User:KBlott/Definition/In

Definition

Preamble

Let $(B_{\bot}^{\top}, \vee, \wedge, -, \bot, \top)$ be a Boolean algebra.
Let $(U_Z^I, \cup, \cap, -, Z, I)$ be a universe algebra.
Let $C \in $$U^I$ be a class in the given universe.
Let $x \in $$U_Z$ be an object in that universe.

Definition

Then, in ($\in$) is a mapping

$U_Z \times U^I \to B_{\bot}^{\top}$

such that

$x \in C \implies \top$.

Notice that the definition of $\in$ is circular. The symbol $\in$ is used to define itself. This technically makes the definition an axiom. Also, the definition is very abstract and requires a somewhat generalized notion of Boolean algebra and what a universe is. Note, however, the axiom it is quite elegant.

Comments

Notice that if the Boolean domain contains only two elements (ie

$B_{\bot}^{\top} = \{\bot, \top\}$

with

$\bot \neq \top)$

then there is a dual mapping

$-(\ni): U_Z \times U^I \to B_{\bot}^{\top}$

such that

$-(x \ni C) \iff x \in C$.