A shallow investigation of a single axiom of the Real numbers leading to a realization about the Complex numbers
It is the case that the numbers we as mathematicians use constantly have no gaps in measurable distance between sequential elements. Why is this the case? In particular why is it obvious that the numbers used for (almost) all of mathematics provably have no measurable gaps. I have made it my business to relearn analysis in more depth and I found myself coming back to this question. There is an axiom of completeness associated with $\mathbb{R}$, necessarily this axiom is made obvious via any construction of $\mathbb{R}$. Here I will investigate this central axiom and make an attempt to examine its finer details. With this newly discovered intuition I investigate $\mathbb{C}$ to address the same question, with particular interest to the contrasting answer.
Intuition is key to every field of mathematics, one does more and more problems until their intuition blossoms. If one builds their intuition upon assumptions that are not completely understood, this can be dangerous to later concepts. Thus is the purpose of my investigation into $\mathbb{R}$. The completeness axiom did not entirely make sense to me which I found interesting enough to write about. This axioms states simply that the real numbers are complete, meaning there is no measurable gap between any two sequential elements.
\begin{equation} \label{comp-axiom} (a, b \in \mathbb{R}) \land (\nexists c \in \mathbb{R} \mid a < c < b) \implies \forall \epsilon > 0, |a - b| < \epsilon \end{equation}
It is particularly interesting that it is vacuously proven through both common constructions of $\mathbb{R}$. Reading through
When we intuit numbers in general we think of the naturals first (naturally). These being the numbers beginning at one incrementing to infinity at a step of one. After we have pondered the naturals for some time we can begin to understand, there exist natural
\begin{equation} \label{rationals} \mathbb{Q} = [ \frac{z}{n} \mid (\forall z \in \mathbb{Z})(\forall n \in \mathbb{N}) ] \end{equation}
We observe from Eq. \eqref{rationals} that the rational numbers include the Integers and every fractional value between sequential Integers. In the event that all numbers of a desired space are fractional $\mathbb{Q}$ presents no problem. Once numbers such as $\pi$ and $e$ appear the problem becomes self-evident. There does not exist a fractional form of these numbers. Formally this concept is presented below in Eq. \eqref{rational problem}.
\begin{equation} \label{rational problem} (\forall a_{n},a_{n+1} \in \mathbb{Q} \mid a_{n} < a_{n+1}) (\nexists h \in \mathbb{Q} \mid a_{n} < h < a_{n+1}) \implies (\exists z \notin \mathbb{Q})(a_{n} < z < a_{n+1}) \end{equation}
We observe that there exist numbers outside the Rational ring, numbers that exist measurably in the world. Thus in an attempt to completely capture the numbers we can observe metrically we create another set of numbers. A union of the Rationals designed above and those numbers which are not included in the rationals but can be detected. The question now becomes how do we construct such a ring
Of the three constructions I present here this one makes the least sense to me. Thus in perfect mathematics fashion I shall present it first. The crux of Cantor’s construction is the understanding of the Cauchy sequence. Not a particular set but (at least in my mind) an attribute of a given set.
\begin{equation} \label{cauchy-seq} (\{ a_n \} \subset \mathbb{R})(\forall \epsilon > 0)(\exists N \in \mathbb{N}) \implies (|a_n - a_m| < \epsilon)(n,m > N) \end{equation}
In Eq \eqref{cauchy-seq} we understand that a Cauchy sequence is one that after some element $a_n$ (assuming $n \le m$) changes very little. That is all the elements of a Cauchy sequence after a particular element converge to the same number
Thus by including those supremums of these sequences we can provably include all irrational values. This is of course an oversimplification, c’est la vie. Thus we can conclusively prove the completeness of $\mathbb{R}$ such that we include all rational Cauchy sequences and their respective supremum. With this construction it should be logically impossible to have gaps in the number line. A colloquial analogy might look something like this: “if you stand in a room with limited light, take into account first what you can see, then use what you can see to infer whats between what you can see and the walls”
Dedekind cuts, the explanation rooted in frustration
flowchart LR
x_dot[...] <---> x[x] <---> z[z] <---> y[y] <---> y_dot[...]
Dedekind proves that each cut produced by no rational number must be produced by an irrational number
We can see this is a valid cut as $\forall a_1 \in A_1, \forall a_2 \in A_2 | a_1 < a_2$. It can then be seen that There exists no rational number whose square is $D$ thus this cut is produced by a non-rational value.
Let us assume the following:
\[\begin{equation} \label{lemma-init} \exists t, u \in \mathbb{Z} \mid (\frac{t}{u})^2 = D \end{equation}\]Note here we can assume that $u$ is the smallest integer such that $u^2 D = t^2$. We know this given the following.
\[\lambda^2 < D = (\frac{t}{u})^2 < (\lambda + 1)^2\]We can now find the integer $u’$.
\[\lambda < \frac{t}{u} \implies u' = t - \lambda u\] \[0 < u' < u \implies 0 = \lambda u - \lambda u < t - \lambda u < (\lambda + 1)u - \lambda u = u\]We can also find $t’ = Du - \lambda t$ by way of the equation $\lambda t < \frac{t^2}{u} = Du$.
\[\begin{equation} \label{lemma-eq} \begin{aligned} t'^2 - Du'^2 = D^2u^2 + 2Du\lambda t + \lambda^2 t^2 - Dt^2 + 2Dt \lambda u - D \lambda^2 u^2 = (\lambda^2 - D)(t^2 - Du^2) \end{aligned} \end{equation}\]We know that $u$ is the smallest value where Eq \eqref{lemma-init} should be true. We have proven that another number $u’$ exists such that the property holds with Eq \eqref{lemma-eq}$^\blacksquare$. The above proof is taken with appreciation from
Thus the square of every rational number $x$ is either greater than or less than $D$. It follows that there is no greatest rational numbers in either $A_1$ or a least rational number in $A_2$. Let us consider the following rational number.
\[\begin{equation} \label{theorem: y def} y = \frac{x(x^2 + 3D)}{3x^2 + D} \end{equation}\] \[\begin{equation} \label{theorem: y-x} \begin{aligned} y - x &= \frac{x(x^2 + 3D)}{3x^2 + D} - x * \frac{3x^2 + D}{3x^2 + D} \\ y - x &= \frac{2x(D-x^2)}{3x^2 + D} \end{aligned} \end{equation}\] \[\begin{equation} \label{theorem: y2D} \begin{aligned} y^2 - D &= \frac{(x^2 - D)^3}{(3x^2 + D)^2} \\ &= \frac{x^2(x^4 + 6Dx^2 + 9D^2) - D(9x^4 + 6Dx^2 + D^2)}{(3x^2 + D)^2} \end{aligned} \end{equation}\]If we assume $x \in A_2$ that is $x^2 > D$, we must also note that $y \in A_2$ as $y-x > 0 \implies y>x$ from Eq \eqref{theorem: y-x}. We can show that $x$ is not the greatest values of $A_1$ by Eq \eqref{theorem: y2D} that $y^2 - D < 0$ implying $y \in A_1$. Once we have shown that there is no greatest rational value of the sequence $A_1$ we show there is no least rational value of $A_2$. Assume $x \in A_2$, we know that $y < x \land y > 0$ and $y^2 > D$ thus $x$ is not the smallest value of $A_2$. Thus no rational value is the smallest value of $A_2$ showing that $(A_1, A_2)$ can not be generated by a rational value$^\blacksquare$
Dedekind has defined irrational numbers using his cuts, now we must define the set $\mathbb{R} = \mathbb{Q} \cup (\mathbb{R} \setminus \mathbb{Q})$. The way he does this is by stitching together all possible cuts
There exists only one number $a_1’$ such that $a_1’ \in A_1 \land a_1’ \notin B_1$. In this second case, it is true that for each value $b_1 \in B_1$ that $b_1 < a_1’$. Thus there must exist some $b_2’ \in B_2$ such that $b_2’ = a_1’$. This is true given that all values of $B_1$ are less than $a_1’$; recall all numbers are contained one of the two sides of a cut thus the number $a_1’$ if not contained in one side of the cut must be in the other side. Given that there exists only one $a_1’ \in A_1$ such that $a_1’ \notin B_1$ but all other values of $A_1$ are in $B_1$. We deduce that $a_1’ = b_2’ \in B_2’$ is the smallest element of $B_2$ or the infimum. That is $\alpha = \beta$ such that the generating numbers of the two cuts must be equal. In particular we say that these cuts are unessentially different
It is the case that the inverse of situation two reveals the supremum property or the least-upper-bound property necessary for completeness. Revealing such a property satisfies my question. With that information we move to the next method, however it is worthy of note that both
Karl Weierstrass also has a construction of a real numbers. One that particularly presents as less of a set theoretic approach. Weierstrass describes that we he terms an aggregate, see
Weierstrass goes on to describe rationals in terms of these aggregates using $\Lambda = \mathbb{N}$. In particular outlining a more combinatoric definition of the field of rational numbers. Specifically with Proposition 3.5 in
It is the case that Weierstrass crafts an explicit construction of the real number described in