Helios

Heavily LIGHT
Explodes continually
Like a scream. The sun
Is the signature
Of its maker? Under a
Sunny moon I dream.

Of Mice and Mengen

As mentioned in my last post, the Burali-Forti paradox seems to be, from the standard perspective (see the aforementioned paper), the most fundamental and paradoxical of the set-theoretical paradoxes, although from my own perspective it is the least fundamental, which is why I’ve already looked at Russell’s and Cantor’s paradoxes.
......I’ll just briefly describe the standard paradox, which arises from the standard presupposition that the natural numbers are a standard set, that they form an actual (or finitesque) totality—in more picturesque terms, one that can, when they (or rather, their tokens) are envisaged spatially, be thought of as being all together, in the same space; or one that can, when their endless sequence is thought of more temporally (in the intuitive sense, i.e. not as part of a quasi-spatial space-time), be completely run through. (For an independent, scientific reason why that standard presumption is likely to be false, see my forthcoming paper.)
......Beginning with standard set theory and its von Neumann ordinals (0 = the empty set, 1 = {0}, 2 = {0, 1}, 3 = {0, 1, 2}, and so forth), we have that each ordinal is the set of the preceding ordinals. The first transfinite ordinal is omega = the set of all the natural numbers, and it is followed by omega + 1 = {0, 1, 2, …, omega}, and so on. There is no set of all such ordinals because if there were it would similarly be an ordinal, and it would be greater than (since arising higher in the set-theoretic hierarchy than) all its members, whereas they were presumed to be all such ordinals. Paradox arises because our intuition that there should be a set of all the natural numbers (and hence our first transfinite ordinal) seems also to tell us that there should be such a set of all those ordinals (and hence some last ordinal, usually denoted by capital-omega).
......Cantor originally needed transfinite inductions, and hence his set (Menge) theory, because he was investigating arbitrary functions of the real number line, as part of the development of Fourier analysis. Although the functions usually met with in physics are well behaved (smooth, differentiable etc.) almost everywhere, one generally expects there to be points (i.e. real numbers) where they are discontinuous etc. Analysis is simplest when there are not infinitely many such points in any finite interval, but if there are then they will have at least one accumulation point, and there might be infinitely many such accumulation points, with accumulation points of their own, etc.
......Cantor was therefore considering sequences of point-sets, each set of points obtained from the preceding one by keeping only the accumulation points. Often, for a given function, the Nth point-set in that sequence would, for some natural number N, contain only a finite number of points, with the (N + 1)th point-set being empty, but the particularly troubling (and therefore mathematically interesting) cases were the functions for which the Nth point-set contained infinitely many points for every natural number N (e.g. the function whose value is 1 at each rational point, and 0 elsewhere). So Cantor was considering what was left after all the naturally numbered stages of that process had been gone through, i.e. at the omegath stage, and (since that might also contain infinitely many points) beyond.
......And one reason why the Burali-Forti paradox remains so paradoxical (see the aforementioned paper) is that today’s mathematicians seem to find a use for inductions that range over ordinals that are (in some mysterious way) even bigger than capital-omega, e.g. in mouse theory. So, what got us using standard ordinals in the first place does seem to take us beyond them, paradoxically. And so perhaps the epistemic possibility that the natural numbers do not actually form a set (which does not presuppose a constructivistic view of numbers, but only that they are an objective structure, whose properties are to be discovered rather than stipulated) should not be too casually dismissed.
......If the natural numbers happen not to form a set, but are rather only potentially infinite, then although we could still consider ordinals insofar as they are the order-types of well-orderings (i.e. as reifications of certain structures) we would not have the same use for them. Cantor’s naturally numbered stages could not have been completely run through, to get us to an omegath stage, but that would not matter because geometrical lines would not then be isomorphic to real number lines, so there would not be such arbitrary functions to investigate in the first place.
......That is not to say that investigating them was not useful, that the standard approach has not given us a nice formal model of geometry, adequate for most applications, but only that it would then be (as it seems to be) no more than that. Note that Cantor did dismiss that possibility rather casually—remarking that a potential infinity (thought of as a variable) presupposes a logically prior actual infinity (the domain of the values that the variable could take)—perhaps too casually (or Hegelian?) given that he was aware of these paradoxes.

More Carnivals

The day of the 50th Philosophers' Carnival, and time to use the carnival submission form to tell me of interesting posts related to the philosophy of mathematics (or the philosophies of logic and science) and mathematical philosophy, which is the use of mathematics (including formal logics, probabilities and scientific models) to tackle any philosophical topic.
......I don't want to exclude any interesting stuff, so any recent post will be considered, but the more mathematical it is the more extensively will “recent” be taken, and the less exceptionally interesting it will have to be. Your post doesn't need to be anything earth-shattering - it just needs to be something that other philosophically-minded people might enjoy reading.
......At the moment I'm thinking about the Burali-Forti paradox, as I'm reading All Things Indefinitely Extensible (from the book being reviewed over on Logic Matters). (PS, July 19 sees the 12th Philosophia Naturalis.)

Metaphysical Progress

Heraclitus: Everything is made of water, and the world is full of gods. Thousands of years later and we reach David Lewis: Everything supervenes upon the local properties of space-time points, and the worlds are full of miracles. Hmm…

Physics and Ethics

This post is a bit long (fourteen hundred words) because I’m asking a question that strikes even me as odd—and if I seem to be defending the popular answer, that is only to emphasise that there is a genuine question here—i.e. who are the experts on ethics? For an example of why that may not seem to be much of a question, consider Anthony Grayling’s commonplaces about Faith (as opposed to science) in his The Meaning of Things (page 122):
......“Contemporary science hypothesises an evolutionary tale of
......physical forces. I say ‘hypothesises’, note; hypothesises on
......the basis of good evidence, severely tested, with many
......aspects of the accompanying theory successfully applied to
......daily life – as exemplified by the light you read by, the
......computer you work on, the airplane you fly in. The great
......advantage of science’s careful and thorough hypotheses,
......always ready to yield if better evidence comes along, is that it
......makes use of no materials or speculations beyond what the
......world itself offers. Religions, in sharp contrast, offer us eternal
......certitudes on the basis only of ancient superstition.”
But to begin with, surely our sciences do not only utilise “what the world itself offers” us—e.g. for most people (including most scientists) the standard model of those physical forces is taken on trust from particle physicists—not, that is, unless such things as the reputations of such people are included within “the world,” as perhaps they ought to be since we are, each of us, part of, rather than apart from, the world—although incidentally, if they are included then religious beliefs, about the non-physical aspects of the world, might not be so very different in kind from our scientific beliefs, about its physical aspects (cf. Polkinghorne’s defence of the Trinity as the best explanation of our written records; with which, incidentally, I disagree). Anyway, particle physicists base their theories upon data taken (almost exclusively) from places that hardly anyone else could gain admittance to—or could understand the workings of (in order to check what was going on) if they did. I’m not suggesting that such people are not to be trusted, just observing that trusting their authoritative testimony is what we do.
......In fact, although the daily life and work of most scientists (and many others) may well provide plenty of support for our beliefs about chemistry and electromagnetism (and thence in quantum mechanics), this “evolutionary tale of physical forces” is hardly so straightforward. For starters, alternative hypotheses are not always eliminated on the basis of the evidence, but surprisingly often because they are just uninteresting, for various reasons—for a glimpse of the sheer range of unconsidered possibilities, consider the following. Particle (or high-energy) physics research has surely (for the last half-century, at least) been of great interest to the military wings of the great world powers, who of course believe that it is of the utmost importance that certain classes of potential discoveries run little risk of falling into the hands of potential enemies—so my question is, what evidence does the world itself offer that the standard model of particle physics is not a fabrication designed purely for public (and thence potential enemy) consumption?
......Now, that particular hypothesis is not to be taken seriously, of course (it is not in capital letters, on a more popular blog), but clearly there are many possibilities, of various kinds, some quite reasonable (e.g. some non-string-theoretic approaches to particle physics may have been neglected for socio-economic reasons), and my point is not that particle physicists are not (or should not be) the experts about physical forces—far from it—but rather that few sorts of knowledge do not require a considerable degree of faith. There may be some, e.g. elementary mathematics, and our everyday knowledge of the world, and of our own minds, but those are hardly examples of scientific knowledge. Now, I personally favour a hitherto unconsidered (and hence unfalsified) hypothesis about physical continua (see my web pages) over the standard set-theoretic hypothesis—and for such reasons I’m undecided about physics beyond elementary quantum mechanics—but even I recognise that it is important what the experts on mathematics and physics favour (for what are presumably good, if not overwhelming reasons). But then similarly, I’m thinking, surely it is important what the experts on ethics think—whether we like what they say, or not—about what the moral facts are (e.g. that would be important in law and politics). So the question is, who are those experts?
......Since the popular answer would be something like, our religious leaders, consider the last line of Grayling above. As a one-liner about Galileo’s troubles it is not too bad, but surely religions are actually what dragged humanity away from the primitive (not to say primate) morality that is just what a purely evolutionary tale would leave us with, but to which few of us (and certainly not most physicists) would wish to return. Over the intervening centuries, the ethical views of those religions have been widely applied, and hence severely tested, both by daily life (as social structures have risen and fallen) and by internal disputes (as obscure as those within modern physics)—in short, why should the experts about ethics not be found (for the most part) within the major religions, just as the experts about physics are (for the most part) in the better-funded departments? I hasten to add that I’m not a member of any religion—in fact, I disagree with many standard religious opinions, e.g. about abortion (where, as a substantial dualist, I don’t see why our rights should not begin with our first breath), and so a good argument that I’m wrong about this answer would not be inconvenient—but (unfortunately) such disagreements are not the issue.
......Similarly, Stephen Hawking is clearly an expert on physical matters irrespective of whatever I think about strings (unfortunately). My worry is that many Naturalists are simply supposing that, just because they believe (on the basis of relatively little evidence at present) that moral facts would, if they exist at all, be best explained by an evolutionary tale, therefore religious thinkers are not experts on ethics. Naturally we think that we know what is right, independently of those religious systems that we are not part of, and of course we do not need those systems in order to function socially (at least not in the short term)—but then, we also know about the physical world independently of the hypotheses of the physicists; and most of us are physically competent, athletes and craftsmen more so than most physicists, who are nonetheless the experts on physics (for good reasons). And since I mean ethics and not meta-ethics (much as I meant above, physics and not metaphysics), so the experts are unlikely to include many philosophers—what philosophy has shown (e.g. via the paradoxes of the trolley and of distance) is that the activity of our consciences no more eliminates our need for experts on ethics than our physical intuitions eliminate our need for physicists.
......But I’m no expert, and maybe I’ve made the question clear enough, so I’ll return to reading Grayling, who is even better than Russell... But on page 100: “The religious attitude is marked by a robust refusal to take things at face value if inconvenient.” That proposition would be even truer were ‘religious’ replaced by ‘naturalistic’ (think of your own feelings and choices, and the physicists’ 10-dimensional strings, etc.)... And on page 101: “Why can we not be prompted to the ethical life by our own charitable feelings? The existence of a god adds nothing to our moral situation,” which inspires me to suggest sarcastically that surely we know about the physical world by bumping around inside it, that surely the existence of strings adds nothing to our physical situation! That is, I could say much the same for mathematics and the existence of sets, or physics and the existence of strings—but even I recognise that if strings exist then they would explain not only how we bump into things—how we actually bump into them, irrespective of what we think we are doing—but also the finer (and more paradoxical) details of the world around us. (And of course, even if they don’t exist, they may—or may not—be a good way of modelling reality.)... Etc.

An Empty Set?

My problem with singletons was (as previously mentioned) that although it is intuitively clear how a plurality (many things) could also be a unity (one plurality)—e.g. one deck of 52 cards—it is consequently obscure how a single object could similarly be a unity different from that object. Nonetheless there is a related obscurity with pluralities, e.g. how do two pairs of socks differ from those four socks?
......Although our 52 cards (think of any particular deck) are exactly the same as those 4 suits (hearts, clubs, diamonds, spades) of 13 cards each—an application of 4 times 13 being 52—those are two different pluralities (52 cards and 4 suits), two different numbers of (different sorts of) things. In one sense they are the same thing, the same deck, but in another sense they are two different things, two different ways of partitioning the cards. When we refer to a collection—e.g. to a par of socks—we are usually referring not to that partition but to those things—to those two socks—but still, two pairs of socks are one collection, while the same four socks are another.
......Perhaps that is why collections are regarded as abstract objects, because they are akin to partitions (which are clearly abstract objects). And when we compare partitions it makes sense to compare partitions of some given things, so perhaps we should be thinking of partitions of some totality of things that are not collections, some universe (of discourse). After all, it remains fairly mysterious what something being a thing (standing apart from everything else) amounts to, but presumably whatever it is will involve those other things (that other stuff) to some extent.
......Of course, such partitions are just collections of collections, so I’m presupposing collections (of many into one), but if collections are indeed akin to partitions then that might shed some light on singletons—e.g. were each collection all the parts but one of some such universal partition, with everything else (in the universe) in the remaining part, so that a singleton would be a part of a partition. Or perhaps the singleton would be the whole partition (an abstract object), containing its single member (that part of the partition) within the rest of itself (the set-theoretic lasso).
......Now, so far I’ve overlooked overlaps, e.g. the collection of pairs (two aces, two fours etc.) is not a partition of our 52 cards, but of three identical decks—or rather (that being impossible) I should not have excluded collections from the universe. Things would get pretty complicated were they all included to begin with, but something like the standard cumulative-hierarchical approach might work out; and if so then maybe the (non-formal) empty set would be the degenerate partition of (i.e. nothing but the lasso that is) some such cumulative-hierarchical universe?

No Use?

I probably won't be posting much until I start my PhD in September (elsewhere I've noticed some interesting answers to the question What use are philosophers of mind?) although I'll be hosting the Philosophy Carnival on August 6 (with a logical/mathematical/scientific theme).

No Credence

The end of a very wet June (not yet so weird as the butterflies last January though), but hopefully I'll be spending less time at the computer next month... Anyway, today (via TAR) I discovered Williamson's recent paper (in Analysis) in which he showed just how problematic is the assumption of regularity: that a credence (or subjective probability, or degree of belief, or of acceptance) of 1 implies absolute (not just everyday) certainty; that, conversely, if your credence in some proposition is 0 then you believe that it is impossible. But surely that assumption is clearly fallacious, like assuming that 0 length of a line means no part of a line, whereas points seem to be possible; so I'm wondering why it's so common: Is it something to do with the meaning of subjective probability?

Carnivals

Yesterday's Carnival of Physics contained lots of philosophically interesting stuff; and today's Carnival of Mathematics includes a couple of interesting views of the totality of the natural numbers, not only mine (as a realistic potential infinity) but also via categorification. (And speaking of ancient philosophical topics, there is a very funny digest of the Nicomachean Ethics.)

Memo on Zero

Mention of Relative Identity (and fractional objects) reminds me of Kessler's Millian approach to the natural numbers: basically (as far as I can recall) the number zero is thought of as a relationship between a property (e.g. being an actual aardvark) and some stuff (e.g. what's now visible). Although I'm not a Millian, that strikes me as a pretty realistic place to start from (and seems prima facie to go with that relative identity approach in logic) because something like a set-theoretical lasso has to capture stuff and turn it into units...?

The Empty Set?

Standard mathematics is based upon an empty set (a set with no members), which is identified as the whole number 0. But while there is only one whole number 0 (the number of things that we have when we have nothing), there are many set theories. There is only one standard theory, but standards change (while 0 does not) and anyway, even were one particular theory especially elegant, coherent and useful, that would not make it true (cf. ideal fluids and Newtonian astrophysics). So the question arises, what could sets be if not formal? Well, what is a chess set? It is 32 chess pieces (and possibly also a checkerboard) of the right kind. If one of them is missing, then that is not a chess set, although it is still a collection of chess pieces—what logicians mean by ‘set’ is something like a collection.
......So, what are collections? Well, what is a stamp collection? It is a collection of stamps—all of the stamps collected by, owned by some individual. It does not include the stamp albums, just the stamps; and if it is sold then it remains the same collection of stamps, belonging to someone else. And if the collector had to sell most of his stamps, and had only one left, then his collection would consist of just that stamp—his collection would be that stamp.
......Michael Potter would presumably say (judging by his 2004 “Set Theory and its Philosophy,” page 22) that while that collection would have one member, that stamp would have a value, a weight, a design, but no member, but I think that it would only be misleading to say that it had one member. Our way of talking about a collection’s members is taken from the paradigm case of a plurality, and there the collection has members in the way that a composite object has component parts; and it is not exactly wrong to say of a simple object that it has only one (improper) part. (Similarly a particular photon has a particular energy, but nonetheless that photon is that energy; and a person both has and is his or her personality, his or her mind, etc.)
......Furthermore if our collector got rid of his last remaining stamp, then although he might have an empty stamp album, he would surely no longer have a stamp collection. And yet there is (supposed to be) an empty set, so a set is not just a collection of things—it is something more, but what? That something more is presumably like that album, but less specific; it is presumably something that all collections have as a matter of logic (rather than physics, or convention etc.).
......Now we have, in logic, classes of things (e.g. the class of men, the class of men now in this city, the class of classes), and we have empty classes, e.g. the class of aunts that are now in this room, and the class of ants that are now in this room. In one sense (intensional) those are two classes, with different modal properties (e.g. had I left the window open, ants might have got in), but in another sense (extensional) they are the same because they do not have different members. And that latter sense is the one we want because there is only one whole number 0. But surely the extensional class is just the collection of the class’s members (and we have already rejected collections), so the empty set is not a logical class. But I cannot think of anything else that sets could be, if not formal.

Potential Infinity?

Clearly whole numbers are objective abstract objects, whose properties are independent of our theories (judging by the obvious objectivity of, for example, these three words). And the natural numbers clearly comprise, collectively, a definite totality (which is therefore another thing, one collection) because a very simple, finite rule (of adding 1 to the previous number, starting with 1) completely determines what they are. Consequently their totality has qualities that are in many ways finitesque (i.e. it has properties that are paradigmatically possessed by the finite collections that we more easily envisage). We can, for example, consider how there are as many rational numbers as natural numbers, by considering a bijection between those two totalities.
......Nonetheless such simply infinite collections differ from finite collections in going on and on, in being generated by their finite rules endlessly. So although mathematical realists (such as myself) do think that they form actual collections (objective structures), if their properties happen not to be as finitesque as is commonly believed (which must be discovered, not stipulated), then their totality will be more like the potential infinity of the constructivists, in many ways, than the actual infinity of the set theorists. The underlying cause of such infinitesque behaviour (if it exists) would not be that our numbers are mental constructions (which is the usual reason for thinking of the natural numbers as potentially infinite) but that they are given by an endless reiteration. That obvious fact about the natural numbers (which all can agree upon) might naturally lead some collections of natural numbers (some sub-collections of that definite totality) to be relatively indefinite—it might, or it might not (and for a realist, such obscure facts must be discovered, not stipulated).
......If we were repeatedly tossing a coin forever, with heads meaning membership (of N in the collection, when obtained on the Nth toss) and tails non-membership, then clearly we would never obtain a definite sub-collection—and whether or not we would in principle be able to obtain one (e.g. by tossing ever faster, or by simultaneously tossing aleph-null coins, etc.) is an open question. It is generally assumed (e.g. by most mathematicians) that we could, but why? Even if set theory is consistent, that would not make it true; and there are indications that we could not (e.g. my forthcoming paper). That the potentially infinite does not depend upon a metaphysically prior constructivism is also indicated by the infinitesque behaviour of the proper classes that inevitably accompany the usual sets.
......But I shall end with a devilish detail, another way in which my notion of potential infinity departs from the norm. I see no compelling reason why lines should not exist (as objective structures) and be full of points, but then a point on a line might determine (in conjunction with two points labelled ‘0’ and ‘1’) an endless sequence of binary places. Then even a random point would determine a sub-collection of the natural numbers. Still, even then the totality of such sub-collections would not exist (so saving my resolution of Cantor’s paradox) because even such pre-determined sequences would be relatively incomplete. They would not be unities in their own right but would depend, for their definition, upon their generating points—in themselves they just go on and on, endlessly, without the intrinsic definition that the existence of their totality would require (cf. Russell's paradox). That is, each generating point would contain far more information (in its precise position relative to the points 0 and 1) than would fit into such a potentially infinite sequence (and while those points would of course form a totality, a line, that would not be a set of points, but rather as described in my last paper).

Tidy Physics?

Recent comments on a recent post by Peter were (inevitably) inappropriate, so I thought I'd better post something here, regarding the causal closure of the physical world. That closure seems to be widely presumed (e.g. by Searle, and Chalmers, and those I've read less recently), presumably because of (what is said to be) overwhelming evidence (although some physicists disagree), but I'm wondering why? I doubt that it is simply a matter of a lot of evidence: cf. how not so long ago there was apparently overwhelming evidence that space was Euclidean (e.g. the huge explanatory success of the extremely neat Newtonian physics of the day), so much so that Kant (and Poincare etc.) could reasonably say that it was a matter of logic, not empirical evidence (rather paradoxically); and of course, modern physics (and the associated philosophy) is hardly so tidy.
......What there was (we now know) was a lot of evidence that space was approximately Euclidean (as we already knew) and about where to look for the non-Euclidean stuff (for Einstein). Similarly a (substantial) dualist about mind and body (such as myself) might expect non-closure to show up only in the details of the mind-brain interaction (e.g. via an explanatory gap remaining even with a completely detailed theory of the brain, or via observations of exceptional neurones directly and therefore unlikely, or via theoretically related phenomena beyond the brain, and so forth). The "overwhelming evidence" cited in defence of closure seems to lie far from where that is likely to be.

Cantor's Paradox

Given some things, we have not only those things but also that collection (the plurality as another unity, the many as one), so the question arises, is a collection (so to speak) with just one member also a further thing, over and above its only member? Intuitively it is not, but it is often (as below) tidier mathematically to say that it is, and then (if we also outlaw self-membership, and potentially infinite collections, etc.) then our collections are sets (and their members are elements). Cantor’s paradox concerned his sets.
......Another question is, when does one collection (a whole number of things) have as many members as another? Cantor realised that it was when the members of one collection correlate one-to-one with those of the other, when they have the same cardinal number of members. The most basic numbers are the natural numbers, 1, 2, 3 and so forth (adding 1 ad infinitum), so how many of them are there? The classical answer was that, since they are generated endlessly (are potentially infinite) they have no number, but Cantor’s answer was that there are aleph-null of them, where aleph-null was the first of his transfinite cardinal numbers.
......Similarly there are aleph-null rational numbers (since they correlate one-to-one with the natural numbers) but more real numbers—there are as many endless sequences of 0s and 1s as there are real numbers (which we may think of as endless decimal or binary expansions), and hence as many subsets of the set of natural numbers (since we may think of each 1 as signifying membership of that set, and each 0 non-membership), and hence there are more real numbers, as follows.
......Counting a singleton (a set with a single element) as different to its element tidies up the mathematics of the set of all the subsets (the power-set) of a given set. E.g. if S = {a, b, c} then its power-set, P(S) = {{a, b, c}, {a, b}, {b, c}, {a, c}, {a}, {b}, {c}, {}}. In general, if the size of S is # (e.g. 3) then the size of P(S) is 2 to the power of # (e.g. 8). Cantor showed that the power-set of any set (even one of transfinite size) is bigger than that set by showing that there could not be any bijection between that set and its power-set (via his diagonal argument), so that since, for each element of that set, there is a singleton that is an element of its power-set, hence the power-set is bigger.
......Cantor’s paradox concerned the set of all the sets. Although it would be the biggest set (by its definition), its power-set would be bigger (by the diagonal argument), whence there is no such set. And although that seems no worse than Russell’s paradox, consider the pure sets. They are built out of (by taking collections of (collections of, etc.)) empty sets, where the empty set does not have a single element (e.g. {} above). Within any particular set theory, the unique existence of the empty set, as one thing, accords with the intuition that a singleton is a different thing to its element (although I would rather begin with a pair of definite things). But that there can be no set of all the pure sets is worse because we cannot resolve this paradox by pointing to the intrinsic fuzziness of (e.g.) the totality of the possible red chairs.
......The totality of all the pure sets is a pure proper class, a class being a collection of all and only those things satisfying some definite predicate, and a proper one being one that is not a set. Mathematicians who are interested in sets tend to regard mathematics as set theory, and so they can ignore proper classes, but the philosophical puzzle remains, motivating us to develop a plausible theory of classes—to point to the definitive property that sets have, and that classes lack. To begin with, if each pure set exists, as a definite thing, then all of them will, their totality existing as some sort of collection. And clearly, the totality of the pure sets and the totality of the cardinal numbers are two proper classes, and each has as many members as the other.
......In short, whatever intuitions led us to transfinite mathematics are likely to lead us towards trans-set mathematics, and inconsistency. In particular, if all the pure sets exist then not only would that totality seem to exist, each collection of pure sets would seem to exist already, as a sub-collection of that totality, whose power-class would therefore also seem to exist—much as the (standard full) power-set of the natural numbers seems to exist. So note that most of the subsets of the natural numbers (which collectively make that power-set larger than aleph-null) correspond to random sequences of 0s and 1s. Although the totality of the natural numbers is clearly a definite collection, a random sequence of 0s and 1s, which is like the probable result of endlessly tossing a fair coin, may well not be a definite thing, because its elements are not determined by a finite rule.
......If the natural numbers are like that (a potential infinity) then, collections being collections of things rather than of fuzzy stuff, numerical collections (at least) avoid Cantor’s paradox—there is only really a paradox when one has (with Cantor) the intuition that the natural numbers form an actual infinity. For Cantor, each potential infinity presupposed an actual infinity (since the former was for him like a variable ranging over the latter) so he thought of proper classes as being actual infinities that were nonetheless inconsistent, a view that is unattractive to most philosophers (who would conclude from an inconsistency the falsity of one or more presuppositions).
......So the problem is, if it is not the endlessness of the simply infinite sequence that makes some infinite collections (those whose members are not determined by any finite rule) unfit to be members of other collections, then what is it? (Note that even if the natural numbers are potentially infinite, their totality may nonetheless be regarded one thing, with a definite number of elements—the same as the number of the rational numbers, via the standard bijection between those two collections—although the devil is in the details).

Benevolence

Though I speak with the tongues of men
......and of angels, and have not benevolence,
............I am become as sounding brass, or a tinkling cymbal.
And though I have the gift of prophecy, and understand all mysteries, and all knowledge;
......and though I have all faith, so that I could remove mountains, and have not benevolence,
............I am nothing.
And though I bestow all my goods to feed the poor,
......and though I give my body to be burned, and have not benevolence,
............it profits me nothing.

Benevolence suffers long, and is kind; benevolence envies not;
......benevolence vaunts not itself, is not puffed up, does not behave itself unseemly,
............seeks not her own, is not easily provoked, thinks no evil; rejoices not in iniquity,
but rejoices in the truth;
......bears all things, believes all things,
............hopes all things, endures all things.
Benevolence never fails: but whether there be prophecies, they shall fail;
......whether there be tongues, they shall cease;
............whether there be knowledge, it shall vanish away.

For we know in part, and we prophesy in part.
......But when that which is perfect is come,
............then that which is in part
shall be done away.
......When I was a child, I spoke as a child,
............I understood as a child,
I thought as a child:
......but when I became a man,
............I put away childish things.

For now we see through a glass, darkly;
......but then face to face:
............now I know in part;
but then shall I know
......even as also I am known.
............And now abides
faith, hope, benevolence, these
......three; but the greatest of these
............is benevolence.

Chapter 13 of Paul's first letter to the Corinthians, translated in 1611 (minor changes in 2007).

Russell's Paradox

The basic concept in mathematics is the concept of a plurality, a number of things. Prima facie each plurality, being one plurality, is also a unity, one thing. E.g. when we think of the extension of the concept of a plurality, i.e. the totality of all the pluralities, we are clearly regarding each plurality as one thing, one member of that totality. That totality must have, as one of its members, itself (since it is a plurality), but clearly many pluralities are not (in that way) self-membered. So now consider the totality of all the non-self-membered pluralities—is that totality a member of itself? Russell’s paradox is that if it is then (by its definition) it should not be, and if it is not then (similarly) it ought to be.
......By analogy with the Grelling-Nelson paradox, perhaps we should have considered the totality of all the other non-self-membered pluralities—but then what of the plurality whose members are that totality and that totality’s members? This paradox is not so easily resolved. Still, the Grelling-Nelson paradox may nonetheless indicate the best resolution. Think of the ordinary pluralities, of cats and dogs, and tables and chairs—whilst any such plurality will be some definite number of definite things, the totality of all the pluralities would be unlikely to be so well-behaved, because even a merely possible object is a thing of some kind, and would therefore be a member of various pluralities.
......E.g. even though some chairs are definitely red, we cannot consider, not as a definite plurality, the totality of all the definitely red possible chairs, because of the fairly obvious borderline cases, such as we are not thinking of when we think of something that is definitely a red chair. Clearly, in order to justify our initial belief, in the totality of all the pluralities, we would need to rule out even the possibility of continuous transitions between individual things (such as a single white cloud, in an otherwise blue sky) and stuff that is too fuzzy to be one or more things, to rule that out for all possible kinds of thing. So, it may well be that Russell’s paradox just shows that such a belief cannot be justified. (2nd Aug: More of the same posted.)

Dusty Stars

The links in this post weren't working
when I checked them in 2026, so here
are some newer links to dusty stars:
......Angel Nebula
......Rho Ophiuchi
......Carina Nebula
......Wolf-Rayet 124
......Pacman Nebula 1
......Pacman Nebula 2
......Rampaging Baboon

The Grelling-Nelson Paradox

The meaning of ‘heterological’ is most easily shown by examples, e.g. since ‘long’ is not a long word, ‘long’ is heterological, whereas since ‘short’ is short, ‘short’ is not heterological, and similarly ‘monosyllabic’ is heterological because it is not monosyllabic, and ‘polysyllabic’ is not heterological—but what about ‘heterological’? If it is heterological then (by its intuitive definition) it is not, whereas if it is not then (similarly) it is.
......This paradox was designed to shed light on Russell’s paradox (of the collection of all the collections that are not members of themselves) and indeed, the best way to see its triviality is to consider the idea of a club for all and only those who don’t belong to any other club. Whilst such a club is perfectly feasible, if the word ‘other’ were missing from its definition it would become paradoxical unless nobody was not in some club.
......Similarly the intuitively heterological adjectives would just be the other adjectives that don’t describe themselves. After all, while it is quite clear what ‘ugly’ being an ugly word would amount to, we have no similarly intuitive grasp of how ‘heterological’ could be heterological. (Furthermore, how heterological are ‘ugly,’ ‘round,’ ‘necessary,’ etc.? Unlike ‘collection,’ ‘heterological’ does not even seem to name a definite concept.)

Curry's Paradox

......Let A = ‘C, if A,’ where C is a contradiction. E.g. “If this statement is true then 0 = 1.” If indicative conditionals were material (if, from a falsity, anything followed) we would be able to get C even from a false A and then, since we’d also get C from a true A (as that is just what A says), we would have C, rather paradoxically; but are they? If they are suppositional (if ‘C, if A’ asserts nothing if A is not true) then we could only deduce (correctly) that since if A were true then C would be true, hence A is not true. Still, the subjunctive version remains puzzling, as follows.
......Let S = ‘If S were true, something that is not true would be true.’ Much as before, S is not true because if it were true then (its antecedent being true) it would be true that something that is not true is true (which is a contradiction). But S therefore seems to be saying that if something (i.e. S) that is not true were true, then something that is not true would be true, which appears to be correct. So although S is certainly not true, it therefore appears to be true. Still, since that appearance must be deceptive, we might resolve this paradox along the lines of the Liar paradoxes, as follows.
......Let L = ‘L is not true.’ By the definition of ‘not,’ L is either true or not true, and if L is true then L is (as it says) not true, whence L is not true. But since we know that L is not true, L therefore appears (paradoxically) to be saying something true. The best explanation for that seems (in my opinion) to be that, whilst another expression (i.e. not L) could be sensibly used to assert that L is not true (as that did), L cannot itself say that because it is nonsense. So perhaps the best explanation for why S does not appear to be nonsense is that it resembles (very closely) a true expression?

Skull Star

When the stars threw down their spears,
And watered heaven with their tears,
Did he smile his work to see?
Did he who made the Lamb make thee?

That's from Blake's The Tyger, 1794 (For the Love of God)