On constructing instants from events (Q761436)

This paper gives precise conditions for the construction of instants of time from the ordering of events - a task first attempted by Bertrand Russell in 1914. An event ordering is a structure \((W,<)\), where the relation \(<\) (''wholly precedes'') satisfies (1) \(a<\!/a\), (2) \(a<b\&c<d\Rightarrow a<d\vee c<b\). The set of instants T(W) is then constructed from an event ordering W as the set of three-fold partitions (P,C,F) of W, where P contains the past events, C the current ones, and F the future ones. An ordering \(<\) in T(W) is defined by \((P,C,F)<(P',C',F')\Leftrightarrow P\subseteq P'\). The main theorem states that T(W) is isomorphic to the continuum if and only if \(W\neq \emptyset\) is dense and has a denumerable dense subset.