactly one element, its only element is a least upper bound (LUB).  Consider S with more than one element and an upper bound.  Let an upper bound be B1. Since S is nonempty and has more than one element, a real number A1 ∈ S exists that is not an upper bound for S.  Define A1 A2 A3 … and B1 B2 B3 … as follows.  Check if (An + Bn) ⁄ 2 is an upper bound for S. If so, let An+1 = An and Bn+1 = (An + Bn) ⁄ 2.  Otherwise, s ∈ S exists such that s ≥ (An + Bn) ⁄ 2.  Let An+1 = s and Bn+1 = Bn.  Then A1 ≤ A2 ≤ A3 ≤ ⋯ ≤ B3 ≤ B2 ≤ B1.  An – Bn converges to zero by construction.  Both sequences are Cauchy.  An converges to L1, Bn converges to L2.  As n  → ∞ An = (An – Bn) + Bn → L2.  L1 = L2 = L.   Suppose L’ < L.  Since An converges to L, An can be found that is arbitrarily close to L, i.e. closer than L – L’.  Hence there are terms, An larger than L’ in the set S.   Hence L’ is not an upper bound of S.   L is the LUB of S.

Note that the rationals do not have the LUB property.  The LUB of a set of rationals is the smallest rational, a rational upper bound.  Let A be the set of all rationals less than √2.  Suppose q < √2 is the LUB of A.  There is a rational s such that q < s < √2.   But s ∈ A, q can’t be a LUB of A.  Suppose q > √2 is a LUB.  Then there is a rational s, q > s > √2.  Hence q is not the LUB of A.  Any rational number is either < √2 or > √2.  No rational can be the LUB of A.

The LUB property is completeness expressed differently.

This leads to another method of defining real numbers called Dedekind Cuts.

Dedekind Cuts

A partition of the rationals into two non-empty sets, A and B, such that A has no greatest rational, and all the elements of B are greater than all the elements of A, is called a Dedekind Cut.

Only A needs to be defined, as B are all the elements not in A.   If A is a proper subset of the rationals with no largest rational, such that all the rationals not in A are greater than all the rationals in A, then A also defines a Dedekind Cut.

Let A and B be a Dedekind Cut.  Then the LUB of A is some number R.   Hence every Dedekind Cut defines a real number.  Suppose a different Dedekind Cut A’ had the same LUB.  WOLOG A’ is a proper subset of A.  Let R’ be the LUB of A’.  Since A has elements greater than those in A’, R’ < R.  Hence each Dedekind Cut defines a unique real number.

Conversely, let A be the set of all rationals less than a real number R.   A is a Dedekind Cut with an upper bound of R.  Suppose there is an upper bound of A, R1 < R.   Hence there is a rational Q, R1 < Q < R.  But Q ∈ A, hence R1 cant be an upper bound of A.   R is the LUB of A and defines a unique Dedekind Cut.

Every real number defines a unique Dedekind Cut; conversely, every Dedekind Cut defines a unique real number.   There is a 1 to 1 correspondence between the reals and Dedekind Cuts.   The set of reals and the set of Dedekind Cuts are the same.   Indeed they can be made the same elements by assigning the Urelement assigned to the reals to the corresponding Dedekind Cut.

Also, the real number of a Dedekind Cut, A and B, is the LUB of A.

Hyperreals

Similar to rational sequences, sequences of reals have the same definitions of equality etc., as the rational sequences.    As before, the real number A is the sequence An = A A A A……………  Two hyperreals, A and B, are equal if An = Bn except for a finite number of terms.  As usual, they are treated as a single object.  We define A < B  and A > B similarly, An > Bn, and An < Bn except for a finite number of terms.

A + B  is defined as An + Bn.  A – B = An – Bn.  A*B = An*Bn.  A/B = An/Bn.   In the definition of division, if Bn = 0, the term An/Bn is set to zero.  Of course, B ≠ 0.

The hyperreals are all all hyperreal sequences that are >, < or = to all real numbers.

This rectifies the ‘holes’ in the hyperrationals.

A sequence that converges to an actual number can be infinitesimally close to a real number but, under the definition of equality, not equal to it.  However, as we will see, it will not cause the same issues as in the hyperrationals because real numbers have the LUB property.

If F(X) is a function defined on real numbers, then F can easily be extended to the hyperreals by F(X) = F(Xn).   This property of the hyperreals is frequently needed in using hyperreals to do calculus.

Unique Decomposition of Hyperreals

Let X be a bounded hyperreal.  Let A = {Q| Q rational Q < X }.  A is defines a partition of the rationals.   We claim R-X is infinitesimal.  If R-X = 0 then R-x  is infinitesimal.  If R-X > 0 and not an infinitesimal there is a positive real S, S < R-X.  X < R-S; hence R-S is greater than any element of A.  R is the LUB of A.  Thus R-S ≥ R.  Contradiction.  If R-X < 0 and is not infinitesimal, there is a negative real S,  R-X < S.  R-S < X, then R – S < R.  Since S is negative, R + S’ < R, where S’ = -S is positive.  Contradiction.   Hence R-X is an infinitesimal.  This implies X – R is also infinitesimal.  Let r = X – R.  X = R + r.   Hence a bounded hyperreal can be written as the sum of a real number and an infinitesimal.  Is it unique?  Suppose X = R1 + r1 = R2 + r2.  R1 – R2 = r2 – r1.   r2 – r1 is infinitesimal.  Hence R1 – R2 is infinitesimal.   But the only infinitesimal real number is zero.  Hence R1 = R2 and r1 = r2.   The decomposition is unique.

The Hyperreals contain every real number.  Let X = R + r where r is any hyperreal infinitesimal.   Hence X is a hyperreal and R + r → R.  Therefore the finite hyperreals are all the numbers of the form where X = R + r, R any real and r any infinitesimal.  They are all the sequences of reals that converge to a real number.

R is called the standard part of X and is written as st(X).  X = Xn converges to R = st(X).   Every hyperrational X can be uniquely written as st(X) + r, where r is infinitesimal.   This resolves the .9999999…. issue in the hyperreals.   In the hyperrationals .99999…. is infinitesimally close to 1.   That is still true in the hyperreals except st(.9999999….) = 1.

The hyperreals are also an ordered field, but not complete because the limit of Cauchy sequences of hyperreals are not unique.  However, the details of such subtleties will be left to the reader to investigate.

Conclusion

The reals have, step by step, been constructed from ZFC set theory.   Along the way, some interesting new number systems have been investigated; the hyperrationals and the hyperreals.   Both contain actual infinitesimals that have properties the early calculus pioneers wanted.   Namely, if |x| is infinitesimal |x| < X where X is any positive number.   Such numbers can legitimately be neglected, and calculus can be developed without limits using infinitesimals.   I am working on an article doing just that, combined with the study of a precalculus textbook.   We also see that the link between infinitesimals and limits is that x = xn is infinitesimal if and only if n → ∞ xn → 0, i.e. xn converges to zero.  Calculus using limits and infinitesimals is the same thing, the only difference being the terminology used.