re-learning to count to infinity

Re-Learning to Count to Infinity

For the last 100 years students have been puzzled by the methods used to count to infinity. It is not surprising that they were confused because the method used was both illogical and inconsistent. By applying one simple rule, the field of counting is restored to a self-consistent and logical method that can be agreed upon by both mathematicians and lay people.

In summary all we do is count within a finite set and then extend that finite set to infinity. This simple change makes a world of difference to the results achieved so that now we can quantify the sizes of different infinite number sets with ease.

Contents

The Basic Infinite Set...................................................................................................................... 2Limits and Number Space...............................................................................................................3Ratios..............................................................................................................................................4

–Rules......................................................................................................................................... 5Calculation of Values...................................................................................................................6Rational Numbers...........................................................................................................................6Table of Results...........................................................................................................................9Infinite Areas................................................................................................................................ 10Ratio of Areas Above and Below a Sloping Line............................................................................11Ratio of Areas Above and Below a Parabola.................................................................................13Further Reading............................................................................................................................14Objections to the Accepted Methods...........................................................................................14Appendix 1: Computer Code to Count Rationals..........................................................................15Appendix 2: Proof of Uniqueness of Rationals.............................................................................16Version History.............................................................................................................................17

Leslie Green CEng MIEE 1 of 17 v1.10: Dec 2015


The Basic Infinite SetThe starting point for this work consists of an infinite set of numbers.

For our purposes we will define a set as a collection of items, separated by commas and contained within curly braces. As an example we could define the set S to be

S { 1, 2, 3 }

We use the “identically equal” symbol (3 bar equals sign) to show that the left hand side is just a symbol representing the right hand side.

We are interested in the number of items (elements) in the list. We will pretend this is evaluated using a computer function, count(). For this case count(S) =3.The count function naturally returns only “counting numbers”.

“Counting numbers” go by various other names such as positive integers, whole numbers, and natural numbers. There is inconsistent usage to say whether or not zero should be included, but we will exclude it for simplicity.

The next basic symbol we will use will be the symbol for infinity, , which we will define as a positive counting number which has been increased indefinitely.

We now introduce a very important new symbol, derived from the earlier definitions.

count ( { 1, 2, 3, 4, … } )

In words, phi is the number of elements in the infinite set of counting numbers.

Some might then get worried that we have introduced a circular definition in the sense that we have effectively said

= count ( { 1, 2, 3, 4, … } )

to which we reply that we have already adequately defined ; the equality above is merely an observation, and not the definition.



Limits and Number SpaceThe notion of limits is a powerful mathematical method which we will use throughout our discussions. Likewise the idea of a finite number space is critical to these discussions.

In all discussions we will start from a finite number set and then look at what changes as we increase the size of this set. Let’s look at a concrete example.

Consider the simple finite number set S { 1, 2, 3, 4, … N }

The last element in this set is N and this represents the current boundary of what we are going to call the “number space”. We then consider what happens as this number space increases without bound.

The way we are going to use this finite number space is to include it in the set definition like this …

S(N) { 1, 2, 3, 4, … (r: r N) }

Which we read as follows: The set S, limited to the finite number space bounded by N. In other words we allow no values larger than N in this set. The final element is written as (r: r N) which is read as having the value r, such that r is less than or equal to the bounded space value N.

But it would be very tiresome to have to write things like (2r+1: 2r+1 N) or worse still to work it out to (2r+1: r (N-1)/2) ). Therefore we are going to leave the finite number limit as a parameter in the set definition and it is to be understood that the last term in the series is implicitly limited by this value.



RatiosSince is simply an infinitely large number, it has been said that the number of counting numbers and the number of even counting numbers is the same, namely infinite. We shall show that this is unhelpful, illogical, and easily remedied.

Let S(N) { 1, 2, 3, 4, 5, 6 … (2r+1) }Let SE(N) { 2, 4, 6 … (2r) }

In order to compare the sizes of these sets we are allowed to pair off members (or sub-groups of members) in the sets, but crucially we are not allowed to exceed the bounds of the number space.1

We can count the elements of an infinite set by comparison with our basic infinite set. We do this for larger sets by linking groups of numbers in the target set with one member of the basic set which we will refer to as a one-to-many pairing. Alternatively, for a smaller target set we can map several elements from the basic set to one member in the target set, a many-to-one pairing.

Let’s take a concrete example: We take 1 and 2 from the set S and pair them with 2 from the set SE. Likewise 3 and 4 pair with 4, and so forth. It is clear that for these two finite sets there are twice as many elements in S(N) as in SE(N).

for any N 2

Given that every large value of N gives this same ratio of 2, there is no particular need to worry about taking N to an infinite limit. The ratio is always 2. It would be illogical to invent a discontinuity at an arbitrarily large value of N wherein the ratio suddenly dropped to 1, and yet that is apparently the current thinking on the subject!

We can now compare this type of infinite set with and say that its size is /2, or /3 if we uniformly discard 2 out of 3 elements, and so forth. This forms a natural

and intuitive counting methodology.

Suppose instead that we insert elements uniformly within the gaps between the counting numbers. For example we could insert one number between 1 and 2 at 1.5. Another would be inserted between 2 and 3 at 2.5. By pairing these with the lower integer in the basic set we see that the set size has doubled. In general, if we insert P 1 It is at this early point that we part company from Mathematicians.



elements uniformly between each counting number the size of the resulting set is increased by a factor (P+1).

The definition of real numbers is all counting numbers plus all those that have parts after the decimal point. It should be clear that the set of positive real numbers can be created by inserting an infinite number of elements between each adjacent pair of counting numbers, but we would then need to include zero.

count( { real positive numbers} ) =

This is a very large number!

–Rules

Note that any non-zero multiple of is an acceptable infinity. However adding anything to (which is not itself a multiple of ) does not increase the result at all.

multiplication by a constant is valid

addition of non-infinite values is irrelevant

addition of the same powers of is valid

higher powers of overwhelm lower powers

higher powers of can be created

can be divided and cancelled

is not a valid operation



Calculation of Values

When we get to more complicated sets, where the density of elements reduces with increasing position in the series, we immediately run into a problem with the simple one-to-many or many-to-one pairing schemes.

Consider the finite set of squares of counting numbersSS(N) { 1², 2², 3², 4² …. r² }

count( SS(N) )= floor( ) and so we get the obvious consequence thatcount( SS() ) =

from which we can generalise the result to give

SK(N) { }count( SK() ) =

Rational Numbers

Rational numbers are those of the form where n and m are counting numbers.

There are of course “an infinite number” of rational numbers, but in the context of this work we need to define just exactly how many that is.

Whilst the counting numbers are strictly a subset of the rationals, we shall see that they form a very small subset. For our purposes it is convenient to count only the rationals that do not include the counting numbers in order to avoid double counting.

If the finite set has a limiting value of N then we will have N2 possible ratios. Terms with a numerator of zero are all neglected. Likewise terms with denominators of

zero are all neglected. Terms of the form are of course equal to 1 and therefore

should not be counted. Terms of the form for k>1 should not be counted

because they duplicate the ratio . It seems that the size of the infinite set is of

the form (a > 3 ) and by the -rules we can simplify this to . But we cannot yet be certain if some duplicated terms have size ( b < 1 ) that also need to be subtracted from the initial set.



Looking back to the finite set it is clear that for the limiting value N, the greatest ratio value will be N/2 and the numerator range from 1 to N fits within this range. As the denominator is increased to 3 the whole 1 to N range of numerators fits into the reduced ratio range from 0 to N/3. Excluding duplicates, it is clear that the density of rationals increases as we get closer to the zero end of the real number range.

We can investigate further with some experimental computation. Even with a limiting N value of 1000 we get 1E6 elements before the duplicates are eliminated. Since the computation scales with N2 we cannot go very many decades away from this starting value before handing unreasonably large data sets. We cannot, for example, realistically store and sort the data set to ensure that we have eliminated all duplicates. The best we can do is to use Euclidian division to establish that the numerator and denominator of the rational are co-prime, meaning they have no common-factors other than 1. (To be clear, this also eliminates the case where the numerator and denominator are equal).

In this table are the results from a computer program where LIMIT is the maximum value of N allowed. The Valid column tells us what percentage of the LIMIT2 combinations give rational number pairs that are co-prime, and therefore not giving duplicated values.

Computed Distribution of Rationals for Finite Data Sets

1.0E+00

1.0E+01

1.0E+02

1.0E+03

1.0E+04

1.0E+05

1.0E+06

1.0E+07

1.0E+08

1.0E+09

1.0E+10

1 10 100 1000 10000 100000

Value of Rational Number

Qua

ntity

100000100001000


LIMIT Valid100 59.870%

1000 60.738%10,000 60.785%

100,000 60.792%


The computer code has clearly established that the duplicated ratios do in fact form a significant proportion of the available ratios. However, we cannot yet be sure that we have eliminated all possible duplicates.

Specifically, if we define p, q, r & s as counting numbers, if then can there be

duplicates of the form ?

The answer is no, but the proof requires some knowledge of analytical number theory so we have pushed the proof into Appendix 2 for the interested reader.

Using the definition that { reals } = { rationals } + { irrationals }

we can immediately write down the count of the irrationals as well.



Table of ResultsWe are now in a position to write down a table of values to clarify the results. Notice that the table is sorted into size-order, with the smallest sets listed first. (In some cases, however, the exact ordering is dependant on the parameters chosen.)

set count()the cubes of all positive counting numbersthe cubes of all integersthe squares of all positive counting numbersthe squares of all integersthe squares of all real numbers with finite precisionall prime numbers (from the Prime Number Theorem)all positive counting numbers that are evenly divisible by Nall positive counting numbers that are evenly divisible by 3all even positive counting numbersall odd positive counting numbersall positive counting numbersall even integersall odd integersthe squares of all positive real numbersall integersall positive real numbers with finite precisionall positive real numbers with D digits after the decimal pointall real numbers with finite precisionall complex numbers with positive counting numbers for the real parts and squares of the counting numbers for the imaginary partsall positive irrational numbers all positive rational numbersall positive real numbersall real numbersall complex numbers with integer real and imaginary partsall complex numbers with real values for both the real & imaginary parts All of these results are infinite, but the sizes of the infinities have a huge range.

This then completes the tools we need to classify the size of any infinite set as being between limits defined by some multiple of some power of .



Infinite AreasIf we can compare infinite counts (a 1-dimensional number space), we should be able to compare both infinite areas and infinite volumes by an extension of the same method. We will, however, just stick to infinite areas for simplicity.

We start from a trivial case where we are asked to consider the ratio of the areas above and below the X-axis. We start off with a limited Cartesian space whose x extent is from –N to +N and whose y extent is also from –N to +N.

The required ratio of areas is

Increasing N to infinity has no effect as it divides out. We have a unique and well defined answer to the question.

We could have been asked to calculate the ratio of the areas of the infinite quadrant B to the other three quadrants and we would have similarly achieved a result of 1/3.


A B

C

x

D

y

N

N


Ratio of Areas Above and Below a Sloping Line

If we want to deal with a sloping line through the origin we only have to do the drawing to recognise that the triangular area missing from one side of the Y-axis is reclaimed on the other side. The ratio of areas remains at 1, regardless of the slope of the line up to a 45° angle. After a 45° angle the question becomes one of the ratio between the left and right sides of the line, but the ratio still holds as 1.

If the line does not pass through the origin we can readily see that if the scales are increased (N increased by ×10 for example) the line will eventually look as if it does in fact pass through the origin. Therefore the areas above and below in the limit become 1 again.

More formally we have shown that slope of the line is not relevant to the areas above and below so we could just consider a line parallel to the X-Axis with an offset of A. The ratio is then


x

y

N

N


We now ask: what is the ratio of areas above and below the red lines as the number space tends to infinity?

for

Again there is no requirement for convergence when N becomes infinite because there is no dependence on N.


x

y

N

N aN


Ratio of Areas Above and Below a ParabolaNext we ask for the ratio of areas above and below the parabola . If you feel that the cyan shaded areas are not strictly “below” the parabola then realise that we have two and only two areas, bounded by the curve. Anything that is not above is necessarily the other side of the boundary – and therefore below.

Since we restrict the number space to ±N in each axis, the parabola is constrained between the two vertical blue lines. We neglect the actual area under the curve that is between the blue lines for simplicity.

Without much effort we have shown that the ratio of areas above and below the parabola tends to zero as N tends to infinity, even though the area above the parabola is actually infinite! If the parabola is not exactly at the origin it is evident from our previous discussions that the relative offset will become negligible as we zoom out from the plot and the ratio of areas will remain at zero.



Further ReadingSince the methods presented here are generally in disagreement with the prior literature, the best you can do with further reading is to see the Mathematicians’ viewpoint. There is a vast range of mathematical literature available which can be accessed using the search terms below.

Georg Cantor (1845-1918)injection / surjection / bijectionAleph-null, Aleph-onecountably infinite, uncountable setscontinuum hypothesis, axiom of choicecardinality, cardinality of the continuumDedekind-infinite set

There is also a term called density for an infinite series, but again, whilst this has similarities to the methods presented here, it diverges rapidly giving illogical results such as that the density of squares and primes is zero. The values we have calculated for these values are non-zero and very different to each other.

Objections to the Accepted MethodsWe “hid” the primary objections here at the end so as to not confuse the reader. In the accepted method, two sets are found to be of the same size by using different number spaces. The sets S and SE are considered to be the same size because they can be paired as shown below, pairing vertically adjacent elements.

Let S { 1, 2, 3, 4, … r }Let SE { 2, 4, 6, 8, … 2r }

It is this fundamental step which we stated as being logically inconsistent and intuitively wrong. “Everyone knows” that there are twice as many counting numbers as there are even counting numbers. It takes years to convince mathematics students that their intuition is wrong, coupled with the fact that they will fail their exams unless they agree!

It is important to note that all we have done here is to add one rule to the counting step, namely that we must count within a common number space, and all the sets mentioned readily become countable.

Actually there is one other thing we have done. We also state that which no doubt will make a lot of people happy; the old counting method being counter-intuitive.



Appendix 1: Computer Code to Count Rationals



Appendix 2: Proof of Uniqueness of RationalsThe computer code eliminated integer and some duplicated ratios by insisting that the numerator and denominator were co-prime, by which we mean the numerator and denominator have no common divisors excepting 1.

We now wish to prove that this rule in fact eliminated all possible duplicates.

If we define p, q, r & s as counting numbers, the elimination of non-co-prime pairs removes a significant fraction of the N2 possible ratios. However, we cannot yet be

sure that if then .

Suppose that there are duplicates such that for .

Then

Whilst is clearly composite, must have a unique prime factorisation (provided the primes are listed in ascending order) according to the Fundamental Theorem of Arithmetic. Since p, q, r & s are not defined as prime, we can factorise them into unique prime products

We have defined p and q as being co-prime, meaning that none of the p’s can be equal to any of the q’s. Likewise none of the s’s can be equal to any of the r’s. But since has a unique prime factorisation we would have to conclude that and

, a situation which we explicitly excluded at the beginning. The contradiction assures us that there can be no duplicate rationals created after our elimination of non-co-prime pairs.



Version Historyv1.00: 22 Dec 2015 – First publication on scribd.V1.10: 31 Dec 2015 – Changed definition of and other sets to exclude 0.


re-learning to count to infinity

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