Lately, I've been thinking a lot about the nature of the real numbers. There are several issues that play into this, like how to have an intuitive understand of continuity in the reals when there is no notion of a "next" real to a given one. Much of my thinking was an attempt to understand how for every two rational numbers there is an irrational between them, and for any two irrations there is a rational between them. Yet there are far more irrationals than rationals. So what are the reals, really?
It's not simple to obtain even a given real. We start with an integer, and the simplest integer is zero, obtained by a count of the null set. That is, 0 = NullSet (I don't see how to get the proper symbol for the null set in this editor). The next integer is formed by a set containing a single instance of the null set, that is, 1 = { NullSet }, and 2 = { NullSet, { NullSet } }, etc. There are a countably infinite number of integers, and each one is unique. Personally, I find it completely amazing that this is the same zero that's the identity element of addition and even more amazing that this one is the identity element of multiplication, but that's a subject for another post.
With the integers established, we can build the rationals by the introduction of a unary reciprocal operator. This gives an inverse over multiplication for all elements except zero (and the necessity of this exception is an interesting point worth another post). The rationals are formed from dividing one integer with another (except zero). This allows us to solve any linear, or first order equation, of the form 0 = ax + b, where a and b are integers. It should be noted that the integers are not exactly rational numbers, though this is a very fussy point. The rational number 4/1 has the same magnitude as the integer 4, but it's not quite the same. A similar distinction may be more interesting later.
It's been known since ancient times that the integers and rationals weren't adequate, since there is no ratio of integers that can produce two when squared. Only recently have I understood why this so upset the Pythagorians, and I'm sure my distress concerning the real numbers is only a fractions of what they felt then. As far as I know, there are only two ways to form a meaningful irrational number, Dedekind cuts and Cauchy sequences. I haven't read Dedekind's papers yet, and I don't know much about his approach. I'm more familiar with Cauchy sequences, which uses an infinite number of rationals to approximate an irrrational, which is obtained as the limit of the infinite sequence. Technicaly, a Cauchy sequence is a series of rational numbers (a0, a1, a2, a3, ... ) where there exists some N such that for any two distinct i, j > N, there is a delta (I'll have to use d) 0 < | aj - ai | < style="font-style: italic;">The Calculus Gallery that the rationals are very poor approximators of the irrationals, from Liouville's number. That is, for any irrational x, any and all rational numbers are a measurable distance away from it
More later.
Notes about builds from integers to rationals to Cauchy Sequences and irrationals. Divide the irrationals into algebraics and transcendentals, the difference being the infinitude of their formula. Distinguish further those transcendentals which can be fully specified with a finite expression and those that cannot. The later class is the unnameable transcendentals. Expand on that.
