In Chap 3 of GTM53, there is an interpretation of L2Real where the symbols of numbers is interpreted as a random variable. I cannot finish the proof of the completeness axiom of reals $\forall f(\exists y\forall x(f(x)\le y)\to \exists z\forall y(\forall x(f(x)\le y)\Leftrightarrow z\le y))$ . Here is how I tried:
We need to prove that forall $\bar{f}\in\bar{R}^{(1)}$, $$\bigvee\limits_{\bar{y}\in\bar{R}}\bigwedge\limits_{\bar{x}\in\bar{R}}\|\bar{f}(\bar{x})\le\bar{y}\|\le\bigvee\limits_{\bar{z}\in\bar{R}}(\bigwedge\limits_{\bar{x}\in\bar{R}}\|\bar{f}(\bar{x})\le\bar{y}\| \Leftrightarrow\|\bar{z}\le\bar{y} \|)$$.
It suffices to prove that there exists $\bar{z}=\bar{z}(\bar{f})$ such that $$\bigvee\limits_{\bar{y}\in\bar{R}}\bigwedge\limits_{\bar{x}\in\bar{R}}\|\bar{f}(\bar{x})\le\bar{y}\|\le\bigwedge\limits_{\bar{x}\in\bar{R}}\|\bar{f}(\bar{x})\le\bar{y}\| \Leftrightarrow\|\bar{z}\le\bar{y} \|$$
and I tried some expressions of $\bar{z}$, but none of them make sense.