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).


Standard hosts research into Bergsonian axioms for physics.

Note that a main conjecture underlying most recent work on this project has been disproved. Consequently the project will be in limbo for the foreseeable future.

The goal is to axiomatize Henri Bergson’s insight using mathematics that were not available to him during his lifetime, then to find whether structures used in physics supply models of the axioms, and (if so) whether they supply all such models, which could help explain some of the universe’s more baroque structural properties.

Bergson’s central insight was that the world is the continual creation of new possibilities, rather than the successive realization of pre-existing possibilities. He argued that this insight alone shows a way out of the block-universe philosophy’s paradoxes. The tools needed to make this insight rigorous did not exist during Bergson’s lifetime, but starting in the 1960s, set theorists developed a robust account of how the universe of mathematical possibilities could expand. For example, there is a core set of real numbers that must be possible because they are definable (more precisely, constructible in the sense established by Gödel); yet it is consistent to suppose that further real numbers with various properties could be added to the continuum.

This suggests a way to revive Bergson’s insight as a precise theory: identify each spacetime point with a distinct continuum, in such a way that more real numbers are possible at later points than at earlier ones, and write axioms to capture the idea that the new real numbers are constructed out of older ones in a smooth, philosophically innocuous way. This is carried out in the paper “Self-Constructing Continua.”

The resulting Bergsonian axioms are clear enough; the matter of whether any structure can satisfy them is not. The technique that lends itself to producing candidates for the axioms is set-theoretic forcing. Applied to our project, it would associate spacetime points with boolean algebras, nested so that an earlier point’s algebra is always a subalgebra of a later point’s; a generic filter G on the outermost algebra would then be postulated, and we would identify a spacetime point with the set of real numbers constructible from G‘s restriction to that point’s algebra.

There are two immediate difficulties with this plan.

The mathematical difficulty is that none of the well-known types of boolean algebras works. While some of them (like the Cohen real and random real algebras) can be used to associate each point with a different continuum in the manner just described, these continua will fail to satisfy the Bergsonian axioms. (This failure is shown in the “Self-constructing continua” paper.)

The scientific difficulty is that our whole setup seems alien to any actual theory of physics. Or, at least, to any physics that an undergraduate is likely to encounter. Algebraic quantum field theory (AQFT) does have similarities to our setup. It too associates regions of spacetime with nested algebras, taken to represent the observables within those regions. These are von Neumann algebras rather than boolean algebras, but algebras of the latter type can be derived from them (more precisely, from their projection lattices) in a natural way.

The chief “coincidence” that has motivated research on Bergsonian axioms is that, under a certain conjecture about von Neumann algebras, the nested boolean algebras derived from AQFT models can have a property (rigid inclusion) that lets them evade the mathematical difficulty noted above. The way in which they would evade the difficulty is shown in “AQFT as a possible source of self-constructing continua.” Moreover, forcing with these algebras has been shown to be a noncommutative analog of random-real forcing that induces a new “random state” on an algebra of observables in AQFT; this suggests that we may eventually be able to argue that Bergson’s insight can be scientifically vindicated if and only if the world grows through the emergence of random quantum states — which it manifestly does!

The central question of our research project is thus: Can we find a variant of random-state forcing that yields continua satisfying the Bergsonian axioms? We will continue to post work on this project as it becomes available.