The Bergsonian Axioms

Standard

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 $X$.

continuum is a set $X$ of real numbers that is constructively closed, meaning $\mathbb{R}(X) = X$.

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 continua is one that satisfies the following:

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

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 the following:

Ordinal Creation Ax.: $(\exists X,Y \in \mathcal{F})(\lambda(X) > \lambda(Y))$, where $\lambda(X)$ is defined as the smallest ordinal $\alpha$ such that $L_\alpha(X)$ satisfies $ZF^-$, that is, the axioms of Zermelo-Frankel set theory without the power-set axiom.

 

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 is just the last “step” in a process of combining, each of whose infinitely many previous steps are already complete. Note that if $X$ is the continuum of a Cohen-real extension or of a random-real extension, it cannot have an $\mathcal{N}$ that satisfies this axiom. In both cases $X$ has an unbounded increasing sequence of sub-continua; however, the sequence’s union is itself constructively closed, hence a continuum (of a model of ZF that notably does not satisfy the Axiom of Choice) that is smaller than $X$.

The Foundation Axiom is a “well-foundedness” guarantee ensuring 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 $\mathcal{F}$.