2021-12-13

Let ${Q}^{2}$ be the rational plane of all ordered pairs (x,y) of rational numbers with the usual interpretations of the undefined geometric terms used in analytical geometry. Show that axiom C-1 and the elementary continuity principle fail in ${Q}^{2}$. (Hint: the segment from (0,0) to (1,1) can not be laid off on the x axis from the origin.
Axiom C-1: If A, B are two points on a line a, and if A' is a point upon the same or another line a′ , then, upon a given side of A′ on the straight line a′ , we can always find a point B′ so that the segment AB is congruent to the segment A′B′ . We indicate this relation by writing $AB\stackrel{\sim }{=}A\prime B\prime$. Every segment is congruent to itself; that is, we always have $AB\stackrel{\sim }{=}AB$.

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Step 1
Let A, B lies on Horizontal axis.
Lets

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Suppose that the Elementary Continuity Principle holds for this interpretation. Let
Let C be the circle in ${Q}^{2}$ with center O and radius OA. Note that C consists of all points (x, y) in ${Q}^{2}$ such that ${x}^{2}+{y}^{2}=9$.
Since $O\cdot P\cdot A,OP. Hence, P is inside C. Let $B=\left(\frac{9}{5},\frac{12}{5}\right)$. Then . This implies that $OA and, hence, that R is outside C. Hence, by the Elementary Continuity Principle, there exists a point Q in ${Q}^{2}$ such that Q lies on P R and C.
Since Q lies on $PR,Q=P+t\left(R-P\right)$ for some real number t such that $0\le t\le 1$. Thus, $P=\left(1,0\right)+t\left(2,4\right)=\left(1+2t,4t\right)$. Since P is in ${Q}^{2}$, we conclude that t is a rational number. Since P lies on C, $OP\stackrel{\sim }{=}OA$. Hence, ${\left(1+2t\right)}^{2}+{\left(4t\right)}^{2}=9$. In other words, $5{t}^{2}+t-2=0$. Since $0\le t\le 1$, it follows that $t=\frac{-1+\sqrt{41}}{10}$ and, hence, $10t+1=\sqrt{41}$. Since t is a rational number, it follows that $10t+1$ is a rational number. In other words, $\sqrt{41}$ is a rational number. On the other hand, since 41 is not a perfect square, $\sqrt{41}$ is not a rational number. This is a contradiction. Hence, the Elementary Continuity Principle fails to hold for this interpretation

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