Dear Eckard,
You wrote:
"I am arguing: Real numbers must be understood homogeneous, i.e. without distinction between rational and irrational ones. Only then they are truly essentially different from the rational ones. This requires to admit that numbers of infinite precision within a continuum of such truly "real" numbers cannot be subject to trichotomy. In other words: I consider Dedekind's extension from rational to real numbers reasonable on condition we do not try to enforce trichotomy for the real numbers too. This paradise is elusive."
Unless I'm missing something, the idea of homogeneous R is acceptable, because any magnitude whatsoever can be set equal to the unit value of R. Thus, what was an irrational magnitude in association with another unit value, can be transformed into a rational magnitude by fiat.
Of course, as soon as the transformation is made, another set of magnitudes, different from the first, becomes irrational. The process is repeatable ad infinitum. So, why then do we speak of rational and irrational numbers in an absolute sense? A given magnitude may be considered rational or irrational, depending upon our choice of reference. In this sense, geometric magnitudes are relative, not absolute.
In my opinion, the really ancient (i.e. more ancient than the Greeks) had it right, when they said that, where two quantities exist, one greater than another, then there shall be one greater still. They made no mention of "irrational" magnitudes.
However, it's not that we can ignore them, or else we wouldn't be having this contest. But the more important question is the one of trichotomy, which the ancients held as fundamental. When we consider that the elements of R have two interpretations, then we can consider the trichotomy of numbers two ways. The first sub-divides the unit number (the quantitative interpretation of number), and the second compares two sets of unit numbers (the operational interpretation of number).
In the first interpretation, the numbers 1/2 and 2/1 differ by 2 operations, not by 2 units. This is the basis of the concept of octaves, or a doubling/halving operation, we might say. If we double 1/2 once, we get the quantity 1, double it again, we get the quantity 2. Conversely, if we divide 2 into half once, we get the quantity 1, half it again, we get the quantity 1/2. Performing the operation twice in either "direction," shows the symmetry of the numbers, with respect to the number 1.
In the second interpretation, the numbers 1/2 and 2/1 differ by two units, in a a single comparison operation. We are evaluating them to determine which number is greater than the other number, or if there is no difference between them. In this case, the operation is like the balance scale, rather than the knife, and the quantities are discrete. Although we could choose to include fractions of discrete units, but then we must decide whether or not to include both the equal divisions (i.e. rational parts) of a single unit, and the arbitrary divisions (i.e. irrational parts) of a single unit, or only one of these.
Choosing to exclude fractions, we can use arbitrary signs (e.g. & -) to indicate our perspective relative to which side of the number 0 we might refer to. However, in this case, we must remember that the number 0 is actually the number 1 again, as in the first interpretation, but with a different meaning.
This time, the number 1 is both 1 and 0, at the same time. It is interpreted as 0, because 0 represents the result of the comparison evaluation between the two numbers, when there is no difference between them. However, 0 is also interpreted as 1, because 1 means that the relative number of units in the comparison of the two sets is equal. It's just two different ways of regarding the same thing.
So, dear Eckard, I believe that you are right in that no trichotomy exist in R, natively, but we must recognize that we can clearly employ the elements of R to produce a trichotomy, in at least two ways.
Sincerely,
Doug
