Here is the latest (pdf) version of the formalization (within set theory) of the idea of a self-constructing mathematical universe. The axioms are presented along with a schema for generating candidates to satisfy the axioms; the main necessary condition for a forcing set to succeed in this schema is derived.

by Henri Bergson

Translated by DVM, from “Le possible et le réel”, pp. 99-116 in La pensée et le mouvant, P.U.F./Quadrige, 6th edition, according to which it first appeared in the Swedish journal Nordisk Tidskrift in 1930. The French text is online here.

I would like to come back to a subject on which I have already spoken, the continual creation of unpredictable novelty that seems to go on in the universe. For my part, I believe I experience it at every instant. My attempts to represent to myself the details of what will befall me are in vain: how weak, abstract, diagrammatic they are in contrast to the event that actually happens! Actualization brings with it the unforeseeable little nothing that changes everything. Say I have to attend a meeting; I know what people I’ll find there, around which table, in which order, for the discussion of which problem. But once they come, sit down, and start chatting as I expected them to, once they are saying what I thought they would — the whole scene gives me a new and unique impression, as if it had now been drawn in one creative stroke by the hand of an artist. Goodbye to the image of it I’d made myself, that simple juxtaposition, thinkable in advance, of things already known! This scene may not have the artistic worth of a Rembrandt or a Velasquez; very well, it is every bit as unexpected, and in this sense every bit as original. Now I may be charged with simply not having known the details of the situation, with not having had the people, their gestures, and their attitudes at my disposal, and therefore with mistaking a simple overflow of details for true novelty in the whole. But I have the same impression of novelty in the unfolding of my inner life. And I experience it more vividly than ever when I exercise my will, bringing about an action of which I am the sole master. If I deliberate before acting, the moments of deliberation arise in my mind like the successive sketches, each unique, that a painter would make for his picture; the act itself, becoming accomplished, can realize as precisely as it likes what has been willed and hence foreseen, it will nonetheless take an original form. —Fine, one might say; there may be something original and unique in your mental state, but matter is repetition: the external world obeys mathematical laws; a superhuman intelligence that knew the position and velocity of all the atoms and electrons in the material world at a given moment could calculate any future state of this world, as we ourselves do for solar and lunar eclipses. —Ultimately I’d agree with that, if it were only a question of an inert world (and despite the controversy that has sprung up on this question, at least for elementary phenomena). But such a world is only an abstraction. Concrete reality contains living, conscious beings, that are framed all around by inorganic matter. I say living and conscious, because I take whatever lives to be conscious in principle; it becomes unconscious in practice when consciousness goes to sleep, but, down to the very regions where consciousness slumbers, in the plant kingdom for example, there is structured evolution, definite progress, aging — all the external signs of abiding that characterize consciousness. And why should we talk of inert matter into which consciousness can only be inserted, like a picture within a frame? By what right does the inert come first? The ancients imagined a World Soul that would ensure the continued existence of the material universe. Pruning this concept of its mythical aspects, I would say that the inorganic world is a series of repetitions, or of infinitely fast quasi-repetitions, which sum to visible and predictable changes. I’d compare them to the oscillations of a clock pendulum; the one ticks off the progress of a spring’s progress as it unwinds, the other sets the tempo of life for conscious beings and measures their abiding. This is how living beings abide essentially — they abide precisely because they are constantly developing something new, and because nothing can be developed without research, which always begins by slowly feeling one’s way forward. Time is this very hesitation, or it is nothing at all. If you take away consciousness and life (and you can only do this through artificial abstraction, since the material world, once again, may necessarily imply the presence of consciousness and life), you will indeed get a universe whose successive states are theoretically calculable in advance, like juxtaposed images on a film reel before it unwinds. But in that case, what good is the unwinding? Why does reality deploy itself this way? Why couldn’t it decline to be deployed? What good is time? (I’m talking about real, concrete time, and not this abstract time that is only a fourth dimension of space.) Such was the long-ago starting point for my reflections. Some fifty years ago, I was very keen on Spencer’s philosophy. One fine day I realized that time had no purpose, that it did nothing. Now, whatever does nothing is nothing. And yet, I said to myself, time is something. Therefore it does something. But just what can it do? Basic common sense answers: Time is what prevents everything from being given all at once. It holds back, or rather, it is identical to holding back. It must therefore be development. Wouldn’t it then be the vehicle of creation and choice? Would time’s existence not prove that things are undetermined? Would time not be this very indeterminacy?

If this opinion is not shared by most philosophers, it’s because human intelligence is made precisely to take things the other way round. I say intelligence, not thought, not mind. For besides intelligence, each of us has the immediate perception our own activity and of the conditions under which we exercise it. Call it however you like; it is the feeling we have of being creators of our intentions, of our decisions, of our acts, and thereby of our habits, of our character, of ourselves. As craftsmen of our own life, and even its artist when we so desire, we continually work at kneading the matter given us by the past and present, by heredity and circumstances, into a unique form, new, original, and unforeseeable as the form a sculptor gives to clay. We are doubtless aware of this work and its uniqueness while it goes on, but the essential thing is that we do it. We don’t need to delve into it deeply; we need not even be fully mindful of it, any more than the artist need analyze his own creative power. On the other hand, the sculptor must know the technique of his art and everything that can be learned about it. This technique concerns above all that which his work will have in common with others’; his materials’ requirements dictate it, to him and to all artists. It concerns that part of art that is repetition or manufacture, and not creativity proper. On this the artist focuses his attention, or what I would call his intellectuality. Likewise in the creation of our character we know precious little about our own creative power; to learn more would require us to turn back to ourselves, to philosophize, to swim against the current of nature — for nature wants action, and has hardly ever thought of speculation. As soon as we go beyond simply feeling our inner élan and thereby assuring ourselves that we can act, as soon as we bend thought back on itself so that it might grasp this power and capture this élan, our difficulty becomes great, as if we had to reverse the normal direction of consciousness. On the contrary, we have the greatest interest in familiarizing ourselves with the technique of our action, that is to say in extracting, from the conditions in which it operates, everything that might furnish us general rules and recipes to base our conduct on. Our acts will have newness only thanks to whatever repetitive sameness we have succeeded in finding in things. Our normal faculty of knowledge is therefore essentially a power to extract whatever is stable and regular in the flow of reality. Is this a matter of perception? Perception takes up the infinitely repeated perturbations that are heat and light, and contracts them into relatively unvarying sensations: trillions of external oscillations crystallize in our eyes, in a fraction of a second, into a vision of color. Is this then a matter of mental conception? To form a general idea is to abstract from diverse and changing things a common aspect that does not change, or which at least offers our activity a solid handhold. The constancy of our attitude, the sameness of our potential or virtual reaction to the multiplicity and variability of represented objects — there you have the hallmark of the generality of ideas. Is this, finally, a matter of understanding? That would simply be to find links, to establish stable relations among passing facts, to draw out laws — a task that admits of perfection, insofar as the relation is precise and the law mathematical. All these functions are constitutive of intelligence. And intelligence, being a close friend of the stable and the lawlike, attains truth insofar as it sticks to whatever is stable and lawlike in reality, to materiality. In this way it touches one side of the absolute, just as our consciousness touches another when it grasps within us a perpetual blooming of newness, or when, more expansively, it sympathizes with the boundlessly regenerative work of nature. The error occurs when intelligence claims to comprehend this latter aspect as it comprehended the first, and employs itself in a task for which it was not made.

I think that the big metaphysical problems are generally ill-posed, and that they resolve themselves when one corrects their givens, or maybe that they are problems posed in phantom terms, which vanish as soon as one inspects the formula’s terms up close. They are the offspring of our tendency to regard creation as mere assembly. Reality is global and indivisible growth, gradual innovation, persistence — like a rubber balloon inflating little by little and taking on unexpected forms at each instant. But our intelligence represents its origin and evolution as an arrangement and rearrangement of pieces trading places; it could then, in theory, foresee any state of the system. For in positing some definite number of stable elements, we give ourselves all their possible combinations in advance. And that’s not all. Reality as we perceive it directly has a fullness that never ceases to swell and that knows no emptiness. It has extension, just as it has persistence, but this concrete extension is not the infinite and infinitely divisible space that intelligence takes for its own workspace. Concrete space has been extracted out of things. They are not in it; it is in them. Only, as soon as our thoughts reason about reality, they make space into a container. Just as their habit is to bring pieces together in a relative void, they imagine that reality fills up some kind of absolute void. Now, if the misunderstanding of radical newness is the origin of poorly posed metaphysical problems, the habit of going from emptiness to fullness is the source of vacuous problems. And it is easy to see that the second error is already latent in the first. But I would first like to define it more precisely.

I am saying that there are pseudo-problems, the very problems that have been the most agonizing in metaphysics. For me they come down to just two. One has spawned the theories of being, the other the theories of knowledge.

The first consists in asking oneself why there is being, why something or someone exists. The nature of what exists doesn’t matter: just say that there is matter, or mind, or both, or say that matter and mind aren’t enough and manifest a transcendent Cause — in any case, once you have considered existences, causes, and causes’ causes, you feel yourself being dragged down the road to infinity. You only stop to ward off dizziness. People are always confirming, believing themselves to have confirmed that the difficulty remains, that the problem is still open and will never be solved. It’s true that it won’t, but it should never have been posed. It is only posed on the assumption of a nothingness that precedes being. One thinks “there could have been nothing” and then one is shocked to discover that there is something, or Someone. But analyze this sentence: “there could have been nothing.” You see that you’re dealing with words, not with ideas, and that this “nothing” has no meaning. “Nothing” is a term of everyday language that can only have meaning on humanity’s home turf, in the domain of activity and manufacture. “Nothing” designates the absence of whatever we were looking for, or wanted, or expected. Even supposing that experience did in fact present us an absolute void, it would be limited, it would have contours, it would in short be something. But in reality there is no void. We perceive and conceive only fullness. One thing disappears only because another has replaced it. Disappearance thus means substitution. Only we say “disappearance” when we see only one of substitution’s two halves, or rather, one of its two faces, the one that interests us. By doing so we note our preference for directing our attention to the object that has departed, and away from that which has replaced it. Then we say that nothing remains, meaning by this that what is does not interest us, that we are interested in what is no longer there or in what could have been there. The idea of absence, or of nothingness, or of nothing, is thus inseparably linked to that of disappearance, real or possible, and that of disappearance is itself only one aspect of the idea of substitution. These are all ways of thinking that are useful in practical life; it is particularly important to our productivity that our thinking be able to fall behind reality and remain fixed, when necessary, on what was or on what could be, instead of being monopolized by what is. But when we move from the domain of manufacture to that of creation, when we ask ourselves why there is being, why something or someone, why the world or God exists and why not nothingness, when in short we pose ourselves the most agonizing of metaphysical problems, we virtually accept an absurdity. For if all disappearance is substitution, if the idea of disappearance is only the truncated idea of substitution, then talk of the disappearance of everything is talk of a substitution that isn’t a substitution, a self-contradiction. Either the idea of the disappearance of everything has exactly as much existence as a circular square — the existence of sound, flatus vocis — or, if it does represent something, it gets across the motion of an intelligence going from one object to another, preferring the one it just left to the one it finds before it, and means by “absence of the first” the presence of the second. The whole was put forth, and then one by one each of its parts was made to disappear, without any attention paid to what replaced it. It is thus the totality of present things, only configured into a new pattern, that one has before oneself when one wants to totalize the absences. In other words this so-called mental picture of an absolute void is, in reality, a picture of absolute fullness in a mind forever jumping from part to part, having vowed never to consider anything but the emptiness of its own dissatisfaction, rather than the fullness of things. Which amounts to saying that the idea of Nothing, when it isn’t just a word, implies as much matter as that of All, with, in addition, an operation of thought.

I’d say as much for the idea of disorder. Why is the universe orderly? How has law imposed itself upon the lawless, form upon matter? How is it that our thought finds itself at home amid material things? This problem, which has become for the moderns the problem of knowledge, after having been for the ancients the problem of being, is born of the same kind of illusion. It vanishes if we consider that the idea of disorder has a definite sense only in the domain of human industry, or, as we have said, of manufacture, but not in that of creation. Disorder is simply the order we weren’t looking for. You cannot make one order disappear, even in thought, without making another spring up. If there is no free will or teleology, it’s because there is mechanism; when mechanism lets down its guard, then will, caprice, and teleology rush in. But when you expect one of these orders and find the other, you call it disorder, formulating what is in terms of what could or should be, and reifying your disappointment. Disorder thus always comprises two things: first, external to ourselves, an order; second, internal to ourselves, the picture of a different order that alone interests us. Making something disappear again means substitution. And the idea of making all order disappear, that is, of achieving absolute disorder, contains a veritable contradiction, since it consists in letting only one face of an operation remain which, by hypothesis, has two. Either the idea of absolute disorder is only a combination of noises, flatus vocis, or, if anything does answer to it, it is the motion of a mind jumping from mechanism to teleology, from teleology to mechanism, and marking its place at each step by indicating the place where it is not. So, in order to make order disappear, you give yourself two or more of them. Which amounts to saying that the concept of an order coming to add itself onto an “absence of order” implies an absurdity, and that the problem vanishes.

The two illusions that I have just pointed out are in reality only one. They consist in believing that there is less in the idea of void than in the idea of fullness, less in the idea of disorder than in that of order. In reality, there is more intellectual content in the ideas of disorder and nothingness (when they do represent something) than in those of order and existence, because they imply many orders, many existences, and, beyond that, a mind juggling unconsciously with them.

Well, I find the same illusion in the case before us. There are plenty of misunderstandings, plenty of errors, at the bottom of the doctrines that misunderstand the radical newness of each moment of evolution. But above all there is the idea that the possible is less than the real, and that, for this reason, the possibility of things precedes their existence. They would thus be representable in advance; they could be thought before being realized. But the opposite is true. If we leave to the side closed systems, subject to purely mathematical laws and locked off due to the fact that abiding never gets a foothold in them — if we consider the whole of concrete reality or simply the world of life, and even more so that of consciousness, we find that there is more, and not less, in the possibility of each of the successive states than in their reality. The possible is simply the real with, in addition, a mental act that casts its image into the past once it has been produced. It’s just that our intellectual habits prevent us from seeing this.

During the course of the Great War, newspapers and magazines sometimes turned away from the terrible worries of the present to think of what would happen later, once peace was re-established. The future of literature in particular preoccupied them. One day someone came to ask how I pictured it. I declared, somewhat confused, that I did not picture it. “Don’t you perceive at least,” the questioner continued, “certain possible directions? Let’s grant that the details can’t be foreseen; at the very least you, as a philosopher, have some idea of the whole. How do you conceive, for example, the great dramatic works of tomorrow?” I will always remember the surprise of my interlocutor when I responded: “If I knew what the great dramatic works of tomorrow will be, I would write them.” I saw very well that he conceived the future works as shut up, already, within some kind of cabinet of possibilities; and I, considering my already long-established relations with philosophy, must have obtained the key to the cabinet. “But,” I said, “the works you’re talking about are not yet possible.” — “Yet they must be, for they will be realized.” — “No, they aren’t. I grant you, at most, that they will have been possible.” — “What do you mean by that?” — “It’s very simple. When a man of talent or genius springs up, he creates a work: it is real, and precisely thereby does it become retrospectively or retroactively possible. It would not be possible, would not have been possible, if this man had not sprung up. That’s why I tell you today the works will have been possible, but they aren’t yet.” — “That’s going a bit far! You’re not going to maintain that the future influences the present, that the present introduces something into the past, that action retraces the march of time and leaves its mark backwards?” — That depends. That one can insert reality into the past and thus work backwards in time, I have never claimed. But that one can lodge possibility there, or rather that possibility is at every moment lodging itself there, that is indubitable. As reality creates itself, unforeseeable and new, its image reflects behind it into the indefinite past; it finds itself having been, for all time, possible; but it is at this precise moment that it begins to always have been possible, and that’s why I said that its possibility, which does not precede its reality, will have preceded it once its reality has appeared. The possible is thus the mirage of the present in the past; and since we know that the future will wind up being the present, as the mirage effect continues relentlessly, we say to ourselves that in this present, which tomorrow will be the past, tomorrow’s image is already contained, although we can’t quite grasp it. That is precisely the illusion. It’s as if, having stepped to the front of a mirror, and seeing your image, you figured that you could have touched it had you remained on the other side. By judging, in a similar way, that the possible does not presuppose the real, you accept that realization adds something to mere possibility: you suppose the possible to have been there for all time, a ghost waiting for its day to come; it would then become reality through the addition of something, some kind of blood transfusion or life transfusion. We fail to see that it’s exactly the other way round, that the possible implies a corresponding reality along with, in addition, something that attaches to it, since the possible is the combined effect of reality that has appeared and an apparatus that casts it back into the past. The idea, immanent in most philosophies and natural to the human mind, of possibilities becoming real through an acquisition of existence, is thus pure illusion. One might as well claim that a flesh-and-blood person arises through the materialization of his image seen in a mirror, on the pretext that the real person has everything that is to be found in this virtual image, with, in addition, the solidity that lets you touch him. The truth is that it takes more to get the virtual than the real, more for the person’s image than for the person, because the image can never be drawn if you don’t start with the person — and moreover you need a mirror.

That’s what my interlocutor forgot when he questioned me on the theater of tomorrow. He may also have been playing unconsciously on the meaning of the word “possible”. Hamlet was surely possible before it was realized, if you mean by this that there was no insurmountable obstacle to its realization. In this particular sense, we call possible whatever is not impossible; and it is self-evident that a thing’s non-impossibility is the condition for its realization. But possibility understood this way has nothing to do with virtuality, with ideal preexistence. Close the gate and you know no one will cross the road; it does not follow from this that you can predict who will cross when you open it. Yet from the completely negative sense of “possible” you pass surreptitiously, unconsciously, to the positive. Just now possibility meant “absence of obstacle”, and now you make it into a “preexistence in the form of an idea”, which is something completely different. In the first sense of the word, it was a tautology to say that something’s possibility precedes its reality: you simply mean by this that the obstacles, having been surmounted, were surmountable. But in the second sense it’s an absurdity, because clearly any mind in which Shakespeare’s Hamlet were composed in the form of possibility would thereby have created it in reality: it would thus have been, by definition, Shakespeare himself. In vain you imagine that this mind could have sprung up before Shakespeare. What you forget to think about are all the play’s details. To the extent that you fill them in, this predecessor of Shakespeare finds himself thinking everything that Shakespeare will think, feeling everything that he will feel, knowing everything that he will know, and so perceiving everything that he will perceive — consequently, occupying the same point in space and time, having the same body and the same soul: Shakespeare himself.

But I am probably insisting too much on something self-evident. All these considerations are unavoidable when you talk about a work of art. I believe we ultimately find it obvious that the artist creates possibility at the same time as reality when he carries out his work. How comes it then that we will probably hesitate to say the same of nature? Is not the world a work of art, incomparably richer than that of the greatest artist? And is there not as much if not more absurdity in saying that the future is composed in advance, that possibility preexists reality? I grant, once again, that the future states of a closed system of material points are calculable, and can consequently be seen in their present state. But I repeat that this system is extracted or abstracted from a whole which comprehends, beyond inert and unorganized matter, that which is organic. Take the concrete and complete world, with the life and consciousness that it shelters; consider the whole of nature, generating new species with forms as creative and new as any artist’s picture — then focus within these species on individuals, plants or animals, each of which has its own character — I almost said personality (for one blade of grass resembles another no more than a Raphael resembles a Rembrandt); then raise yourself, above the level of individual people, to societies, where actions and situations comparable to those of any drama unfold: how can one still speak of possibilities preceding their own realization? How can one not see that if the event is always explained, after the fact, by such and such earlier events, a totally different event would have been explained just as well, in the same circumstances, by differently chosen earlier events — what am I saying? by the same events sliced differently, distributed differently, in short seen differently by hindsight? A constant front-to-back remodeling is carried out by the present on the past, by the cause on the effect.

We don’t see this — always for the same reason, always as victims of the same illusion, always because we treat as more what is less, and as less what is more. Let us return possibility to its proper place, and evolution will become something other than the execution of a program; the doors of the future open wide; freedom is given a limitless field. The fault of those doctrines — rare indeed in the history of philosophy — that have made room for indeterminacy and for freedom in the world, is that they have failed to see what their affirmation implied. When they have spoken of indeterminacy and freedom, they have meant by indeterminacy a competition between possibilities, by freedom a choice between possibilities — as if possibility were not itself the creation of freedom! As if any other hypothesis, giving to reality an ideal existence in possibility, would not reduce newness to a mere rearrangement of old elements! As if in doing so it would not, sooner or later, have to be led to consider it calculable and foreseeable! To accept the postulate of the opposing theory is to let the enemy right into the camp. We must take a side: it is reality that makes itself possible, and not possibility that becomes real.

But the truth is that philosophy has never frankly admitted this continual creation of unforeseeable newness. The ancients abhorred it straight away because, platonists to one degree or another, they supposed Being to have been given once and for all, perfect and complete, in the changeless system of Ideas. The world unfolding before our eyes could thus add nothing to it, and was on the contrary nothing but diminution or degradation; its successive states measured the growing or shrinking gap between what is — a shadow cast in time — and what ought to be, Idea enthroned in eternity; the world’s states would record changes in a deficit, the variable form of a void. And Time would have spoiled all of this. The moderns put themselves, it’s true, in a totally different point of view. They no longer treat Time as an intrusion, a perturber of eternity; but they willingly reduce it to a mere appearance. The temporal viewed this way is just the confused form of the rational. What is perceived by us as a succession of states is conceived by our intelligence, once this fog descends, as a system of relations. The real becomes once more the eternal, with the one difference that it is the eternity of Laws in which phenomena are resolved, instead of the eternity of Ideas serving as their model. But, in one case as much as the other, we’re dealing with theories. Let us stick to the facts. Time is immediately given. That is enough for us, and, at least until someone proves its non-existence or its wrongheadedness, we will simply affirm that there is this effective surging forth of unforeseeable newness.

Philosophy would do well to find some absolute in the moving world of phenomena. But we would do well too, to feel stronger and more joyful. More joyful, because reality inventing itself before our eyes will give each of us, endlessly, certain satisfactions that art supplies occasionally to fate’s chosen few; beyond the stasis and monotony that we first see in this reality, with senses hypnotized by the constancy of our wants, reality will reveal to us ceaselessly regenerating newness, the moving creativity of things. But we will above all be stronger, for in this great work of creation that has been there from the beginning and keeps going before our eyes, we will feel ourselves to be participants, creators of ourselves. Our power to act, taking hold of itself once again, will intensify. Having been abased until then in a submissive attitude, slaves of some natural necessity, we will stand again, masters in allegiance with a greater Master. Such will be the conclusion of our study. Let us beware seeing a mere game in speculation on the relations of the possible and the real — it may well be preparation for living well.

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Notes on translation: I have rendered Bergson’s key terms dure, durée as “abide” and “abiding”, which connote more activity than “last” or “persist” or “endure”. These other options would also suggest “keep going”, or “fail to expire”, which is not what Bergson wants to get across. Finally, there is a Buddhist idea that one often sees translated as “abide”; I have no idea what the original Sanskrit or Pali is, but the idea seems close to the one Bergson tries to get across with durée.

I put “teleology” for finalité but confess to not having given much thought to this. There may be a technical philosophical meaning better served by some other word.

I think I put “surge” for élan once, but I mostly left it untranslated, since its use in English is close enough to what Bergson means by it. It doesn’t come up much in this essay.

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).

The earliest (2016-17) reasonably complete exposition of Bergsonian axioms for physics.