01 Infinity
Infinity is, in German and Dutch, called unendlich and oneindig respectively, literally meaning unending, 'something' does not terminate, it has no end (it is 'endlos', endless), which gives the right emphasis to the essential idea.
The essence is, there is no end to be named infinity, or an end that has properties. All references to an entity at, or beyond, such an end, are meaningless.
That 'something', that has no end, is an array, a series, a sequence etc. The forming process is unending, it has no end. E.g. the set of all positive integers is unending, (the process here: add one to the previous number), but there is no 'last' number, to call infinity or whatever.
Writing about the infinite and its associated problems has been going on for some 2500 years. The earlier writers struggled with the idea of infinity and illustrated the difficulties with some paradoxes. Modern writers use those same paradoxes, or variations thereof, to make the subject either appear more difficult and mysterious or, reduce it, with a simple statement, to nothing extra-ordinary.
For instance with statements like: "The modern view is that Achilles runs one minute, because we know that the sum, of that infinite geometric progression, equals one minute". This avoids the essence of Zeno's argument, that there is apparently no end to the number of distances Achilles has to cover, and it misstates the idea of a limit.
When one is first introduced to the idea of infinity, it is emphasized that infinity is neither a limit, nor a number, but this is often ignored, and not just by students. Hilbert's hotel with its infinite number of rooms is an example of this. Later more about this hotel.
02 Zeno's paradox
To solve the paradox it is necessary to emphasize the difference between a continuous process and a staccato process. To show the difference, seemingly trivial, examples are used.
The paradox arises due to the fact, that while Achilles runs a continuous race, it is described (intentionally?) as a staccato race. It does so by stating, misleadingly: "Achilles must first reach the point where the tortoise began etc". "Reach" instead of "pass"!
If Achilles does indeed stop (instead of merely pass) at all those points, and 'waits' for the tortoise to move a certain distance, Achilles will indeed never overtake the tortoise.
Achilles does not have to wait for the tortoise, merely stopping, even for the shortest possible interval t ( t > 0 ) at those points, is sufficient to make it a staccato 'race', with or without a tortoise. In the actual race Achilles does not stop and the distance to be run l, is only divided, and the sum of the parts remains equal to l, regardless of the number of (imagined) divisions.
In other words, a staccato race becomes a continuous race when t equals zero. Ignoring this difference produces the paradox.
[02-01] The continuous case :
[02-02] The staccato case:
[02-03] Or, as it is usually stated:
The limit equals one, the sum of the series, never equals one.
03-Galileo's paradox
Consider the following example, used in one form or an other, to illustrate the peculiarities of an infinite set:
Row (1) is the list of the positive integers.
Row (2) the list of true squares in list (1).
Row (3) another list that counts the squares in list (2).
The lists are all 'man-made', and there is no reason to assume that the squares come out of the third list rather than the first, or to assume that there is only one such list. Also, can such lists that are in an unfinished, a nascent state, be considered, and treated as sets, as if as sets, they become an entity with finite properties.
Treating the three rows, nevertheless, as sets, we get:
* Cantor's definition of an infinite set:
Counting the members of set (2), starts a new sequence (3), with its own properties. When the sets (2) and (3) have N members, then the set (1), that 'progresses' faster, has, at least, N2 members, and this is true for all N, however large, or whatever N may be called.
Using only the sets (2) and (3) in the definition of an infinite set, leads to erroneous and amazing results.
I quote from an April 1995 SA-article on the subject:
It implies that (3) can be used without the existence of the set of positive integers. We have a very good and powerful number system, and, with a few algorithms, we can work out e.g. the square of any number, without referring directly to the basic number line, (49 does not exist without 48, and so on and on).
Trying to work out the square of LXXVII, without translating it first to the decimal system, will make clear that this cannot be done without knowing the convention, the sequence, of the Roman number line.
In a number system that uses the alphabet as basis, B-squared would be, BB = D, arrived at by measuring AB, B times along the base line, and finding that D is the matching 'number'.
For E-squared we find in this way EB = Y, for 'numbers' larger than E we first have to make arrangements how the basic set, the alphabet, is to be continued, e.g. Z can be a symbol for the 'number' after Y or it can indicate the absence of a 'number'.
The difficulties are, to a large extent, caused by the (sub-conscience?) use of infinity as a number, (notwithstanding statements to the contrary), and taking the sets (1) and (3) as identical, which they, clearly, are not.
04-Hilbert's Grand Hotel
From Wikipedia:
A hotel with an 'infinite number' of rooms, when full, also has an 'infinite number' of guests, whatever 'infinite number' may mean.
Still, the suggested solution, of adding one more guest to the full hotel, consists of having, contrary to the above, one more room than guests, but, since the hotel was full before, apparently only so when a new guest arrives.
Treating infinity plus one, for the rooms, as equal to infinity for the guests, makes infinity indeed a, very special 'number'. The elaborate way of accommodating the extra guest serves only to hide this, because taking this new guest directly to room infinity plus one, would give the game away.
05-The 'number' of points on a line
Take two concentric circles, one with a radius of one millimeter, and the other with a radius of one meter. Form an angle of one radian with two straight lines through the centre. The length of arc is then, 1 millimeter and 1 meter respectively.
Now when it is stated, correctly, that all points on the smaller circle can be paired of with a point on the larger circle, it appears, wrongly, that there are more points on the larger circle than on the smaller one. By implication: the number of points on those lines determine the length, and again there is a paradox.
When 999 evenly spaced points on the outer arc, are connected with lines to the centre, we have arc-lengths of one millimeter on the outer-, and of one micrometer on the inner circle. Both arcs now are divided in a 1000 parts, and this process! can be repeated without end.
When a line (in mathematics, of zero! width), intersects another line, a point of zero dimensions results. For the two arcs it means simply, that N * ( L / N ) = L (in millimeter or meter). The implied belief, that the length of a line is related to the sum of a number of points, is wrong. It forms the basis of the erroneous idea that a line must have points in proportion to its length.
06-Repeating decimals
Things clearly unequal can be 'proven' to be equal, again by using the extra-ordinary properties of infinity.
In this example, ( 0.999 . . . = 1 ), it is done as follows:
This result, the equality, is obtained by using two different numbers. When 0.999 . . . is treated as a number and is multiplied by 10 it changes to 9.99 . . ., but it changes to 9.999... when 9 is added to it, that may not seem to be important, but it holds the clue to this paradox.
Multiplication and subtraction can only be performed on ordinary numbers, however long the tail of decimals may be, in the calculation it must have a definite number of decimals, and that same number must be used throughout.
Using the same number in the calculation, slightly changing the notation, changes it to:
This is the correct result for any N repeated decimals*, N is the number where the unending process was, temporarily, stopped, to do the calculations. Again, in a sequence without an end, there is no reason to believe that this pattern will not be maintained.
When the 'calculation' is shown in its true form, that is, ADD! nine to both sides, it proves that 1 = 1. In the subtraction the decimal parts cancel, and that is the case whatever its nature.
Infinity is not a number and is not a limit, but in demonstrations one reads regularly about a hotel with an infinite number of rooms, and an infinite number of 9 s', and so on.
Note*:07-Cantor and infinity
Cantor proves that infinity comes in different sizes, 'infinite numbers' and 'transfinite numbers'. It is stated that it is e.g. impossible to write an algorithm that includes all decimal numbers between zero and one. The so-called diagonal argument proofs that.
The binary numbers (the decimal points have been omitted) in the frame, --horizontally n and vertically 2n--, show that all numbers between zero and one can be listed to any degree of accuracy. Here all numbers, with four decimals, are present in the list, and therefore on a 'Cantor' diagonal.
( The list, with its clear pattern, goes back to Euler ).
Using decimal numbers gives the same pattern, but then we have for the number of rows 10n. It is obvious that n is smaller than 10n, e.g. for n equal to six for the decimals, one million rows are needed.
Every additional decimal increases the number of columns by one, but the rows increase by a factor ten, both unending processes, but the rows 'progress' much faster, meaning that the diagonal argument, that rests on equal numbers for rows and columns is not applicable. All decimal numbers with n digits are included.
It shows that, when taking N for the number of decimals the number of rows equals 10N, and that one can use that pattern to do calculations because there is no reason to assume that this pattern will not be maintained. Cantor introduces for the rows a different type of infinity.
To paraphrase Orwell:
It arises from considering infinity a number. A number with some very special properties, but a number, nevertheless. It is equally obvious that infinity then no longer means boundless or endless, it has become a very large number, with a varying degree of largeness, notwithstanding what is said to the contrary.
Infinity is then a range of very large numbers, of indefinite numbers, denoted by
etc.
08-Remarks
The set theory as developed by set theorists is of course a typical product of professional mathematicians. Dividing mathematics in three sections helps understanding the different treatments.
I find the following division helpful:
AM uses PM, but it always helps to know and be aware of the real nature of the application. To give a simple example: the area of square field with three meter sides is not 3m x 3m = 9m2, but 3 x 3 x 1m2 = 9 m2. Of little consequence, but take three rows of three beauty queens, (indicated by q), the result is then, 3q x 3q = 9q2, producing 9 square! beauty queens.
A better example is the following, well known, anecdote:
Trivial business indeed, but when one reads the extra-ordinary interpretations given by theoretical physicists, one can only wonder if they are --at their level-- not making Johnny's type of mistake.
Points on a line, can also do with a proper interpretation.
In articles on infinity, the number of points on lines of different length is used as an other paradox of the infinite. Schoolboys knowledge of Euclid gives the answer:
Breadthless has as a consequence that as long as there is a distance, however small, between lines, there are two separate lines, and two points when they intersect an other line. Such lines can, however, not be next to each other, with no distance between them; when that happens, we have a single line, and consequently a single point, 'which has no part'.
Cantor's set theory is, of course, for the professional mathematician, but in my opinion they have not succeeded in really taming infinity. The idea of infinity is difficult to understand, and it is even more difficult to work with for the ordinary every day user of mathematics. If one likes to really understand the subject one is working with, like I do, quite a few questions come to the fore. I dealt with some of them, and like to hear the opinion of others about this.
snr@casema.nl
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