The Bergsonian Axioms


Bergsonian axioms for physics formalize the idea that the world is the continual creation of new possibilities, rather than the successive actualization of pre-existing possibilities; they assert, under the supposition that each point in space-time can be identified with the set of all real numbers possible there, that new real numbers are created in a smooth, philosophically innocuous way.

We take a real number to be a set of natural numbers (it is easy enough to put such sets into correspondence with points on the real line, but we will have no reason to do so.)

The constructive closure of an arbitrary set $X$ of real numbers, written $\mathbb{R}(X)$, is the set of all real numbers in the constructive hierarchy $L(X)$ “seeded” with $X$ (more precisely, with the transitive closure of $\{ X \}$ ).

continuum is a set $X$ of real numbers that is constructively closed, meaning $\mathbb{R}(X) = X$. If it equals $\mathbb{R}(\{ x \})$ for some real number $x$, we call it singly-generated

A set $\mathcal{N}$ of continua self-collects into another continuum $X$ if the following hold: (i) $X \not \in \mathcal{N}$;  (ii) $\mathcal{N}$ is a directed set under the inclusion ordering; (iii) $\mathcal{N} \in L(X)$; (iv) for no $x \in X$ does $\mathbb{R}(x)$ lie between $\mathcal{N}$ and $X$ in the sense that $\bigcup \mathcal{N} \subseteq \mathbb{R}(x) \subset X$; and (v) $X = \mathbb{R}(\bigcup \mathcal{N})$.

self-constructing family $\mathcal{F}$ of singly-generated continua is one that satisfies the following:

Self-Collection Ax.: $(\forall x \in \mathbb{R}(\mathcal{F}))(\mathbb{R}(\{x\}) \in \mathcal{F} \iff (\exists \mathcal{N} \subseteq \mathcal{F})(\mathcal{N}$ self-collects into $\mathbb{R}(\{x\})) )$;

Foundation Ax.: $(\forall x \in X \in \mathcal{F})(\exists Y \in \mathcal{F})
(x \in Y \subseteq X$ and $(\forall Z \in \mathcal{F})(Z \subset Y \Rightarrow x \not \in Z))$.


A Bergson history is a self-constructing family that also satisfies an axiom intended to provide a robust solution to the Russell-type paradoxes. When $X = \mathbb{R}(\{x\})$ for some real $x$, we write $\lambda(X)$ for the smallest uncountable ordinal in $L(X)$, and we write $L^-(X)$ for $L_{\lambda(X)}(x)$. Note that $L^-(X)$ is well-defined (does not depend on choice of $x$).

Anti-Paradox Ax.: $(\exists X,Y \in \mathcal{F})(L^-(X) \in L^-(Y) )$.


Brief explanation of the axioms:

The Self-Collection Axiom can be construed as saying that a continuum $X$ arises as a member of $\mathcal{F}$ if and only if $X$ is the constructive closure of the union of some smaller continua in $\mathcal{F}$ that “naturally form a collection.” This is the case when these smaller continua have already been brought together in $\mathcal{F}$ in all finite combinations (directedness), so that taking their union and closure is just the last infinitesimal “step” in a process of combining whose infinitely many previous steps have already been completed.

The Foundation Axiom ensures that each real number $x$ “first arises” in an $\mathcal{F}$-member that is the constructive closure of some other continua in $\mathcal{F}$ that do not have $x$ as a member; thus $x$ does not “slip in unaccountably” during the growth of $\mathcal{F}$.

The Anti-Paradox Axiom asserts that the set of all sets “at” some $X \in \mathcal{F}$ — understood to be $L^-(X)$ — is itself a member of the set of all sets “at” some $Y \in \mathcal{F}$ that is larger than $X$. This arguably allows us to avoid the Russell-type paradoxes without introducing an ad-hoc “class”/”set” distinction (so long as the universe of sets “keeps expanding” this way).