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 $\{ X \}$ ).

A *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) $X = \mathbb{R}(\bigcup \mathcal{N})$.

A ** 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).