In many posts to this forum, and most recently in a couple of posts about fractals — A Question for Philosophically Inclined Mathematicians and Fractals and the Banach-Tarski Paradox — I have discussed the cultivations of novel forms of intellectual intuition that allow us to transcend our native intuitions which make many demonstrable truths counter-intuitive. The cultivation of intuition is a long and arduous process; there is no royal road to it, just as Euclid once informed a king that there was no royal road to geometry.

The good news is that the more people work on difficult ideas, the easier they can make them for others. That is why it is often said that we see farther because we stand on the shoulders of giants. I have pointed out before that the idea of zero was once very advanced mathematics mastered by only a select few; now it is taught in elementary schools. People who are fascinated by ideas are always looking for new and better ways to explain them. This is a social and cultural process that makes difficult and abstract ideas widely accessible. Today, for example, with the emphasis on visual modes of communication, people spend a lot of time trying to come up with striking graphics and diagrams to illustrate an idea, knowing that if they can show what they are saying in an intuitively clear way, that they will make their point all the better.

What is required for this intuitivization of the counter-intuitive is a conceptual effort to see things in a new way, and moreover a new way that appeals to latent forms of intuition that can then be developed into robust forms of intuition. Every once in a while, someone hits upon a truly inspired intuitivization of that which was once counter-intuitive, and the whole of civilization is advanced by this individual effort of a single mind to understand better, more clearly, more transparently. By the painfully slow methods of autodidacticism I eventually came to an intuitive understanding of ε0, though I’m not sure that my particular way of coming to this understanding will be of any help to others, though it was a real revelation to me. Someplace, buried in my notebooks of a few years ago, I made a note on the day that I had my transfinite epiphany.

My recent discussion of the Banach-Tarski Paradox provides another way to think about ε0. I don’t know the details of the derivation, but if the geometrical case is anything like the arithmetical case, it would be just as easy to extract two spheres from a given sphere as to extract one. I’ve drawn an illustration of this as a branching iteration, where each sphere leads to two others (above but one). Iterated to infinity, we come to an infinite number of mathematical spheres, just as we would with the one-by-one iteration illustrated above. But, if for technical reasons, this doesn’t work, we can always derive one sphere from every previous sphere (I have also attempted to illustrate this (immediately above), which gives us a similar result as the branching iteration.

Notice that the Banach-Tarski Paradox is called a paradox and not a contradiction. It is strange, but it in no way contradicts itself; the paradox is paradoxical but logically unimpeachable. One of the things are drives home how paradoxical it is, is that a mathematical sphere (which must be infinitely divisible for the division to work) can be decomposed into a finite number of parts and finitely reassembled into two spheres. This makes the paradox feel tantalizingly close to something we might do without own hands, and not only in our minds. Notice also that fractals, while iterated to infinity, involve only a finite process at each step of iteration. That is to say, the creation of a fractal is an infinite iteration of finite operations. This makes it possible to at least begin the illustration of fractal, even if we can’t finish it. But we need not stop at this point, mathematically speaking. I have paradoxically attempted to illustrate the unillustratable (above) by showing an iteration of Banach-Tarski sphere extraction that involves extracting an infinite number of spheres at each step.

An illustration can suggest, but it cannot show, an infinite operation. Instead, we employ the ellipsis — “…” — to illustrate that which has been left out (which is the infinite part that can’t be illustrated). With transfinite arithmetic, it is just as each to extract an infinite number of arithmetical series from a given arithmetical series, as it is to extract one. If the same is true of Banach-Tarski sphere extraction (which I do not know to be the case), then, starting with a single sphere, at the first iteration we extract an infinite number of spheres from the first sphere. At the second iteration, we extract an infinite number of spheres from the previously extracted infinite number of spheres. We continue this process until we have an infinite iteration of infinite extractions. At that point, we will have ε0 spheres.

In my illustration I have adopted the convention of using “ITR” as an abbreviation of “iteration,” each level of iteration is indicated by a lower-case letter a, b, c, …, n, followed by a subscript to indicate the number of spheres extracted at this level of iteration, 1, 2, 3, …, n. Thus ITRanbn refers to the nth sphere from iteration b which in turn is derived from the nth sphere of iteration a. I think this schemata is sufficiently general and sufficiently obvious for infinite iteration, though it would lead to expressions of infinite length.

If you can not only get your mind accustomed to this, but if you can actually feel it in your bones, then you will have an intuitive grasp of ε0, a visceral feeling of epsilon zero. As I said above, it took me many years to achieve this. When I did finally “get it” I felt like Odin on the Day of the Discovery of the Runes, except that my mind hung suspended for more then nine days — more like nine years.

Odin was suspended for nine days upon the world tree Yggdrasil in his quest to know the secret of the Runes.

I will also note that, if you can see the big picture of this geometrical realization of epsilon zero, you will immediately notice that it possesses self-similarly, and therefore constitutes an infinite fractal. We could call it an infinite explosion pattern. All fractals are infinite in so far as they involve infinite iteration, but we can posit another class of fractals beyond that which involve the infinite iteration of infinite operations. We can only generate such fractals in our mind, because no computer could even illustrate the first step of an infinite fractal of this kind. This interesting idea also serves as a demonstration that fractals are not merely artifacts of computing machines, but are as platonically ideal as any mathematical object sanctioned by tradition.

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Fractals and Geometrical Intuition

1. Benoît Mandelbrot, R.I.P.

2. A Question for Philosophically Inclined Mathematicians

3. Fractals and the Banach-Tarski Paradox

4. A visceral feeling for epsilon zero

5. Adventures in Geometrical Intuition

6. A Note on Fractals and Banach-Tarski Extraction

7. Geometrical Intuition and Epistemic Space

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Grand Strategy Annex

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Extraordinary Sets

3 March 2010


Zermelo built upon the work of Georg Cantor, who was one of the great intellectual revolutionaries of all time.

Usually in this forum I write for an intelligent general audience assuming no particular background other than an interest in the world. Today I’m going to write about a topic in set theory and I will assume a certain familiarity with the concepts of set theory. If set theory isn’t your thing, I appreciate that and I hope you’ll come back tomorrow for something different. But from the hints I’ve dropped about my admiration for Georg Cantor (for example, recently in Salto Mortale) the careful reader (if I have any) will already have guessed my interest in this area.

A little-known concept in set theory is that of what are sometimes called extraordinary sets, which are sets that have infinite descending membership chains. In other words, an extraordinary set is a set with a element that is a set that has an element, and so forth, ad infinitum. In most familiar forms of set theory there are no extraordinary sets, as they are ruled out ab initio by an axiom of foundation. This is one way to prevent the possibility of the paradoxes of self-membership, but in my opinion this simply is a case of throwing out the baby with the bathwater. While we do eliminate one kind of paradoxical set in this way, we also eliminate a lot of potentially interesting sets that are neither paradoxical nor contradictory.

It has occurred to me that sets could be extraordinary in three ways:

(1) a set has an infinite descending membership chain,

(2) a set has an infinite ascending membership chain, and

(3) a set has both infinite ascending and descending membership chains.

Now, we know that (2) is a part of orthodox set theory, as this is nothing other than Zermelo’s infinite set. I find this interesting as this schematic approach puts Zermelian set theory, at least in so far as the strategy it employs to secure an infinite set, in the context of extraordinary sets. One familiar axiom of infinity, viz. “(F) \/x\/y(x in y & /\z(z in y => zU{z} in y)” is usually associated with VNB and results in an ordinal infinite set in so far as it iterates the set theoretical equivalent to ordered pairs. Zermelo’s axiom is simpler, yielding the empty set embedded in an infinitude of brackets.

Postulating only that there is at least one set with an infinite descending membership chain — (1) above — is sufficient to secure a beginning for the cumulative hierarchy. In What is Mathematical Logic? (by Crossley, et al.), Chap. 6, pp. 62-63, the only rationale given for the axiom of foundation is the elimination of infinite descending membership chains, and the only rationale given for this is the elimination of the possibility of self-membership. (Cf. also Foundations of Set Theory, by Fraenkel, et al., North Holland Pub, Chap. II, sec. 5. I highly recommend this latter book; it is well written and easy to understand.) Now whether self-membership must be construed as constituting an infinite descending membership chain I consider an open philosophical question, but I will grant the point for purposes of argument. In any case, this seems to me a minor matter that could be dealt with within the context of an axiom which would postulate at least one set with an infinite descending membership chain.

In what follows I will use “(Ex)” for the existential quantifier, “(Ax)” for the universal quantifier, “⊂” for the membership relation, and “≠” for non-identity (with a few other obvious symbols).

Suppose we have an axiom like:

1* (Ex)(Ay)(y x . (Ez)(z y) . x≠y≠z)

…which I take to mean that there is a set which contains elements which themselves all contain a set.

or, more simply,

2* (Ex)(Ay)(Ez)(y x → z y . x≠y≠z)

…meaning that every element has an element in at least one set.

By replacing E’s above with A’s we have a more radical formulation in which every set is infinitely embedded.

Putting the matter schematically puts Zermelian set theory in a different perspective, and I can take this shift in perspective further. From the trichotomy of infinite sets above we can move directly to a numerical model, specifically the number line, where (1) is associated with infinities in the small, and (2) is associated with infinities in the large. For example, for (1) we may pair left brackets with the series ¼, 3/8, 7/16 . . . , and right brackets with ¾, 5/8, 9/16 . . ., and for (2) we may pair left brackets with the negative integers and right brackets with the positive integers. (To make the correspondence more obvious we could do it all within one interval, with the same kind of convergent series, representing (2) instead by left brackets at ¼, 1/8, 1/16, . . . , and right brackets at ¾, 7/8, 15/16, . . .) In this way we can demonstrate a one-to-one correspondence between these two methods of infinitely embedded sets, and from this I conclude (1) is as good a way as the orthodox (2) for jump-starting infinite sets. But don’t ask me to prove this in an axiomatic setting, because the whole question here is what axioms we ought to use for set theory.

This creates an immediate intuitive connection between the ZF infinite set postulated by the axiom of infinity and the infinity of the natural numbers. In any case, this provides a way of going immediately to object-formalism without detouring through the linguistic formalism of a logical or axiomatic system — object-formalism being really just another form of appealing to arithmetical intuitions, but don’t tell that to dedicated metamathematicians. (Note: Object formalism is the formalization of the mathematical object itself rather than the formalization of the language in which the mathematical object is mentioned, which is the traditional method of applying logic to mathematical reasoning. The paradigmatic example of object formalism is Gödel’s arithmetization of syntax. Here, numbers go proxy for objects.)

I assume that a set with an infinite descending membership chain doesn’t need to be parlayed into an infinite set because it already is a set with an infinite number of elements, sort of like a mirror image of a set with an infinite ascending membership chain, which is the orthodox introduction of infinite sets into axiomatic set theory. The mirror image pairing I suggested above demonstrates a thoroughgoing parallelism between the infinite set in ZF and an infinite set with an infinite descending membership chain. Therefore, if ZF’s axiom of infinity guarantees an infinite set, then I hold that the negation of the axiom of foundation (in certain forms) can guarantee an infinite set where it allows a set with an infinite descending membership chain.

The notion of sets with infinite descending membership chains is not new. However, I don’t know that anyone has suggested the correspondence between Zermelo’s infinite set and sets with infinite descending membership chains. As I mentioned before, they have been called extraordinary sets (I think this in Introduction to Axiomatic Set Theory by Jean-Louis Krivine, but I can’t find my copy and am still looking for the reference), and more recently they have been called non-well-founded sets by Peter Aczel (in a book of the same title), sometimes also called hypersets by those who have developed Aczel’s theory (like Jon Barwise — Barwise and Etchemendy’s book The Liar applies hypersets to the liar paradox; they don’t bother to rule out loops). The “non-well-founded” refers to the axiom of foundation, which in Aczel’s theory is strongly negated by the anti-foundation axiom (AFA). Also, hypersets may contain cyclical membership chains, which would justify self-membership, and put it in the same category as infinite descending chains though distinguishing the two (which I consider an acceptable response to what I called an open philosophical question above).

With this sanction of recent tradition, professionals in set theory might find this an acceptable option for jump-starting the cumulative hierarchy. Although the literature on hypersets recognizes elements of type zero, i.e., individuals (what Zermelo called “urelemente” and banished from his set theory), they don’t do without the empty set, and they don’t try to generate the cumulative hierarchy from hypersets. But I don’t see any objection in principle to doing so. The simplest solution is to admit individuals, which means having an axiom stating that there is at least one individual. Somewhere I remember reading that Russell early on considered an axiom like this but rejected it because he thought mathematics had no business assuming anything about the world, and presumably an individual is part of the furniture of the universe. This is a philosophical question, and an interesting one, but not part of set theory proper, at least as I understand it.

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Grand Strategy Annex

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A Formulation of Naturalism

16 November 2009


Hallett Cantorian Set Theory

One of my favorite books on set theory is Michael Hallett’s Cantorian Set Theory and Limitation of Size. While I have read some rather critical notices of the book, I am not the only one who appreciates it. I found a customer review at Amazon by William D. Fusfield that reads, “This is BY FAR the best and most INTERESTING book available on how Cantor developed his key ideas about transfinite sets, large cardinals, ordinals etc.”

Since Cantor is remembered for legitimizing the infinite as a mathematical concept, it might be a little surprising to hear that Hallett attributes “finitism” to Cantor, but by finitism Hallett does not mean any of the range of constructivist or strict finitist positions staked out by those who deny the legitimacy of the actual infinite and set theory, but rather it describes what we may call the methodological finitism of Cantor’s approach to the transfinite numbers he defined by way of set theory.

Finite mathematics is largely uncontroversial and commands the consensus of almost all who take an interest in the matter, however much they disagree on other parts of mathematics. Thus Hallett formulates what he calls Cantor’s principle of finitism thus:

“The transfinite is on a par with the finite and mathematically is to be treated as far as possible like the finite.” (p. 7)

This I would call methodological finitism. A little further on, on page 32, Hallett quotes Weyl thus:

“…for set theory, there is no difference in principle between the finite and the infinite.”

Hallett then comments:

“…the unity which Weyl points to is so much a fundamental part of Cantorianism (at least when we substitute ‘transfinite’ for ‘infinite’) that I have called it Cantor’s principle of finitism.”

A week ago I was musing about naturalism while making a longish drive and it occurred to me that something parallel to this approach could be used in a formulation of naturalism. “Parallel” is the key term here as were are talking about very different things with naturalism and the transfinite. What strikes me about Hallett’s formulation is the innate good sense of “as far as possible.” This stands in contrast to polarizing and absolutist definitions that employ formulations like “nothing but” or some equivalent of an extremal clause.

In contemporary science, scientific materialism is largely uncontroversial and commands the consensus of almost all interested parties. But from a philosophical standpoint materialism is as dissatisfying as finitism. If you can focus on the science and not think much about the materialism, you’ll be fine. But if the whole object of your interest in science is to illuminate the world and to come to a better understanding of it over all (as is my own interest), then one cannot only not avoid thinking about scientific materialism, one is obligated to think about it carefully.

At this point, then, I would suggest a methodological naturalism parallel to Hallett’s formulation of methodological finitism in Cantor: “Naturalism is on a par with materialism, and philosophically is to be treated as far as possible like materialism.” Or one could formulate it thus: “The natural is on a par with the material and scientifically is to be treated as far as possible like the material.”

Such a formulation would acknowledge both the success and the limitations of classical materialism that views the world entire as “nothing but” matter in motion — Democritean atoms whirling in the void — a classically reductionist formulation. Methodological naturalism as I have formulated it above, parallel to Hallett, would follow classical materialism as far as possible, and would only depart from materialism when that materialism was unsustainable in light of the evidence. And at this point I do not mean to suggest that one makes a transition from matter in motion to a non-naturalistic account of the world. On the contrary, it is at this point that naturalism shows itself to be as distinct from materialism as the infinite is distinct from the finite. Naturalism takes the spirit of materialistic explanation forward into areas that patently cannot be treated in terms of matter in motion.

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Naturalism: a Series

1. A Formulation of Naturalism

2. Two Thoughts on Naturalism

3. Naturalism: Yet Another Formulation

4. Joseph Campbell and Kenneth Clark: Bifurcating Naturalisms

5. Naturalism and Object Oriented Ontology

6. Naturalism and Suffering

7. Transcendental Non-Naturalism

8. Methodological Naturalism and the Eerie Silence

9. Some Formulations of Methodological Naturalism

10. Darwin’s Cosmology: A Naturalistic World

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Grand Strategy Annex

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