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Suppose $X$ and $Y$ are jointly distributed real-valued random variables and for all outcomes $\omega_1$, $\omega_2$, we have $$ X(\omega_1)\le X(\omega_2)\quad\Longrightarrow\quad Y(\omega_1)\le Y(\o …

The first thing I thought of was order embedding and this is confirmed by an article on monotonicity in order theory.

In a directed graph $G$, what do we call a path, a sequence of edges $$(v_0,v_1),(v_1,v_2),\dots,(v_{n-1},v_n)$$ of length $n$, that intersects every other path of the same length $$(w_0,w_1),(w_1,w_2 …

You could call $$\mathbf r(t)=\langle x(t), y(t), z(t)\rangle$$ the vector function (or vector?) of $(x,y,z)$.

I once heard it was because "body" is too sensual for (Victorian?) England, so "field" was preferred. As @HJRW pointed out, this may be made up. The circumstantial evidence seems to be there, though: …

In philosophy, this would be called family resemblance -- if $E_i\cap E_j\ne\emptyset$ and $E_j\cap E_k\ne\emptyset$ then $E_i$ and $E_k$ have a family resemblance. That is, perhaps I have no common …

$\mathscr F$-indistinguishability. In analogy with Topological indistinguishability.

Yes, $C(s)$ is an example of a cylinder set. More specifically, $C(s)$ is called a basic open cylinder (since other cylinder sets are unions of such sets). See e.g. Andre Nies' monograph Computabili …

Perhaps there is some standard terminology. …

$ x +^{-1} y $ seems like a good notation in that $x^{-1}+^{-1} y^{-1}=(x+y)^{-1} $, and $$ \frac{1}{x}\frac{1}{+}\frac{1}{y} = \frac{1}{x+y} $$ (is this the "Freshman's Dream" in another incarnation? …

It may seem a bit backwards, but one could try to look at it the other way around: pretend the axioms and theorems are the things that we observe. We don't really observe the field $\mathbb R$ of all …

At the risk of stating the obvious, in modal logic the axiom schema $$\Box\varphi\rightarrow\varphi$$ is called the schema T.