What if there is a proof?

Even if a proof that there is a God is, as I believe it is, hidden beneath the foundations of modern mathematics, the experts will, I am sure, not want to waste their valuable time checking whether there is or not. But perhaps it would be different elsewhere.

And perhaps it could have been different here. If Cantor's paradox is essentially a proof that there is a God, then anyone who could have noticed Cantor's generalized diagonal argument could have discovered that proof. Cantor found his paradox by introducing infinite ordinal numbers, and the ancient Greek paradox of Achilles and the tortoise is a reason to introduce infinite ordinal numbers. Could a similar paradox have been discovered by the mathematicians of the ancient world? Might that have been part of the motive for Plato's Form of Forms? Or part of the justification for the doctrine of the Trinity? It is too late to know now (the experts might say that they do know that that is not the case, but if there is such a proof beneath the foundations of modern mathematics then there always was such a proof).

I also wonder whether the work of Russell in the period 1901 to 1906 had any connection with Einstein's 1905 paper. Russell's obscure mathematics flew in the face of logic, and not too dissimilarly, Einstein's obscure mathematics survived being contradicted by empirical observations. Furthermore, Einstein, like Cantor, was in the Leibnizian tradition (not the Newtonian tradition). And after all, the powers of the world would presumably have liked to keep the truth about high energy physics as secret as they could. Still, it is for that reason impossible to know either way now (although the experts would say that they do know that that is almost certainly not the case).

Still, if God did create this universe in such a way that there was this proof, then we might expect the universe to be full of people taking themselves to be living in God's family (and if Einstein was wrong about the light-speed-limit, then that might make a tangible difference to us).

Where have you been?

The One remains, the many change and pass;
Heaven’s light forever shines, Earth’s shadows fly;
Life, like a dome of many-coloured glass,
Stains the white radiance of Eternity,
Until Death tramples it to fragments.

That's from Shelley’s Adonais, 1821.

BLOOM!

Logical Reference

Vienna, 1893, a countess glances at a robin and decides to paint it. A few days later, her terrible painting of a robin is put up in one of the many corridors in her palace, where it remains for half a century. The countess had not signed her painting, with anything other than the year, and so when her painting was found by the Nazis, they thought that it must be valuable because of where it was. These Nazis were from the Ruhr, so they took the painting to be a bad painting of a fat goldfinch. And because they could see that it would make Hitler's paintings look relatively good, they sent it to Berlin. In Berlin, it was catalogued as Der Fette Stieglitz and put with all the other paintings that the Nazis thought were terrible. After the war, it was taken to Moscow, where it remained for half a century until it was sold to an American, who thought it a fine example of early modern art. He hung it in his dining room.

New York, dinnertime, the owner of the painting points to it and says “that bird was very well fed,” starting a conversation that slowly moves on to who the unknown artist might have been a student of. The owner of the painting and the other diners all assumed that “that bird” referred to an imaginary goldfinch dreamt up by some sadly forgotten genius. But we know that it referred to a European robin, do we not? After all, imagine that we are looking through a warped and dirty window at a robin in bad light: if I thought that it was a fat goldfinch, and I pointed to it and said “that bird is fat,” would I be wrong? Maybe not, but I would certainly be talking about a robin, and it might not be a fat robin. And if we had been looking at a very bad photograph of a robin, or at a mediocre painting of a robin, or even at a terrible painting that we did not understand, what difference would that make?

Doubting the existence of Gardeners

What Do Philosophers Do?

For myself, I just notice such facts as:

(A) The overwhelming majority of professional mathematicians are not going to be wrong about what numbers are.

(B) The overwhelming majority of mathematicians assume, in their professional work, that numbers are axiomatic sets.

(C) Numbers are not axiomatic sets.

The conjunction of (A), (B) and (C) would be a contradiction, were the mathematicians of (B) not just assuming that numbers are axiomatic sets for the purposes of proving theorems from axioms, as I suspect they do. But many analytic philosophers deny (C), because of that apparent contradiction. Such philosophers also ask questions like: “Do numbers (or sets) exist? If they do, where are they? If they don’t, then what does ‘2’ refer to?”

The implication is that since numbers (or sets) are abstract objects, hence if they do exist then they exist in some Platonic realm of abstract objects, raising the question: “How is it that we can access that realm, in order to know such properties of numbers as arithmetic?” To see how stupid such questions are, one only has to ask such questions as: “Does value exist; and if so, where is it?” Clearly some things have value, but it makes no sense to ask where it is (or what colour it is); such questions hardly further the analytical task of describing accurately what value is.

A question similar to the one about numbers might be: “Do shapes exist?” Shapes are instantiated in, and abstracted from, shaped things, clearly; and similarly, whole numbers are instantiated in, and abstracted from, numbers of things. That is basically what John Stuart Mill said (in passing); it is only common sense, although his observation was jumped on by a founder of analytic philosopher, Gottlob Frege (incorrectly).

Fissix

F is six

in ancient Greek,
and "fissix" sounds like physics,
and the external physical world is perceived in six basic ways:

Looking at it with our eyes,
hearing it with our ears,
smelling it with our noses,
tasting it with our tongues,
feeling it with our skins, and
knowing which way is up, via our inner ears.

People tend to forget about that sixth way of perceiving the external physical world because people tend to think of the sixth sense as being an ability to see spirits, or some vague sense of impending doom, or a sense of being stared at. Is there a sense of being stared at? Would that be a seventh way of perceiving the external physical world? Or a way of perceiving the external mental world? Are ghosts ectoplasmic, or psychic? I suppose that if psychologists demonstrated that there is a sense of being stared at, and if they got a good materialistic theory of how it worked (involving, say, the nature of quantum-mechanical collapses), then philosophers would say that that was indeed a seventh way of perceiving the external physical world. That is all very iffy though.

What about the sixth way of perceiving the external physical world with our inner ears? Well, "going up in the world," people say, and "feeling a bit down." People naturally associate up with good and down with bad. But physics came of age when the heavens fell under its laws, in the seventeenth century, thanks to Newton.

By pursuing empirical truth, instead of the next epicycle, Britain began the industrial revolution and built the biggest empire the world had ever seen. And the other sciences followed in the footsteps of physics. In the eighteenth century chemistry emerged from alchemy, in the nineteenth century biology became Darwinian, and in the twentieth century physics became Einsteinian. But the twentieth century was a century of cultural wars, as well as actual wars that got bigger than wars had ever been. Soviet biologists doubted the Darwinian foundation of biology. A few mathematicians even doubted the set-theoretical foundation of twentieth-century mathematics. Still, no one doubted the quantum-mechanical foundation of chemistry. And no one doubted the Einsteinian foundation of high-energy physics, which was a little odd because quantum mechanics all but refuted the four-dimensionalism of Einsteinian spacetime.

Almost all of the physical evidence supported the equations of quantum mechanics and Newton's equations, and the rest of it was evidence that could only be perceived by physicists who were assuming Einstein's equations and making very expensive observations, funded by those with the funds to fund the military, observations that could hardly be independently verified.

Physicists are still pursuing the next dark object at the cutting-edge of high-energy physics, and inventing ever more elaborate Einsteinian theories; and philosophers are still giving "empirical truth" new definitions. But if you could choose between appearing professional and being able to learn, which would you choose?

Curry's Paradox

Last year’s new SEP entry on Curry’s paradox followed in Haskell Curry’s footsteps by saying nothing about where our reasoning goes wrong in such informal versions of the paradox as the examples in the introductory section of that entry, the first of which was as follows:

Suppose that your friend tells you: “If what I’m saying using this very sentence is true, then time is infinite”. It turns out that there is a short and seemingly compelling argument for the following conclusion:

(P) The mere existence of your friend’s assertion entails (or has as a consequence) that time is infinite.

Many hold that (P) is beyond belief (and, in that sense, paradoxical), even if time is indeed infinite.

[...]

Here is the argument for (P). Let k be the self-referential sentence your friend uttered, simplified somewhat so that it reads “If k is true then time is infinite”. In view of what k says, we know this much:

(1) Under the supposition that k is true, it is the case that if k is true then time is infinite.

But, of course, we also have

(2) Under the supposition that k is true, it is the case that k is true.

Under the supposition that k is true, we have thus derived a conditional together with its antecedent. Using modus ponens within the scope of the supposition, we now derive the conditional’s consequent under that same supposition:

(3) Under the supposition that k is true, it is the case that time is infinite.

The rule of conditional proof now entitles us to affirm a conditional with our supposition as antecedent:

(4) If k is true then time is infinite.

But, since (4) just is k itself, we thus have

(5) k is true.

Finally, putting (4) and (5) together by modus ponens, we get

(6) Time is infinite.

We seem to have established that time is infinite using no assumptions beyond the existence of the self-referential sentence k, along with the seemingly obvious principles about truth that took us to (1) and also from (4) to (5).

That may look rather formal to you, but formal logic is not even logic (it is mathematics); the above is just very well laid out. Note the two uses of modus ponens, the two sets of three steps, with the first three steps, (1), (2) and (3), all beginning “Under the supposition that”. You should note that because we cannot always use modus ponens within the scope of a supposition, e.g.:

(a) Under the supposition that modus ponens is invalid under a self-referential supposition, (A) implies (C).

(b) Under the supposition that modus ponens is invalid under a self-referential supposition, (A).

With (a) and (b) we have, under the supposition that modus ponens is invalid under a self-referential supposition, a conditional and its antecedent, but it would of course be absurd to use modus ponens within the scope of that supposition, to obtain

(c) Under the supposition that modus ponens is invalid under a self-referential supposition, (C).

There was, then, at least one step in the above argument for (P) that stood in need of some justification, i.e. the step to (3). Were no other step deficient in justification we could conclude, from the absurdity of (P), that the step to (3) was invalid.

Of course, it would be more satisfying to see where precisely that step lacked justification, so presumably we need an analysis of what would in general count as justification for such a step. For now, note that in order to get to (3) we used modus ponens under the supposition that k is true, which was no less self-referential than the supposition that modus ponens is invalid under a self-referential supposition. In the step to (3) we had k being true instead of (A) implying (C), and “k is true” instead of (A).

To progress, we need to step back, I think, because I suspect that the reason why we find (P) to be beyond belief is that the above argument for (P) has exactly the same logical structure as a clearly invalid argument for the obviously false (Q):

Let your friend say instead: “If what I’m saying using this very sentence is true, then all numbers are prime”. Now, mutatis mutandis, the same short and seemingly compelling argument yields (Q):

(Q) The mere existence of your friend’s assertion entails (or has as a consequence) that all numbers are prime.

My suspicion is based on the fact that one could conceivably have a valid argument for

(S) The mere existence of “happy summer days” entails (or has as a consequence) that time is infinite.

For a start, the mere existence of some words can entail the actual existence of something important, as when Descartes proved that he existed: I think, therefore I am. But furthermore, there is a surprisingly valid argument from the existence of “happy summer days” to the probable existence of a transcendent Creator of all things ex nihilo (this links to that), and it might only take some tidying up to get to (S), because such a Creator is an omnipotent being endlessly generating a temporal dimension. (Such a Creator could possibly have a logical existence proof, because of its unique ontological status.) And of course, were there a valid argument for (S), then there would be an identical, equally valid argument for (P).

Anyway, a six-step argument for (Q) that is identical to the Curry-paradoxical argument for (P) would have, in place of k, some such l as “If l is true, then all numbers are prime”. And l is likely to be about as true as not, because (i) it is about as true as not that a contradiction follows from a statement that is about as true as not, since such a statement is about as false as not, and also because (ii) one informal meaning of l is the obvious meaning of the liar sentence “l is not true”, which is, if meaningful, about as true as not, according to my The Liar Proof. And of course, our logic is naturally suited to that part of our language where propositions are either true or else not true, exclusively and exhaustively. For a proposition that is otherwise, we have natural clarification procedures that enable us to construct new propositions that are more suited to logical reasoning. So, it seems likely that propositions that might be about as true as not should be ruled out from the use of modus ponens within the scope of a too-self-referential supposition (to say the least).

Curry’s paradox entered into the analytic philosophy of the Forties, where the logical paradoxes were in general thought to be reasons for replacing our informal logical reasoning with formal logical reasoning (via the mathematical philosophy of formal languages), on such grounds as that (i) one would not expect primates, even highly evolved primates, to be able to reason perfectly, and (ii) the physical sciences use mathematics to get to the underlying physical laws. However, why would such primates not take themselves to be reasoning perfectly adequately; and why should I be doing mathematics when I am really doing philosophy?

Sequence and Consequence

I add grains of sand to the same place, one by one, and eventually I have a heap of sand.

Originally I did not have a heap of sand, of course, and in between having such numbers of grains – just a few originally, and later on lots of them – there are numbers of grains with which I do not obviously have a heap, and do not obviously not have a heap.

Let n be some such number. Perhaps it is the case that when I have n grains of sand I have a heap of sand; but certainly, if that is not the case, then n grains is not a heap. So for all n, either n grains is a heap, or else n grains is not a heap. It follows logically that there must be some n, say N, such that N is large enough for N grains of sand to be a heap of sand, but (N – 1) is not large enough for (N – 1) grains of sand to be a heap of sand.

The problem is that the rather vague meaning of “heap” does not allow there to be such a number. Given any heap of sand, if I take just one grain away from it, then I would still have a heap of sand. Quite generally, if you take one unit away from any very large number of units, then you still have a very large number of units. So it must, after all, be false that for all n either n grains is a heap or else not. That is surprising, but at least we have a picture of how that can be false, with this picture of my adding grains of sand one by one. So while it is surprising, it is not beyond belief; it is not paradoxical (despite what many analytic philosophers seem to think).

It is not, then, always the case that, given some description, either that description is true or else, if that is not the case, the description is not true. It could be that there is no fact of the matter of whether the description is true or not, as when meanings are a little vague and we have a borderline case. Now, in such cases the description cannot simply be neither true nor not true, as that is just to say that it is not true while ruling out its being not true. The only thing left for us to say is that it is about as true as not. In the picture above, the sand changes from not being a heap, to being about as much a heap as not, and then more a heap than not. That is just a fact, of such matters.

One consequence of the possibility of assertions being about as true as not is that there is a relatively simple resolution of the Liar paradox.

Note that it can indeed make sense to say that a proposition is about as true as not. Consider, for another example, how if I say of some artwork that I think good “That is not good” then I am lying. I am saying something false. What would be true would be for me to say that it was good. And if some artwork seems to me to be about as good as not – and you must allow me such a possibility, because such matters are matters of opinion – then it would be true for me to say that it was about as good as not. In such a case, it might make sense (as follows) for it to be about as true as not for me to say that it was good. And if so, and if we all agreed that a particular artwork, say Z, was about as good as not, then it would make sense for “Z is good” to be about as true as not. Does that make sense? Well, were it simply true to say that Z was good, then were “Z is not good” true too, it would follow that Z was good and not good, whereas the symmetry of Z being about as good as not means that we could hardly have one true and the other not true. And if it was instead not true to say that Z was good, so that it would not be the case that Z was good, then there would be a problem with it being not true to say that Z was not good, because that would mean that Z was good.

The Modal Paradox

Here in the actual world A we have a ship, let us name it the good ship Theseus, made of 1000 planks. Our first intuition X is that the same ship could have been made of 999 of these planks plus a replacement for plank #473. In possible-worlds terms that means there is another world B where the same good ship Theseus exists with all but one plank the same as in our world A, and only plank #473 different. But then in world B one has a good ship Theseus made of 1000 planks, and by the same sort of intuition, there must another world C where the same good ship Theseus exists with all but one plank the same as in the world B, but with plank #692 different. That means for us back in world A there is another world C where the good ship Theseus exists with all but two planks the same as in our world A, but with planks #473 and #692 different, so one could have two planks different and still have the same ship. The same sort of considerations can then be used to argue that one could have three planks different, or four, or five, or all 1000. But that is contrary to our other intuition Y [a ship made of a thousand different planks would have been a different ship].
     The modal paradox resembles well-known paradoxes of vagueness, such as the heap and the bald one, for which proposed solutions are a dime a dozen — except that here what seems to be vague is the relation of identity. And the idea that ‘is the very same thing as’ could be vague is for many a far more troubling idea than the idea that ‘heap’ or “bald’ is vague. Indeed, according to many, it is an outright incoherent idea.

From John P. Burgess, "Modal Logic, In the Modal Sense of Modality" pp. 40-1.
     In the fourth line Burgess says "by the same sort of intuition," which is a relatively weak sort of thing to say, and so it might be where the paradoxical reasoning started to go wrong. Our original intuition X was that in the actual world A, Theseus could have had one plank different and still have been the same ship. We know, equally intuitively, that X coheres perfectly well (somehow) with the intuition, Y, that in the actual world A, Theseus could not have a thousand planks different and remain the same ship. Whereas the ship in B is not exactly the same as the ship in A. And the meanings of all our words are rooted in A. The further we get from A, the more vague we might expect our meanings to become.
     In the first line of the second paragraph Burgess observes that proposed solutions to the paradoxes of vagueness are "a dime a dozen" and that means, I think, that anything I might say is bound to pointlessness; but, onward and upwards. And it is certainly the case that "the very same thing" often equivocates, as in the case of the famous clay statue: if that thing is squashed then, while it is the same lump of clay, it is no longer a statue at all. So let us look at a very different formulation of the ship paradox, one that does not involve modality at all. The following is by Ryan Wasserman, "Material Constitution", §1:

[...] the story of the famous ship of Theseus, which was displayed in Athens for many centuries. Over time, the ship’s planks wore down and were gradually replaced. [...] Suppose that a custodian collects the original planks as they are removed from the ship and later puts them back together in the original arrangement. In this version of the story, we are left with two seafaring vessels, one on display in Athens and one in the possession of the custodian. But where is the famous Ship of Theseus? Some will say that the ship is with the museum, since ships can survive the complete replacement of parts, provided that the change is sufficiently gradual. Others will say that the ship is with the custodian, since ships can survive being disassembled and reassembled. Both answers seems right, but this leads to the surprising conclusion that, at the end of the story, the ship of Theseus is in two places at once. More generally, the argument suggests that it is possible for one material object to exist in two places at the same time. We get an equally implausible result by working backwards: There are clearly two ships at the end of the story. Each of those ships was also around at the beginning of the story, for the reasons just given. So, at the beginning of the story, there were actually two ships of Theseus occupying the same place at the same time, one of which would go on to the museum and one of which would enter into the care of the custodian.

For myself, I do not think that the ship in the museum was the famous Ship of Theseus, I think that what was left of that ship is now the custodian's ship. But I concede that it could be that the museum ship is legally the ship of Theseus. It would then follow that the custodian's ship was not, for legal purposes, the ship of Theseus. So I think that there are at least two senses of "ship of Theseus" in play. What we can say about those senses is another matter. Our language is inextricably rooted in the usual events of the actual world. But it could be scientific to know that there are those two senses even before our theories of such senses have become a dime a dozen. And we might find clues as to what we should be saying from related puzzles.
     There are many intuitive puzzles about identity. For one example, suppose that the world that you are in splits into two worlds, so that you are in one while an identical person is in the other. You are the same person as the original you, of course, but so is the person in the other world. So, that other person is the very same person as the original you, who is the very same person as you, and yet that other person is not the very same person as you. So, either personal identity is not always a transitive relation, or else this scenario is impossible. And while the latter is certainly plausible, it seems to me to be the wrong sort of conclusion to draw from this scenario. Knowing whether it is the wrong sort of conclusion to draw or not may well be a prerequisite for getting anywhere with such puzzles as the modal paradox, because intuitions for "is the very same thing as" not ever being at all vague seem to be very similar to intuitions that it must always be transitive.
      For another example, it is conceivable that you change all of the molecules in your body as you eat and shit and eat and shit and so on and so forth; and you would also be aging. You would be the same person, of course; but if another person arrived now, with completely different molecules and a slightly different, older look to you, that would of course be a different person. Being the same person or not is not, it seems, a matter of what one is made of (a thought that is entirely consistent with our being incarnated souls that might possibly reincarnate). And in the real world, where we live our lives, being the same person or not is a matter of identifying someone from descriptions and pictures.

Force and Foreseeability

Some thinkers think that if there is a God, then God will know all about the future, because otherwise bad things might happen. About ten years ago I spent a few years trying to refute one such view as neatly as possible (see my result here), during which attempt I found a new theodicy (which I called "The Odyssey Theodicy") and discovered the mathematical proof that there is a God (who is not immutable) that I have recently been tidying up. Today I thought of this title to go with my original refutation; basically, my original thought was that God's power over God's creation gives God plenty of ability to know that good will definitely happen, without God needing to know all about the future. However, despite now having the sort of title that I like, for my thought, I find that I now have little interest in expressing as neatly as possible such academic thoughts. That is because my thought is so obvious that the view that I was refuting must have existed for some other reason than simply not knowing that thought. Could that view not have been clearer about its reasons, I wonder. But, that is just the academic way, it seems. I also now think that finding new theodicies is pretty pointless, though; consider this analogy: it is the first day of school, and things do not go well. And of course, you learn very little; but of course, that is no reason to have no first day of school. And the evidence that, if there is a Creator of all things, then it is an evil Creator, is a bit like that: if all of this was created by such a power, then there is very likely to be life after death (like further school days after the first, and then life after school, a life enhanced by prior schooling) because that would be better, and no less possible than this life; and so the worse this life is, the more likely there is to be life after death, if there is a God. The logic of such arguments is simple, and undeniable, and so the way the problem of evil is hyped up by mainstream analytic philosophers of religion is, clearly, pure rhetoric.

On the Sorites

          A drop of water falling on a hill does not wash it away.
So, if we start with a hill, then after a drop of water we still have a hill.
After another drop, we still have a hill; and many repeated applications
of the first, italicised line means that after lots of drops the hill remains.
But, after enough drops the hill will, of course, have been eroded away.

That is basically a Sorites paradox. Similarly, all real-world calculations will, if long enough, become swamped by error bounds. All measurements should come with error bounds, and while a short calculation will result in only slightly larger error bounds on the result, a very long calculation will be useless. Now, logic is supposed to be different, more like Geometry, where given certain lengths, geometrical manipulations can be arbitrarily long. But that will only be the case if the terms that the logic is applying to are definite. In the real world, there is a ubiquitous, if usually very slight, vagueness (it is there because it is so slight: nothing has acted to remove it). Consequently logical arguments that are about real things should not be too long. It is an interesting question, how long they can be; but certainly, those of the Sorites paradoxes are too long.

Why Merit Lacks Merit

Everywhere has been invaded, usually several times. As a rule, the invaders steal everything, killing some and enslaving the rest. A few of the natives help them. There is a continuum, between such collaborators and the dead, on which the majority of the original people find themselves. Most of them tend not to volunteer for anything, after invasion. But among the new ruling class there are many volunteers, eager to prove their metal. People pair up, and new generations give way to even newer ones. Now, some people do well, and often it is in large part because their ancestors inherited stolen goods. Of course, such people have still, themselves, done well. Perhaps they also had good genes. Perhaps they were lucky. They hardly thereby deserve advantages over people who suffer, while doing badly, because, in large part, their ancestors had stuff stolen from them, or because their genetic inheritance was also poor, or because they were unlucky. Note that according to standard evolutionary theory, every genetically fit individual owes that fitness to the immense sufferings of a huge number of other individuals who died without leaving any offspring behind. Of course, it is in general much more complicated than that, too complicated for simple words to do it justice; but it would certainly, therefore, be quite unjust, to say the least, to talk up meritocracy. (More details: this from last year.)

The Signature of God

What follows is a proof of the (probable) existence of God.
     Such an extraordinary claim requires extraordinary evidence, of course, and so this post is a bit long, but most of the heavy lifting has already been done by those who have been failing, for over a hundred years, to find atheistic explanations of certain basic mathematical facts.
     Evidence for the existence of God must be extraordinary, and of an appropriate kind. Suppose we saw letters of unearthly fire in the sky, spelling out a claim that there is a God; the most likely explanation would be pranksters, or, at a push, aliens. Evidence for the existence of the Creator of all things, including such things as the human mind, should therefore include something more like a logical proof. There are already several arguments that claim to be such, e.g. the ontological argument, and you might think of the following as another (we could expect there to be several logical proofs because when we find one proof of a mathematical theorem there are usually others to be found).

What follows is based on the nineteenth century mathematics of Georg Cantor, and in particular, his famous logical paradox.
     Logical paradoxes are chains of thought that seem logical but which take us from self-evident truths to contradictions. Nothing, you might think, could be further from a proof; but it is precisely because logical thoughts take truths to truths, not to contradictions, that it follows that in every such paradox there must be some false assumption(s). The harder the paradox is to resolve, the stronger – and more surprising – will be the chain of thought from the false assumption(s) to the contradiction. A very tough paradox can therefore amount to a rigorous chain of thought that takes some very plausible assumption(s) to a contradiction, thereby proving by reductio ad absurdum the assumption(s) to be – surprisingly – false. In particular, Cantor’s paradox refutes atheism (and classical theism, which I take to be the view that there is a being who is omnipotent, omniscient, immutable and so forth).
     Things that are as Cantor’s famous diagonal argument shows them to be could, just possibly, exist within the creation of a Creator of all things (were that Creator not classically immutable). You will see why below; and while that fact may not seem like much, it yields a reason why there is probably such a Creator because there is very probably no other way in which things as we know them to be could exist. That high probability comes from the fact that mathematicians and logicians have been looking for a more intuitively satisfying resolution of Cantor’s paradox for over a hundred years, working within their background assumptions – atheism, for the most part (although also classical theism, especially in Cantor’s day) – and in all that time they have found no better way of avoiding paradoxical contradictions than the formalization of mathematics and logic.

     Cantor was working on Fourier analysis, in the 1870s, when he found it necessary to extend arithmetic into the infinite, despite various paradoxes. He resolved those paradoxes by extending arithmetic in a rigorously logical way, throughout the 1880s, but sometime in the 1890s he found his own paradox. Naturally he worried that he had refuted his own work, but he had been very rigorous, and so there was little the mathematical community could do – given their background assumptions – but formalize the foundations of mathematics. The question of what numbers really are was left to philosophers; in mathematics, there is no paradox: there are formal proofs, in most axiomatic set theories, that there is no set of all the other sets: were there such a set, its subsets would outnumber the sets, via a diagonal argument (see below), whereas subsets are sets. Formalization enables the paradox to be avoided, but it does not resolve the underlying problem: whenever we have a lot of sets, we do have their collection, because a collection of things is, intuitively, just those things being referred to collectively; and since each of its sub-collections is, intuitively, just some of those sets, we also have all of those sub-collections. Intuitive versions of Cantor’s paradox remain, then, to be resolved.

The following version, in particular, works by way of showing that certain possibilities become more and more numerous (see my earlier sketch of this version). Now, if something is ever possible, then it was always possible; but, possibilities of various kinds can grow in number by becoming more finely differentiated, as you will see in the following two paragraphs. But to begin with, an initial worry might be that even if some possibilities were differentiated in the future, those differentiated possibilities would already exist in spacetime (so that their number would actually be constant). So note that while presentism – the view that only presently existing things really exist – is not popular, it is generally agreed to be logically possible. Let us therefore use ‘time-or-super-time’ to name time if presentism is true, and something isomorphic to presentist time – at a mere moment of which the whole of spacetime could exist – if the whole of spacetime really does exist. The point of that definition is that time-or-super-time might exist even if presentism is false; either way, ever more possibilities could, just possibly, be individuated (in time-or-super-time).

     For a simple example of differentiation, suppose that spacetimes come into being randomly, in time-or-super-time, with some of them happening to be exactly the same as our spacetime. Someone exactly the same as you exists in each of those spacetimes. And of course, each of those identical copies of you was always possible in time-or-super-time. As we consider any one of them, it seems as though there must always have been the individual possibility of that particular person; and certainly, that individual was always possible. But what about the copies of you in future spacetimes? How could their individual possibilities be already distinguished from the more general possibility of someone exactly the same as you? Such copies of you do not yet exist, to be directly referred to, and indeed, they may never exist. So for such random beings, in presentist time-or-super-time, it would not make sense for their particular possibilities to exist. So despite our hindsight, the possibilities of such people must originally have been undifferentiated parts of the more general possibility of someone just like you. It is only with hindsight – after differentiation – that we see the differentiated possibility in the past.

     For an example without randomness, suppose that a Creator in time-or-super-time determines to create a ring of equally spaced, absolutely identical objects. None of those objects can be individuated until the ring has been created, because their Creator does not want to individuate them. So before then there is only the general possibility of such an object. Afterwards there is, for each object, the individual possibility of that object in particular, in addition to that general possibility. Once a particular object exists, there seems always to have been that particular possibility – because that particular object was always possible – even though we know, from the description of this scenario, that it was the general possibility that always existed.

I will be describing how certain possibilities might become more and more individuated by a dynamic (as opposed to immutable) Creator of all things ex nihilo. Creation of things ex nihilo is the creation of things out of nothing; it contrasts with the creation of things made out of some already existing substance (like a sentient computer making a phenomenal world out of computers and human brains). Creation ex nihilo is, at the very least, logically possible. After all, the Big Bang was clearly possible, and for all we know it could have followed nothing physical; for all we know, it could have followed some sort of creativity, such as a person. What we know for sure is that in the world there are physical objects and people. It is not easy to see how real people could be made of nothing but chemicals, but physicalism is of course a prima facie logical possibility; and it is similarly possible that spacetime and everything in it was created by a transcendent person.

     Given that such a Creator is logically possible, the following paradox then shows that the possibilities in question probably do become ever more numerous, because that is probably the only way of avoiding the contradiction derived below (other than simply ignoring it, or in other ways rejecting logic). Furthermore, it is very hard to imagine how those possibilities could possibly become more numerous if there is no such Creator. That is why this resolution of the paradox has for so long been overlooked. And that is how this paradox will show that there is probably such a Creator. So, to my intuitive but rigorous version of Cantor’s paradox.

We should begin with a self-evident truth; and clearly, these words are distinct from each other. That fact is self-evident because that is how we were able to read those words. There are, then, numbers of things; for example, ‘I’, ‘am’ and ‘lying’ are three words.
     Note that pairs of those three words – {‘I’, ‘am’}, {‘am’, ‘lying’} and {‘I’, ‘lying’} – are just as distinct from each other as those words were, because those three pairs differ in just those three words. Similarly, pairs of those pairs – e.g. {{‘I’, ‘am’}, {‘am’, ‘lying’}} – are just as distinct; as are pairs of those, and so on.
     Now, because of that ‘and so on’ we will have infinitely many, equally distinct things, if we can indeed count pairs as things. But is there really something that, for any two things, sticks them together to make a third thing? Put that way, it must seem unlikely. But, for you to pick out any two of our original three words, those two words must have already been a possible selection. Such possibilities can be our third things. In general, a combinatorially possible selection from some things corresponds to giving each of those things one of a pair of labels, e.g. the label ‘in’ if that thing is in that selection, or else the label ‘out’. If two of the labels are ‘in’, for example, we have a combinatorially possible pair. Every combination of as many such labels as there are things in some collection corresponds to some combinatorially possible selection from that collection, and vice versa.

So, let us take ‘{‘I’, ‘am’}’ to be the name of the combinatorially possible selection of ‘I’ and ‘am’ from our original three words, and similarly for the other increasingly nested pairs described above, which we may call, collectively, ‘N’. The following intuitive but rigorous version of Cantor’s diagonal argument proves that for any collection of distinct things, say T, the collection of all the combinatorially possible selections from it, say C(T), is larger than T.
     Informally, two collections are equinumerous – they have the same cardinal number of things in them – when all the things in one collection can be paired up with all of those in the other. So suppose, for the sake of the following reductio ad absurdum, that C(T) has the same cardinality as T. Each of the things in T could then be paired up with a combinatorially possible selection from T in such a way that every one of those possible selections was paired up with one of the things in T. Let P be any such pairing. We can use P to specify a possible selection, say D, as follows. For each thing in T, if the possible selection that P pairs that thing with includes that thing, then that thing is not in D, but otherwise it is, and there is nothing else in D. Since the only things in D are things in T, D is a possible selection, and so it should be in C(T). But according to its specification, D would differ from every possible selection that P pairs the things in T with, which by our hypothesis is every possible selection in C(T). That contradiction proves our hypothesis to be false: C(T) does not have the same cardinality as T. Furthermore, C(T) is not smaller than T, because for each of T’s things there is, in C(T), the possible selection of just that thing; so, C(T) is larger than T.

     As well as N, there is therefore the even larger collection C(N), and similarly C(C(N)) – which is just C(T) when T is C(N) – and so forth. All the things in all those collections are as distinct from each other as our original three words were, because they differ only in things that are just as distinct. Let the collection of all those things be called ‘U’: U is the union of N, C(N), C(C(N)) and so forth. U is larger than any of those collections because for each of them there is another of them that is larger and whose things are all in U. And since there are all of those things, there are also all of the combinatorially possible selections from them, which are just as distinct from each other, and which are collectively C(U). And so on: there is always a larger collection to be found; if not another collection of all the combinatorially possible selections from the previous collection, then another union of every collection that we have, in this way, found to be there. Those steps always take us to distinct possibilities that are fully defined by things that are already there. So, there must already be all the things that such steps could possibly get to.
     The problem is that from all of those things existing, it follows that all of the combinatorially possible selections from them also exist – since they are equally distinct possibilities, fully defined by things that are already there – and there are even more of those possible selections, as could be shown by a diagonal argument, which contradicts our having already been considering all the things that such steps could possibly get to.

Since there are no true contradictions – outside formal logic – something that seemed self-evident in the above must have been false. But the above chain of reasoning was a relatively short argument, from a self-evident premise. It is very easy to survey the whole of the argument and see how rigorous it was. The only lacuna is the one highlighted above: the obscure possibility of those combinatorially possible selections being the end results of more general possibilities becoming individuated. The following proof relies on that being the only lacuna, which you can only determine for yourself by trying – and failing – to find another. Perhaps, for example, there are no such things as possibilities? But were there no logical possibilities, logical thought would become impossible (except in some formal sense), and so we must presume that there are such things. It can be argued that there are not; but similarly, there are those who argue that there is only mind, while others argue that there is only matter. It seems to me to be self-evident that there are phenomena – our experiences – as well as physical things (e.g. those that we experience), and, similarly, that a huge range of non-formal logical thought is possible. And in particular it seems to me to be self-evident that {‘I’, ‘am’} is one of three combinatorially possible ways of making a pair of words (from our original three). Consequently the question is where a principled line should be drawn: where are the joints of nature? The reason why {‘I’, ‘am’} is a possible selection is that ‘I’ and ‘am’ are two of our original three words, and that reason generalises in an obvious way: for any things, in any given collection of things, those things are a possible selection. Note that a logically possible being could select those things from that collection.

     Regarding the possibility of the combinatorially possible selections being the end results of more general possibilities becoming individuated, it is conceivable that the Creator of all things ex nihilo would be able to individuate them because of the unique authority of such a being. Much as the individual possibilities of particular people, in the example above, could not be distinguished from the more general possibility of just such people, not until those people were there to be directly referred to, so it might be that the most unimaginably nested of the combinatorial possibilities are not individuated until such a being individuates them (by thinking of them). They need not be individually possible selections until then because who could possibly make such a selection? There is only the Creator, thinking of them in the absolutely definitive way of such a being. Naturally, such possibilities seem as immutable as the laws of physics, to us; but of course, to a God the laws of physics are mutable.
     There is not much more to be said, about such divine differentiation, though. Creation ex nihilo is totally alien to our experience, so it is essentially obscure. But, it is a relatively clear logical possibility for all that. Analogously, it is quite obscure how atoms of lifeless matter could be arranged so as to make conscious life, but that does not stop materialism being a logical possibility (for all that it might make it seem less plausible). Note that such a Creator could have existed prior to any things at all, because such a being could be, in itself, more like a Trinity than a thing. Such a being could have always known of the most general possibility of things as we know them, before choosing to contemplate creating some such things; and could then have known an awful lot about combinatorially possible selections, nested around those possible things, up to unimaginably high levels of an increasingly nested hierarchy (such levels as standard mathematicians would never contemplate). It makes sense that a being that could create things ex nihilo would know so much about them (and might even enjoy finding out more). Standard set theory would therefore be a very good mathematical model of the more imaginable levels (and of how there are unimaginably high levels, not all of which can be assumed to exist already). (Note that none of the properties of the underlying things would be made variable by the higher levels being variable; on the contrary, each level would be completely determined by those things being distinct things.)

So, since a dynamic Creator is, at the very least, a logical possibility, hence our combinatorially possible selections could, just possibly, be growing ever more numerous. And since there seems to be no other way of avoiding the contradiction, hence those possible selections are probably growing in number. Furthermore, outside the context of the absolute dependency upon their Creator of things created ex nihilo, there is no conceivable way in which those possible selections could grow in number. That is why this resolution has, for so long, gone unnoticed. And that is why it follows that there is – at least probably (in view of that long period of modern thought) – such a Creator.
     The big problem with that conclusion is, of course, that the majority of scientists are atheists. You might therefore be quite sure that there must be a flaw somewhere in the above. The most surprising thing about the above, however, is how scientific it could seem to simply ignore it, even if there is no such flaw. Many logicians take the logical paradoxes to be good reasons for not trusting pre-formal logic (and similarly, pre-formal arithmetic), however rigorously it is applied. After all, we would hardly expect primates – even highly evolved primates – to be perfectly logical. Whereas you might expect that a more formal treatment would find there to be no problem; and indeed, there is no formal paradox. Formal logic does not just look scientific, it reliably delivers desired results.

     Nevertheless, logic – our natural, pre-formal logic – is not so much an option as a necessity. Would highly evolved primates reject their own logic just because it gave them something that had seemed too good to be true? Probably not; but more importantly, it is not really an option. It is only because we believe science to be logical – in the pre-formal sense – that we believe science when it tells us that we are highly evolved primates. It is not because scientific results could be written up in a formal logic. After all, there are formal logics in which true contradictions have been formalised. And while most formal logics do not allow true contradictions, the question is: how could we determine which formal logic to use, except by applying our natural logic, as rigorously as we can? Even letting formal criteria decide the matter would be to have decided pre-formally to do so. Note that we should not do that; such formal criteria as simplicity, for example, might tell us to allow true contradictions. Indeed, the logical paradoxes could all be regarded as straightforward proofs that there really are true contradictions, unless we had already ruled that out. And we should of course rule that out, because things cannot be a certain way while not being at all that way. Being that way is precisely what ‘not being at all that way’ rules out, pre-formally.

     It was one thing to reluctantly replace logic with formal logic, and numbers with axiomatic sets, in order to avoid paradoxical contradictions; it would be quite another to jump at the chance to make such replacements just to avoid the refutation of a strongly held belief. The latter would clearly be unscientific. Of course, you may think that there is no such refutation, that God has been invoked to explain something that may well be explained by science one day. And such God-of-the-gaps arguments are indeed unsound. Before it was discovered that we are on the surface of a massive spheroid orbiting a star, for example, a sunrise might have been explained by invoking God, on the grounds that only a God could cause such an awesome event. My argument, however, is more like the Newtonian connection of the motion of planets with the motion of projectiles. That is because there is, in mathematics, a practice of defining mathematical objects in terms of human constructions; such constructivism is not popular, but it is a valid practice. I am explaining the Cantorian property of things by invoking divine constructivism, not a simplistic miracle. Note that there is no perception in modern mathematics – as there was in the early years of the twentieth century – that Cantor’s paradox might be resolved by future research within the mainstream. Rather, our axiomatic set theories and formal logics are beginning to look more and more like epicycles.
     It might be thought that I do have a God-of-the-gaps argument because I do use God to explain something scientific. So note that there were similar objections to Newton’s invocation of action-at-a-distance, in his explanation of astronomical observations, on the grounds that action at a distance is magical action. Physical action was thought to be action by physical contact (even though the physicality of such contact is primarily phenomenal). Of course, any actual action in the external world will fall under physics. And my finding of a scientific use for the hypothesis of a Creator shows that God can be a scientific hypothesis.
     Euclidean geometry was axiomatised, but that did not make it true; space is what is it. Ptolemaic astronomy could have been axiomatised, but the earth still turns. Standard mathematics is axiomatised; nevertheless, there are numbers of things.

The Liar Proof

This assertion is not true.
Let that assertion – if it is an assertion – be called ‘L’.
          If L is an assertion – the assertion that L is not true – then L is an assertion that it is not true that L is not true, and so L is also an assertion that L is true. That is unusual, to say the least; but it is clear enough what is being asserted – how else could we know that it had that unusual property? – and so L is fairly clearly an (unusual) assertion. And if L is as true as not (see below), then it is as true to say that L is true as it is to say that it is not, so there is that consistency. Note that L is not a simple conjunction of those two assertions; it is wholly the assertion that L is not true (if it is an assertion), and it thereby asserts that L is true. And note that no assertions are perfectly straightforward; all are to some extent vague, for example.

          Nevertheless, logic seems to take L to a contradiction. (By ‘logic’ I mean that which formal logics model mathematically. Formal axioms are abstracted from informal but rigorous arguments, arguments so rigorous that we regard them as proofs. Were such a proof to include a step that did not correspond to any axiom, we should have a reason to revise our formal logic; we should have no reason to reject the proof.) If L is true – if it is true that L is not true (and that L is true) – then L is not true (and true). But L cannot be true and not true, of course; the ‘not true’ rules out its being true. And so if L must be either true or else not true, then it follows that L is not true. But if L is not true – if it is not true that L is not true (and that L is true) – then L is true (and not true); and L cannot be true and not true.

          So, logic takes L to a contradiction if – and as shown below, only if – we assume that assertions must be either true or else not true. The negation of that assumption is not logically impossible – see below – and so it is that assumption that logic is taking to a contradiction. That assumption is certainly very plausible, of course. To want the truth of a matter is to want things to be made clear. It is to want the vagueness to be eliminated. Nevertheless, there are a variety of abnormal situations where it would be highly implausible for the assumption in question to be true (and L is not a normal assertion). Suppose, for example, that @ is originally an apple, but that it has its molecules replaced, one by one, with molecules of beetroot. The question ‘what is @?’ is asked after each replacement, and the reply ‘it is an apple’ is always given. Originally that answer is correct: originally it is true that @ is an apple. But eventually it is incorrect. And so if the proposition that @ is an apple must be either true or else not, then an apple could (in theory) be turned into a non-apple – some mixture of apple and beetroot – by replacing just one of its original molecules with a molecule of beetroot. And that, of course, is highly implausible.

          What is surely possible, since far more plausible, is that @ is, at such a stage, no less an apple than apple/beetroot mix, that it is as much an apple as not, so that the assertion that @ is an apple is as true as not. That assertion could not be true without @ being an apple, nor not true without @ not being an apple (and we can rule out neither true nor not true, because that is just not true and true). More precisely, @ is likely to move from being an apple to being as much an apple as not in some obscure way that is, to some extent, a matter of opinion. In between true and not true we may therefore expect to find states best described as ‘about as true as not, but a bit on the true side’, ‘about as true as not’ (a description that would naturally overlap with the other descriptions) and ‘about as true as not, but a bit on the untrue side’. For such abnormal situations, formalistic precision would be quite inappropriate, because the truth predicate is indeed suited to the elimination of vagueness. It is much better to say ‘it is as much an apple as not’ instead of ‘it is an apple’ when the former is true, the latter only as true as not.

          But we cannot express L better, we have to understand it as it is. Fortunately, if we do not assume that assertions must be either true or else not true, then from the definition of L it follows only that (if L is an assertion then) L is true insofar as L is not true, and hence that L is as true as not. There is no contradiction, and so either L is not even an assertion (which seems implausible) or else the Liar paradox is a disguised proof by reductio ad absurdum that it is not the case that assertions must be either true or else not true. Note that there is no ‘revenge’ problem with this resolution. E.g. consider the strengthened assertion R, that R is not even as true as not (which is thereby also an assertion that R is at least as true as not). If R is true then R is false (and true), if R is as true as not then R is false (and true) and if R is false then R is true (and false); but, if R is about as true as not, a bit on the untrue side, then it would be about as false as not to say that R was not even as true as not (and about as true as not to say that R was at least as true as not). Greater precision than that would be inappropriate for an assertion as unnatural as R.

Definitive Selections?

Are definitive selections too odd?
      When we think of some things, and various combinations of them, it seems clear that all those combinatorial possibilities are there already, awaiting our consideration. And yet I am asking you to imagine that when a Creator, some such brilliant mind, considers some things, all those possibilities are blurred together (although none so blurry that it cannot be picked out); or am I?
      I am suggesting that for selections of selections of ... of selections, from some original collections, each possible selection from those will be a particular possibility only as it is actually selected by our Creator, independently of whom no collections of things would exist, were there such a Creator (as there provably is). The possible selections that make S(N) bigger than N (to use the terminology in my Cantorian diagonal argument) are those endless sequences of ‘I’s and ‘O’s that are pseudorandom; to make them, infinitely many selections have to be made, each one of which involves some arbitrarily large finite number of selections. They might be made instantaneously by our Creator, of course; and if so, then typical selections from S(N) could be made arbitrarily quickly.
      What about S(S(N)), which contains more things than infinite space contains points? Well, a Creator might be able to do all of that instantaneously. And similarly for selections from U, and UU, and maybe UUU; but still, you see how our Creator would have to do much more, and much, much more, and so on and so forth, without end. It is therefore quite plausible that for selection-collections that it would take me far more than mere trillions of pages to describe, our Creator would be unable or unwilling (and thence unable) to make all such selections instantaneously. After all, it is logically impossible for all possible selections to be made instantaneously. To will an incremental development of such abstract mathematics, as a necessary aspect of the creation of any things, might be regarded as a price worth paying for some such creations. And it is also quite plausible that were the Creator unable to do something (even as a consequence of such a choice) then that thing really would be impossible, given that the very possibility of it derives from that Creator.
      Solid things are solid; but mathematical properties related rather abstractly to their individuality can be works in progress; why not? Modern mathematics has a weirder story to tell of such matters! It is relatively straightforward to think of Creation as dependent upon a Creator who transcends even its mathematics. So, it may not be too odd to think of a Creator creating number by definitively adding units: 1, 2, 3 and so forth; is that any weirder than a Creator creating something ex nihilo? Number is paradoxical, so that the ultimate totalities of numbers are indefinitely extensible, and so numbers just do pop into existence, somehow; and what more reasonable way than by their being constructed by a Creator? What would be very weird indeed would be their popping into existence all by themselves, what with them being essentially structural possibilities rather than concrete things. It makes some sense to think of us creating them, as we think about the world around us, but there is something very objective about numbers of things. And again, if it makes sense for us to do it, then how can it be too odd to think of a Creator doing it, in a Platonistic way?
      There will be better ways to think of definitive selection, I am sure, but are they needed? Consider the question of how simple brute matter (just atoms, in molecules of atoms, each just some electrons around a nucleus) could possibly have feelings; a common enough answer is: Well, it must be possible, because we have such feelings, in this physical universe.

The Spectre of Logic

A famous example of a logical paradox is: “This is a lie.”
If that is a lie, then it is a lie that it is a lie, so it is not a lie.

But if it is not a lie, then what it says is false, so it is a lie.
Whereas, if it is not a lie, then it is not the case that it is a lie.

Contradiction! So, logic gives us paradoxes. Still, so what?
Would we expect highly evolved apes to be perfectly logical?

Many scientists, having taken logical looks at the evidence, while giving very low prior probabilities to the existence of a transcendent Creator (having read books by Richard Dawkins and Bertrand Russell), do think of themselves as highly evolved apes, in a purely material world that just happens to exist, so they might not be too surprised to see their natural logic being shown to be contradictory. Modern logic is, after all, much more "rigorous" than natural logic ("rigorous" in the sense of mathematical). But if their looks at the evidence were not very logical, then their lack of surprise might not be very logical. And what if there is a logical proof that there is a transcendent Creator? Of course, they would think that there is unlikely to be such a proof, because of those very low priors. But at the end of the day there is such a proof.

Perhaps you think that it would still not be irrational of them to refuse to countenance the possibility of a transcendent Creator, much as this would not be irrational:

I see a tree, so I know it is a tree. That is certainly rational. I cannot rule out its being an alien quasi-stick-insect of a very convincing kind empirically, of course (that is what “very convincing” means); but so what? I naturally assume that it is no such thing, and even now, after thinking of this particular possibility, I still have no idea how unlikely it really is, so I cannot do better than continue to make that assumption. Making that assumption may well make my knowledge that I am looking at a tree a sort of gamble, but such is knowledge.

What if you had evidence that that tree was an alien quasi-stick-insect, though? What if you showed me that evidence? Should I not take a logical look at it? Your discovery would be very important, if your evidence was good enough (extraordinary claims do stand in need of extraordinary evidence). What if I looked at your evidence and saw nothing wrong with it but also saw that given my assumption, it must be defective; would that be logical enough? I might think so, but then, the alien might bite my head off and you would have tried to warn me (in a very scientific way).

Suppose that you had a very good argument for something that I did not like, not at all, but suppose that I gave your argument the time of day. If I let the strength of my dislike trump the logic of your argument, would I seem logical? Suppose that I deduced from my dislike that you must have made a mistake; suppose I say that there is probably something wrong with your argument (to err is human), and that I have better things to think about. Is the latter logical enough, but not the former? Surely we owe it to ourselves to be properly logical.

Even if our logic was flawed, it would still be our logic. Assuming that it was not flawed would, even then, just be a very human error. And maybe our logic is flawless:

    “The assertion that you are now reading is not true.”
    Let us call that assertion “L”, so that     L is true if, and only if, L is not true
    or, if “is true” is a vague predicate,        L is true insofar as L is not true.
    From the latter, it follows that L is as true as not,
    which it can be, if “is true” is a vague predicate.

There is only a contradiction if “is true” is not a vague predicate. And while that might not be a proof by reductio ad absurdum that “is true” is a vague predicate, it is close to being such a proof and very far from being a reason to doubt logic.

Apparently Timeless Possibilities

Apparently timeless possibilities could, possibly,
become more numerous over time, e.g. as follows:

You were always possible,
but had you never existed,
then that possibility would have been
the possibility of someone just like you.
      It could not have been
      the possibility of you in particular 
      were you not there to refer to.
Looking back now,
we can see that there was always
that possibility, of you in particular 
as well as the more general possibility,
even before you came into being.

Now, Presentism is logically possible,
and if Presentism is true then there may
originally have been no such distinction,
even though you were always possible.
      Under Presentism it could have been
      that you might not have existed.
The distinction could therefore have
arisen when you came into being.

It is therefore logically possible
for apparently timeless possibilities
to emerge as distinct possibilities
from more general possibilities.

The Signature of God

I think belief in God reasonable only if it is based on considerations available to all humans: not if it is claimed on the basis of a special message to oneself or to the group that one belongs.

Anthony Kenny ("Knowledge, Belief, and Faith," Philosophy 82, 381-97)
      So what better signature of the creator of homo sapiens than an elementary logical proof that there is a God? In my last post, I described the argument that given some things, cardinally more selections from them are possible.
      That post ended with a brief description of how that means that paradox arises: we naturally assume that each of the possible selections that such endlessly reiterated selection-collections and infinite unions would or could ever show there to be is already a possibility, that it is already there, as a possible selection; it would follow that they were all there already, that they are collectively some impossible collection of all those possible selections.
      Logic dictates that we have made some mistake; and this version of Cantor's paradox arises because we are considering combinatorially possible selections: that is why the sub-collections that define those selections were able to become so paradoxically numerous, why the paradoxical contradiction did not just show that there are not, after all, so many extra things, over and above the original things.
      My resolution begins by observing that apparently timeless possibilities could, possibly, become more numerous over time; it begins that way because if possible selections are always becoming more numerous, then we would never have all of them. A Constructive Creator could, possibly, make the definitive selections; and if that is the only logical possibility, then that is what has been shown.
      Note that serious mathematicians have taken Constructive mathematics seriously, and when constructed by a transcendent Creator the mathematics would be much more Platonic, and much more Millian. Consider, for an analogy, how God's commands could, just possibly, define ethics. And note that such creative possibilities are not that different to the Creating of mere things ex nihilo, if you think about it: how is such Creation even possible? For us, the laws of physics present immutable limits to what can be done; for a God, such laws are, metaphorically, a brushstroke.
      We live in a world of things, and numbers of things; and for us, numbers appear timeless. But logic does seem to say that such numbers are impossible. When we first think of the origin of things, we might think of things that could have been there forever, like numbers. But logic seems to say that there was originally stuff, not things; perhaps mental stuff, perhaps a God that is not exactly one thing. There would have been some possibility of things, and more arithmetic the more that God thought about that possibility.
      I should add a note about what sort of God is being shown to exist. The proof does not show that God could not have created a four-dimensional world in a Creative act above and beyond that temporal dimension. So this God might be what we call "timeless," and might know all about the future; or not. And either way, this God could always have known all of our textbook mathematics, if only because that is essentially axiomatic.

Cantorian Diagonal Argument

Cantor’s diagonal argument that there are more real numbers than natural numbers gets its name from its picture proof (as below), and it generalizes to show that powersets are always bigger than their original sets.

Cantor originally used collections instead of sets, but they gave him a paradox. Now, a collection of things is just those things being referred to collectively, so it is hard to see how that could have been the problem. But Cantor introduced the notion of a set, or consistent collection, and most modern mathematicians use axiomatic sets, for added confidence, and define numbers from them.
      Nevertheless, there really are numbers of things, so I take the logical essentials of Cantor’s paradox, and find a lacuna. My version of the diagonal argument shows that given some things, cardinally more selections from them are possible. As is the case with Cantor’s diagonal argument, it is best to begin with small collections, and build up from there, so that the general case can be more easily understood, by comparison with those simpler cases.

There are clearly three things:
      clearly there are
Given those three words, we can select a couple, e.g. ‘clearly’ and ‘there.’ There are three ways of making a pair – three different pairs that could be made – from those three words: {‘clearly,’ ‘there’}, {‘there,’ ‘are’} and {‘clearly,’ ‘are’}.
      Each of those ways of making a pair derives from, and is therefore defined by, the presence of two particular things in the original collection. Given those two things, there is that way of making a pair, whether or not anyone would or could make it. Because the original things were distinct, those pairs are distinct, and so each can count as one thing, as when we think of those three ways of making a pair. Possible selections are, in that sense, things.
      In a similar sense, collections are things, e.g. those are three collections of words. But given some things, thinking of just some of them as yet another thing can easily seem like the weak link in a chain of reasoning that leads to a contradiction. (After all, the collection of one thing is just that thing.) It is clearer that, given some things, a way of making a pair from them is indeed another thing. (A way of selecting just one thing is a way of selecting.)

Making a pair is just one way of making a selection. There are, in total, 2 to the power of 3, or 2^3= 8 possible selections from a collection of 3 things (cf. its powerset). It is an elementary result in combinatorics that there are 2^3 ways of assigning 2 labels, say ‘In’ and ‘Out,’ to 3 things, e.g.
      for the pair that is ‘clearly’ and ‘there,’
      ‘clearly’ has the label ‘In,’ as does ‘there,’
      while ‘are’ has ‘Out’ because it is not in that sub-collection.
In general, for any collection of things, T, there is a selection-collection, S(T), of all the combinatorially possible selections from T, each corresponding to some combination of as many ‘In’s and ‘Out’s as there are things in T.
      There is the selection-collection of 2^8 = 256 possible selections from the aforementioned 8, for example; and there are, similarly, a further 2^256 possible selections from that collection, and so on.
      Each of those possibilities is distinguished and defined by the presence of pre-existing things in the original collection, and so they are all implicitly there already, with our original 3 things.
      There is therefore an infinitely big collection of combinatorial possibilities, say N (from which further selections might possibly be taken, giving us S(N); and so on).

Two collections of things have the same cardinal number of things when there are one-to-one mappings from each collection onto all of the other. Cardinality therefore captures some of the intuitive sense of there being as many things in one collection as there are in another. Whether cardinal numbers are rightly called ‘numbers’ or not does not matter here; for the purposes of this proof, the main thing about cardinality is that it is an equivalence relation: it is reflexive, symmetric and transitive. Cardinality therefore partitions collections into equivalence classes. In particular, S(N) is not in the same class as N, for the following reason.

For simplicity, the things in N will be given the names
‘1’ (e.g. naming {‘clearly,’ ‘there,’ ‘are’}),
‘2’ (e.g. naming {‘clearly,’ ‘there’}),
‘3’ and so forth. To get the things in S(N) we associate the things in N with either ‘In’ or else ‘Out,’ and so each thing in S(N) can be named by an infinite sequence of ‘I’s and ‘O’s.
      If S(N) had the same cardinality as N, then we could associate each combinatorially possible sequence of ‘I’s and ‘O’s with one of the names of the things in N, and thereby list all the combinatorially possible sequences of ‘I’s and ‘O’s.
      E.g. the element of S(N) whose name is the sequence I, I, O, O, … might be associated with the element of N named by ‘1,’ and so on:

 N         S(N)

 1           I           I         O         O         …

 2          O         O         I          O         …

 3           I          O        O         O         …

 4           I          O         I           I         ...

…         …         …        …         …         ...

If we can name an element of S(N) that is not in that list (for any such list), then that will show that N and S(N) are not the same size; and we can specify one that is different from each of those by specifying that:
its 1st label differs from the 1st label of the element associated with 1,
its 2nd label differs from the 2nd label of the element associated with 2,
and so on (e.g. O, I, I, O, ...). And that diagonal argument generalizes to show that every selection-collection, S(T), is cardinally bigger than its original collection, T, as follows.

We begin by supposing, counterfactually, that S(T) has the same cardinality as T, i.e. that there are one-to-one mappings from T onto all of S(T). Let M be one such mapping.
      We use M to specify a collection D as follows: for each thing in T, if the possible selection that M maps that thing to includes that thing (in other words, if that thing has the label ‘In’ in that possible selection) then D does not include it (it has the label ‘Out’ in D), but otherwise D does; and there is nothing else in D.
      Since the only things in D are things in T, D should be in S(T); but according to its specification, D would differ from every possible selection that M maps the things in T to. Consequently there is no such M; S(T) does not have the same cardinality as T. And since S(T) has at least one element for each thing in T – e.g. the selection of that thing – hence S(T) is bigger than T.

So, given any 3 things, there is an infinitely big collection, N, and an even bigger collection, S(N), and the even bigger S(S(N)) = S^2(N), and similarly S^3(N) and so on.
      All the things in all those collections are collectively the union, U, of those collections; so, there is also U. U is bigger than each of those S^n(N), for natural numbers n, because it contains all the things in each S(S^n(N)). Furthermore S(U) is even bigger, and so on. So there is also the union, say V, of all the S^n(U) for natural numbers n. S(V) is even bigger, and so on; and so forth, past W, say.
      There will be a union, UU, of U, V, W, …, and thence a union, UV, of all the S^n(UU), and similarly UW, and so on, through VU, VV, VW, ..., and so forth. There will be a union, UUU, of UU, VV, WW, …, and a union of U, UU, UUU, …, and so on, and so forth.

Paradox arises because we naturally assume that each of the possible selections that such endlessly reiterated selection-collections and infinite unions would or could ever show there to be is already a possibility, that it is already there, as a possible selection. It follows that they were all there already, that they are collectively some collection, C, of all those possible selections.
      But, there cannot be any such collection, because its elements, simply by existing, would define those of S(C); and by the diagonal argument, S(C) would contain even more things of that very kind. And we cannot – by the meaning of ‘not’ – have both that C does contain all such possible selections and that C does not contain them all.

      [Lacuna]