Proof that Emergence of Something out of Nothing is Impossible
A growing number of physicists working in the area of quantum cosmology express the view, to the dismay of some philosophers, that it is possible to create something out of nothing, although the ‘nothing’ of quantum cosmology is not, strictly speaking, a logical nothing (~ ∀(x)), understood as negation of all physical attributes (x). Here I attempt to construct a proof that creating something out of nothing (in the strict logical sense of the term) is not even hypothetically possible.
The most charitable reading of the phrase ‘to create something out of nothing’ may be emergence of a something possessing physical attributes without any constitutive antecedents. The primary implication is that something did not exist before time t but exists at t. A secondary implication is that the relevant something is not a result of continuous transformation of something existent into something else, a result of nominal reconstitution or reconfiguration of pre-existing ingredients, or a renaming of something existent.
Proof from continuity. To exist is to endure in time, that is, to possess physical attributes continuously over non-zero duration. Creating something out of nothing entails a discontinuity in the history of a thing, an instant of zero-duration at the common limit of two continuous time-intervals: t(not-exist) and t(exist). Given that something that has just emerged ‘out of nothing’ has no duration at the instant of its emergence, that something has no existence at the instant of emergence. Furthermore, an instant of zero duration has no capacity to accomodate change from non-existence to existence, therefore no capacity to accomodate emergence. The same limitation must then affect any subsequent instants, ensuring that the alleged something has no duration of existence, therefore no existence.
More formally,
Premise 1: X exists at time-point t but not before t.
Premise 2: To exist is to persist over non-zero duration.
Implication: Time-point t has zero duration OR, in case of non-zero interval (infinitesimal), includes time-points t(beginning) and t(end), therefore not a time-point.
Consequence: X does not exist at t.
Following the same logic as above it is possible to construct a proof of the impossibility of annihilating anything to nothing. If the interval of time between existence and non-existence of something is zero, and there can be no change in an interval of zero duration, then that something cannot become nothing at any time.
Premise 3: X exists before t but not at t.
Premise 4: Change is possible only over non-zero time interval.
Implication: The time interval between the end-point of X’s duration and t has zero duration, therefore no change takes place at t.
Consequence: X exists at t.
There are two possible objections to Premise 4: a) a point in time can contain superposition of contradictory states, although emergence and nihilation of states faces the same doubts as emergence and nihilation of entities; b) change can be discrete/discontinuous, but this again faces the problem of emergence and nihilation unless time is not continuous but discrete, where all instants are extensive (quanta of duration).