Axiom:Binary Mess Axioms

Definition

Let $S$ be a set.

Let $I$ be the set of all finite subsets of $S$.

Let $\Bbb B$ be a Boolean domain.

Let $M$ be a set defined as:

$\ds M \subseteq \bigcup_{P \mathop \in I} \Bbb B^P$

where $\Bbb B^P$ denotes the set of all mappings from $P$ to $\Bbb B$.

That is, such that $M$ is a set of mappings from finite subsets of $S$ to $\Bbb B$.

$M$ is a binary mess if and only if the following conditions are satisfied:

$(\text M 1)$	$:$			$\ds \forall P \in I: \exists t \in M:$	$\ds \Dom t = P $				Every finite subset of $S$ is the domain of some mapping in $M$
$(\text M 2)$	$:$			$\ds \forall P \in I: \forall t \in M:$	$\ds t {\restriction_P} \in M $				The restriction of every mapping in $M$ to every finite subset of $S$ is in $M$

\((\text M 1)\)	$:$			\(\ds \forall P \in I: \exists t \in M:\)	\(\ds \Dom t = P \)				Every finite subset of $S$ is the domain of some mapping in $M$
\((\text M 2)\)	$:$			\(\ds \forall P \in I: \forall t \in M:\)	\(\ds t {\restriction_P} \in M \)				The restriction of every mapping in $M$ to every finite subset of $S$ is in $M$