Let us hypothesize on the notion of local time.

Existence of temporal order: For each concrete basic thing x ∈ Θ, there exist a single ordering relation between their states ≤.

We now give a name to this ordering relation:

Denotation of temporal order: The set of lawful states of x is temporally ordered by the ≤ relation.

The above is a partial order relation: there are pairs of states that are not ordered by ≤; e.g. given an initial condition (x_{0},v_{0}) for a moving particle, there are states (x_{1},v_{1}) that are not visited by the particle.

Proper history: A totally order set of states of x is called a proper history of x.

The axiomatics do not guarantee the existence of a single proper history: they allow many of them, as in “The garden of forking paths”. The following axiom forbids such possibility.

Unicity of proper history: Each thing has one and only one proper history.

Arrow of time: The axiomatics describe a kind of “arrow of time”, although it is not related to irreversibility.

A proper history is also an ontological history. The parameter t ∈ M has not to be continuous. The following axiom, a very strong version of Heraclitus’ hypothesis *Panta rhei*, states that every thing is changing continuously:

Continuity: If the entire set of states of an ontological history is divided in two subsets h_{p} and h_{f} such that every state in h_{p} temporally precedes any state in h_{f}, then there exists one and only one state s_{0} such that s_{1} ≤ s_{0} ≤ s_{2}, where s_{1} ∈ h_{p} and s_{2} ∈ h_{f}.

The axiom of continuity is stated in the Dedekind form.

Continuity in quantum mechanics: Although quantum mechanical “changes of state” are usually considered “instantaneous”, theory shows that probabilities change in a continuous way. The finite width of spectral lines also shows a continuous change in time.

Real representation: Given a unit change (s_{0}, s_{1}) there exists a bijection T : h ↔ R such that

h_{1} = {s(τ)|τ ∈ R} —– (1)

s_{0} = s(0) —– (2)

s_{1} = s(1) —– (3)

Local time: The function T is called local time. The unit change (s_{0}, s_{1}) is arbitary. It defines an arbitrary “unit of local time”.

The above theory of local time has an important philosophical consequence: becoming, which is usually conceived as evolution in time, is here more fundamental than time. The latter is constructed as an emergent property of a changing (i.e. a becoming) thing.