The Essence of Cantor's Paradox

(1)    There are at least three things

Clearly there are.

(2)    Given some things, there are possible selections from them

E.g. ‘clearly’ and ‘there’ are a pair of words.

(3)    There are all the things given by reiterating (2), given (1)

Note that each possible selection is a thing.

(4)    Given some things, cardinally more selections from them are possible

That is shown by a Cantorian diagonal argument.

(5)    There are cardinally more things of kind (3) than there are things of kind (3)

That follows from (4), given (3), but is contradictory, and hence false.

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 becoming more numerous, then that could easily change the meaning of (3) enough to avoid (5). There is no other way of avoiding the contradiction (the main resolutions were constructivism, with its potential infinities, and going axiomatic, which means not addressing numbers of things directly), which is why this is a paradox.

Consequently this is essentially a proof by reductio ad absurdum that possible selections do become more numerous over time. And how could possible selections become more numerous, if not by a transcendent creator making definitive selections, constructing arithmetic as part of the creation of all things? There is no other way that I can think of; whereas this way is a serious possibility, because
A) mathematicians have taken constructivism surprisingly seriously, and constructivism would only be more Platonistic, and Millian, were the definitive constructions made by a transcendent creator, and
B) theologians have taken the idea of God being beyond our conception of number very seriously, e.g. the Trinity.

Logic Needs That Hypothesis

In the Germany of the eighteen-nineties, Georg Cantor discovered the mathematical paradox that bears his name.
He put it down to the ineffability of God, even though he was only studying numbers; they were very big numbers.
But, the mathematical mainstream has since then replaced our natural conception of a collection with formal (or fictional) sets that are better behaved.
Whereas, the natural conceptions are fundamental to our actual thinking; in particular, if we cannot rely on our best thinking about formal sets, then why should formal sets be any better?
Consequently logical thinkers need to hypothesize God: only that allows those conceptions without paradox (as previously posted, and as sketched in my next post).
Over the next few posts I aim to scrutinize the elements of this, e.g. the essence of Cantor's paradox, and why we do still need logic in this democratic and scientific age.

Is there a Problem with Prefaces?

Suppose that in the preface to some non-fiction book, the author apologizes for whatever false statements there may be in the book, observing that there are bound to be some, even though each statement in the book is believed to be true by the author. Is this a Preface Paradox? Philosophers often "solve" this problem by taking belief to be sufficiently high credence, so that we would not believe large conjunctions of our beliefs; but of course, our believing each of our beliefs means that we believe them all. Belief is simply not sufficiently high credence (as recently posted). In fact, there is no problem here to solve. And of course, not even a philosopher would ask: Why not say that sets are simply such that sets of beliefs are well-behaved? Could we not have all that we want and nothing that we do not want by putting precisely that much into the axioms of our favourite set theory? No: that is simply not what conjunctions of beliefs are. Nevertheless, that is essentially the mainstream response to Cantor's Paradox (which I shall be posting on in my next few posts). The Preface "Paradox" may not be very paradoxical, but it does show how absurd the mainstream foundations of mathematics are.

Mrs Fox's feelings

Basking in the warmth of Heaven, she floats weightless and naked, far far above the factory chimneys and church spires of the world, in the upper reaches of a sultry sky. It's an intoxicatingly fragrant atmosphere, surging and eddying with huge gentle waves of wind and pillowy clouds – nothing like the motionless, transparent oblivion she'd always imagined Paradise would be. It's more like a breathable ocean, and she treads the heavy air, narrowing the distance between her body and that of her man who's flying beside her. When she's close enough, she spreads her thighs, wraps her arms and legs around him, and opens her lips to receive the incarnation of his love.

Michel Faber, The Crimson Petal and the White, 671

Is very high credence belief?

Imagine holding an almost spherical die with thousands of tiny faces. Each of them is highly unlikely to end up on top when you roll the die. For each face, you have a very high credence in the proposition that it will not end up on top. But when one of them does end up on top, you would not be surprised.

When something that we believe turns out not to be the case, we are surprised. So you did not, for any of those faces, believe that it would not end up on top (you believed that it very probably would not). Very high credence is not belief (despite what many philosophers believe).

I see a red car outside my window, and I can see that its red colour is out there, where the car is. If you told me that the red colour that I was seeing was on the surface of the car, where its paint was, I would not believe you: your saying that would remind me that the red that I see is only in my perception, that it is not actually out there where the car is. Still, I would feel some surprise at that thought, for all that it would be a very familiar surprise. Even though I have a very low credence in the proposition that the red that I see is out there, where the car is, I cannot help believing, each time I look at the car, that its colour is out there. Seeing that it is out there is, at least momentarily, believing that it is out there. That is just the way my perception works. And yours? What about when you look out of a window. Consider any particular arrangement of cars, leaves et cetera that might be outside some window that you could easily get to. Presumably the chance of there being that particular arrangement is very low. You would be surprised if that was the view, were you to look out of that window.

It would be like putting a tiny red dot on one of the faces of that almost spherical die, rolling the die and seeing the red dot on top of it when it stops rolling.

Still, it would also be surprising if you had been able to imagine the view from that window in very much detail before you looked. So, suppose that you go to that window and look through it: you would probably have an unsurprising view. And you could then ask yourself whether you believed, before you looked, that that view was not going to be what you saw. Look now. I think that before you looked, if you did look, you did not believe that you would not have that view. Had you had such a belief, would you not have been much more surprised? Perhaps you were surprised (perhaps, for example, there was a red car there, and you were not expecting there to be one); but even then, your surprise would not, I think, have been like the surprise that we feel when something that we believe turns out not to be the case. It would be less, I think, than your surpise at the red of the car not actually being out there where the car is.

If I believed, of each of that die's faces, that it would not end up on top, then I would believe that none of them would end up on top.

That is how my beliefs appear to me (if I believe that an object is a car, and I believe that it is red, then I believe that it is a red car). How do yours appear to you? Beliefs are what they are, and what they are is not a million miles away from us. We can find out for ourselves what our beliefs are, our particular beliefs, and also beliefs in general (beliefs of various kinds) if we are philosophically inclined, and sufficiently logical. I wonder if philosophers who say that belief is very high credence are confusing belief with a mathematical model of belief.

Turri's Maxwell's Car

When Maxwell arrives at work in the morning, he always parks in one of two spots: C8 or D8. Half the time he parks in C8, and half the time he parks in D8. Today Maxwell parked in C8. It’s lunchtime at work. Maxwell and his assistant are up in the archives room searching for a particular document. Maxwell says, “I might have left the document in my car.” The assistant asks, “Mr. Maxwell, is your car parked in space C8? It’s not unheard of for cars to be stolen.” Maxwell thinks carefully for a moment and then responds, “No, my car has not been stolen. It is parked in C8.”

With that example, John Turri's "Epistemic closure and folk epistemology" post at the Certain Doubts blog began, and he went on to add that:

The epistemic closure principle says, roughly, that if one knows that P, and one knows that if P then Q, and one infers Q, then one knows Q.

Maxwell may have been misapplying logic, when he thought carefully: he recalled that he had parked in C8, rather than D8, and so he thought that his car was in C8 (unless it had, as his assistant noted, been stolen), from which he may have concluded that it was not stolen. (But perhaps he took the low chance of his car having been stolen to be reason enough to think that it had not been stolen. And for all we know the archive's windows looked down on C8.)

Most people think that Maxwell knew that his car was parked in C8 (assuming that it had not been stolen, etc.), but not that it had not been stolen, which contradicts Closure: if Maxwell knows that his car is in C8, and he knows that if his car is in C8 then it has not been stolen (he did seem to know that because he did seem to infer, from it being in C8, that it had not been stolen), then Closure says that Maxwell did know that his car had not been stolen.

Logically, if Maxwell's car was in C8, then it had not been stolen; and the whole point of logic is that logical reasoning takes us from knowledge to knowledge. So it seems to be logical, to go from Maxwell knowing that his car was in C8, rather than D8 (which I think he did know), to Maxwell knowing that his car was in C8 (which most people think he did know), to Maxwell knowing that his car had not been taken out of C8 (which most people think he did not know).

But of course, we can see from this example why that is invalid; and so we also have this insight into why skeptical scenarios are not threats to knowledge (and also why they are). If we are BIVs then we do not have real hands, so how can we know that we have real hands but not know that we are not BIVs? If we are BIVs then we think we have real hands (which is good enough for us BIVs).

"Know" means know

When people say "I do not think it, I know it," they are usually saying that they are sure of something. And even philosophers regard what they are sure of as knowledge. What if what they are sure of is not actually the case? Well, everyone thinks that what they personally are sure of is the case: that is what it means to be sure of something.

But what if I am sure of something fairly complicated, and you think that what I am sure of is not actually the case. Would you say that what I had was probably some knowledge beyond your powers of understanding? Or would you think it more likely that I had a complicated kind of ignorance?

Almost everyone thinks that a belief must be true if it is to count as knowledge. And philosophers noticed, a while ago now, that there is more to knowledge than certainty and truth:

What if I am sure that it will rain on my parade, and it does. Would you say that I knew that it would rain? Or do you think it more likely that I was just being pessimistic, and unlucky?

For such reasons, philosophers have been debating how they should define "know"; for my thoughts from nine years ago, click on this link. More recently, I have been thinking about scenarios like this:

By the end of the nineteenth century, physicists were sure that, thanks to Newton, they knew the most general laws governing how physical objects move, but thanks to Einstein, our physicists are now sure that those physicists did not know as much as they thought they knew.

Still, it is possible that physicists, in a thousand years time, will discover that it was only some dark ether, or something even stranger, that had made things look Einsteinian. It is possible that those nineteenth-century physicists did, after all, have a lot of knowledge.

Indeed, all sorts of things that we cannot even imagine might be discovered in a thousand years time. The problem is that we can only judge how likely such future discoveries are by using the knowledge that we are now taking ourselves to have. Philosophers are therefore making sure, as best they can, that what they think they know is indeed knowledge:

Philosophers tend to define "know" by constructing axiomatic models of knowledge, so they do seem to be pursuing the truth in a scientific manner (not too dissimilarly, physicists construct mathematical models of physical systems in order to understand them).

But could philosophers pursue the truth about knowledge if they did not already know what truth and knowledge were (note that physicists do not need to know what matter is in order to discover the laws that govern its movements)?

Is there something about truth and knowledge that philosophers do not understand as part of common sense, and cannot come to understand by logical introspection or by asking psychologists, sociologists and linguists?

Well, I suppose that it would be quite mysterious how people could possibly know truths about the world if people were nothing more than highly evolved animals in a purely physical universe. But then, that is why philosophers should not ignore a logical proof that in all likelihood, that is not what we are: logic dictates that in all likelihood, we are incarnated souls (not too dissimilarly, logic dictates that in all likelihood, Fermat's last theorem is true).

Gettier's Smith's Job

Smith applies for a job, as does X. Smith thinks that X will get the job, and knows that X has 10 coins in his pocket, so Smith thinks that the man who gets the job will have 10 coins in his pocket. As it turns out, Smith gets the job, and also has 10 coins in his pocket, and so his italicized belief was true. Since Smith was justified in thinking that X would get the job (his new boss had told him that X would get the job) his italicized belief was also justified; but, it was not knowledge, according to Gettier.

One problem with all of that is that it is a bit obscure what exactly is going on: there was a bit of inferring going on, and as a rule we cannot rely on such things as, for example, epistemic closure: If you know that P, and also that P implies Q, then even if you infer Q, you do not necessarily know Q (there was a good example by John Turri at Certain Doubts). Still, one thing is obvious: Smith's reasons for believing the italicized belief were no part of the reasons why it was true, and so it was not knowledge.

However, because those reasons turned out not to be that good (X did not get the job), there is also a question mark over whether they really were good enough to count as justification in the sense required for knowledge (only a question mark). A statement known to be false was always a statement that could have been false (that really could, not just could theoretically). When we think of knowledge we think of statements that can be relied upon, that are justified in ways that basically guarantee their truth.

Much Knowledge is Epistemic Luck

In a recent post (linked to here) I observed how we simply assume that we can refer directly to the things around us: we cannot know that their substances are not changing in ways that leave their properties the same, because we can only know their properties. Were their substances changing, reference to them would keep failing (assuming that reference is direct).
     And similarly, we cannot really rule out that we are Brains In Vats: all of our evidence is compatible with our brains having been harvested by aliens (in a real world where such aliens are common) and put into high-tech vats that simulate worldly experiences. While we are unlikely to have been harvested recently (as recently noted (although note that we cannot rule out as unlikely a world where are are frequently, but not too frequently, re-vatted)) it is not unlikely (by the standards of the apparent world) that there are such aliens (what is strange is that we see no aliens).
     But of course, we can and do simply assume that there are not such aliens, that we are not currently asleep in our beds and dreaming, that all of our particles are not always being switched with identical particles, and so forth. It is upon such foundations that our knowledge of the external world is built. And of course, we are not BIVs, we are not dreaming, and so on; or at least, I do assume not. And so we do have knowledge of the external world. But, because those are assumptions, such knowledge is epistemic luck.

Something like Fake Barns

What is adequate justification for holding a belief? It depends on one's context. A true belief that was well-justified when it was formed might cease to count as knowledge within a stricter setting, such as a court-room or a laboratory. And the famous Fake Barns involve unusual contexts. And of course, one needs to be sufficiently rational. Suppose that I see a red car, in a normal road (no fake cars), for example, and so form the belief that there is a red car. But, I also have a lot of irrational beliefs that there are various objects. When there is a red car, I believe that there is, and I am unlikely to have the belief that there is a red car if there is not a red car, although I am quite likely to have some belief that there is something when there is nothing. (It is easy to think of other examples of true beliefs that might seem at first glance to be justified but which do not count as knowledge because they are held by someone who is in some way unqualified.)

What Do Philosophers Do?

What do you know, for sure? Being sure that you have hands,
for example, could be justified ( you may be using them now,
to operate a phone or a keyboard), but you can hardly rule out
the following scenario, according to which you have no hands:

Your brain was recently harvested by aliens and you are now in a vat
experiencing a detailed simulation; your memories have been altered,
but for you "hand" still refers to things outside the vat. And out there,
those aliens have turned the real world into one enormous brain-farm.

You cannot rule that out,
but you can know that it is unlikely
that your brain was only recently harvested
and so you can, of course, know that you have hands.

Magical Reference

Perhaps you are asleep and dreaming now, but if we bracket the question of what physical objects are made of (3-dimensional particles, or our own thoughts in a dream, or 10-dimensional strings, or whatever), then physical objects are definitely possible. Your body, for instance, is actual, so it is possible. And mental states are also possible, of course. And insofar as we do not know for sure that a God is not a possibility, a God is an epistemic possibility. And because it is, in that sense, possible that God created you, it is similarly possible that as well as creating you, a God created an exact copy of you in an identical universe, because if a God could create this universe then a God would be able to create an identical one (even if that God would not want to create an identical one). Now, it is similarly possible (if implausible) that a God who had made you and an identical copy of you might swap you with that other person, and swap you back, and do that over and over. And the same goes for everything else in this universe. Such swaps could conceivably be happening all the time, for all that we could tell just by looking around us, however closely we looked.

Does that make a God seem unrealistic? But this is not a problem with the possibility of there being a God. Do we know for sure that fundamental particles (or whatever substances are physically fundamental) could not be replaced by identical particles (or substances)? We naturally assume that such replacements do not occur, but such replacements would make no difference empirically, so we cannot rule them out empirically. And prima facie, we cannot rule them out on purely logical grounds. And while they may well seem unlikely, that is only because we are implicitly assuming that they never happen. We have no empirical or logical grounds for believing them to be unlikely. For all we know, they could be happening all the time. We simply assume that they do not happen at all.

You and that hypothetical identical copy of you would not be the same person, even though there would be nothing to tell you apart. That should be obvious because the copy of you would now be in another universe. And the same goes for me, of course. You can refer to me by name (my name is in the sidebar), and that reference should be simple and direct: it should be me that you refer to, not someone else. And of course, it is me that you would be referring to. Or so we assume. The way reference works amounts to an assumption that such swaps do not occur. It is almost magical the way your referring to me means that a host of things that could, for all we know, be happening are not in fact happening.

The thing is, the solution to that logical problem may well be the probability of there being a God.

Logic tells us that things are such that there is Cantor's paradox, and from that paradox we can get, in a purely logical way, to the probable existence of a God. Would such a God make such swaps as those considered above? There may well be good reasons why a God would not do that. So, logic may well show that such swaps are unlikely. It is something to think about...

Isserley's Feeling For Snow

The sensation of fresh snow crunching underfoot was deeply satisfying to Isserley. Just the idea of all that water vapour solidifying by the cloudful and fluttering to earth was miraculous. She couldn't quite believe it, even after all these years. It was a phenomenon of stupendous and unjustifiably useless extravagance. Yet here it lay, soft and powdery, edibly pure. Isserley scooped a handful off the ground and ate some. It was delicious.

Michel Faber, Under the Skin, 56

"Him Who Has Understanding"

222111 is the number of The Royal Mint,
and it equals 1 + 2 + 3 + ... + 666, where
666 = 1 + 2 + 3 + ... + 36 is The Number
Of The Beast. And of course, the love of
money is the root of all evil. Note that
36 = 1 + 2 + 3 + ... + 8 and that
8 = 2 x 2 x 2 x 1 x 1 x 1

A mathematical poem

1 × 2 × 3 × 4 × 5 = 360/3

                          12 = 3 × 4
                          56 = 7 × 8
                          90 = 360/4

                    0 + 12 = 3 × 4
                    5 + 67 = 8 × 9
                                = 360/5

Is "no" the answer to this question?

              In other words:

Is “no, it is not” a correct answer to this question?

              Well, “it is not” means
              that “no” is not a correct answer,
              so if “it is not” is correct,
              then “no” is not a correct answer.

        So, if it is the case that “no, it is not” is a correct answer,
              then “no, it is not” is not correct:

                        It would be correct and not correct.

              But if it is not correct,
              then it is not the case that it is correct,
              so it cannot be correct as well as not correct.

        Does it follow that it is not the case that it is correct?

              But if it is not correct
              then “it is not” is correct, so
              it would be correct as well
              (and it cannot be both).

        Could it be that there is no correct answer?

              But then “it is not” in particular
              would not be a correct answer, and
              it would also, as above, be correct
              (and it cannot be both).

        Could it be that logic is not what we think it is?

Do we exist in a purely physical world?
        Would highly evolved apes have evolved
        to be perfectly logical in such a world?
              You might wonder if appeals to logic
              should be taken more seriously than
              political rhetoric.

              As Dr. Who said:
        Logic, my dear Zoe,
merely enables one to be wrong with authority.

Are we left with everything being political?
No, because answers to questions can be
as correct as not. Consider this question:

              Is “it is an apple”
              a correct answer
              to the question
              “What is @?”?
        Suppose that @ was originally an apple,
        and that it is having its molecules replaced
        one by one with molecules of beetroot.

        “It is an apple” was originally a correct answer,
        but it will eventually stop being correct.
        If that answer must be either correct or else not,
        then an apple can be turned into something else
        (such as a mixture of apple and beetroot)
        by replacing just one of its molecules.
        Because that is absurd, it is plausible
        that @ can be as much an apple as not,
        that “@ is an apple” can be as true as not,
        and “it is an apple” as correct as not.

If “no, it is not” is as correct as not,
then the fact that “it is not” means that “no” is not correct
just means that “no, it is not” is as incorrect as not, and that is the same
as it being as correct as not. There is no contradiction, only consistency.
        So, the fact that all of the other conceivable answers
        (except the one that says not to bother being logical)
        are clearly contradictory means that

              “No, it is not” is as correct as not

        (if logical reasoning is better than political rhetoric).

Telling the Truth

To begin with a preface, suppose that the following sentence occurs in the preface of a book:

All the sentences in this book are true.

That sentence is clearly there to say that all the other sentences in that book are true. So, it would make sense for us to take that sentence to be true if all the others are. It would be silly to say that it was not true on the grounds that it might be false (because if it is false then not all the sentences are true). But what about the following sentence:

This sentence is true.

Sentences like that are called Truth-Tellers, because sentences like "this sentence is not true" are called Liars. Now, that Truth-Teller is saying only that what it is saying is true, so it is certainly not saying much. It is unlike the sentence in that preface, in that respect; although in other ways, the two are very similar: it is consistent for the Truth-Teller to be true, and consistent for it to be false. Still, there is not much for it to be true or false about, so it is hard to think of it as being very true or very false. And there seems to be no way of determining which it is, in any case. Could we say that it was not very true and not very false? Could we say that it is about as true as not (and about as false as not)?
......To see why that might be possible, consider a man going bald. As he goes bald, he will not, by the loss of just one or two hairs, become bald, so there might be an intermediate stage at which he is about as bald as not, when it would make sense for it to be about as true as not that he was bald (and about as false as not). Perhaps it will help to look at the following sentence, which occurs in the preface of a work of fiction:

None of the sentences in this book are true.

If that sentence – let us call it Disclaimer – was true, none of the sentences in that book would be true, so Disclaimer would not be true. By reductio ad absurdum, that means that Disclaimer is not true. So, either it is false – in which case at least one of the other sentences would have to be true (and we can assume that none of them are) – or else Disclaimer is neither true nor false. And Disclaimer did not seem to be meaningless. Are we stuck? Hardly. Disclaimer was presumably there to inform us that none of the other sentences in that book were true. If none of them were, then Disclaimer was, in that sense, true. And how could it have had any other meaning?

Proof's Nearest Kin

Paradoxes are akin to proofs: We have a paradox when we have a very good argument for something that is beyond belief, a proof when we have a logical argument for something not too odd. Many a paradox is therefore a proof by reductio ad absurdum of the negation of its weakest premise. Georg Cantor’s famous paradox of the 1890s was exceptional, being a logical argument for a contradiction, but it thereby proved that human reasoning is not perfectly logical. In response the twentieth century saw a proliferation of formal logics, but as we develop such calculi with mathematical precision we might easily forget that some illogicality is unavoidable in our reasoning. To remind us, then, the following is the essence of Cantor’s paradox.

Consider any 3 things, e.g. a chair, a plate and a fork. There are clearly 3 different ways of making a pair from those 3 things (e.g. the chair and the fork are, collectively, a pair), each of which derives from, and is defined by, the presence of 2 particular things in our original collection of 3 things: Given those 2 things, we have that way of making a pair. Now, making a pair is just one way of making a selection. There are 8 different ways of making selections from our original collection because there are 2³ ways of assigning the labels “In” and “Out” to 3 things (e.g. the chair has “In,” the plate has “Out,” and the fork has “In”). Given our original collection, there are also those 8 combinatorial possibilities, those 8 ways of making selections, ways that are entirely grounded in our original things and which are therefore distinguished from each other whether there is a selector who can make those selections or not. Let us call them possible selections, from our original collection, and say that they are collectively the selection collection for our original collection. In general, for any natural number n, if there is a collection of n things, then there is also a selection collection of 2ⁿ possible selections from it.

We can safely assume that there are 2 things (e.g. you and I are 2 people), so there is also the selection collection of 4 possible selections from those 2 things, and the selection-collection of 16 possible selections from those 4, and so on. All those possible selections are there already, intrinsically distinguished from each other, and so there are infinitely many things, which are certainly things in the weak sense that there are numbers of them, in the fairly weak sense of cardinal number. Two collections of things have the same cardinal number of things when there are 1-to-1 mappings from each collection onto the other. Cardinality is fairly weak, e.g. there are clearly fewer prime numbers than natural numbers in some stronger sense, but it is an equivalence relation – it is reflexive, symmetric and transitive – and so it partitions collections into equivalence classes. For any collection of things, T, there are possible selections from T – each corresponding to some combination of as many “In” and “Out” labels as there are things in T – even if T is infinite, and so there is a selection collection, S(T), of all the possible selections from T. And for the following reason (which is essentially Cantor’s diagonal argument) every selection collection is cardinally bigger than its original collection.

Suppose that S(T) has the same cardinality as T. There would then be 1-to-1 mappings from T onto all of S(T). So let M be one such mapping. We might then use M to specify 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 (i.e. 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, and so D would differ from everything in S(T). D is therefore contradictory, and so there is no such M. Consequently S(T) does not have the same cardinality as T. And since S(T) contains at least one thing for each thing in T – e.g. the possible selection whose only “In” label is assigned to that thing – and cardinality is an equivalence relation, hence S(T) is bigger than T.

Given you and I, then, there is some infinitely big collection, say N, and so by the diagonal argument above with T = N there is also a bigger collection, S(N), and by the diagonal argument with T = S(N) there is the even bigger S(S(N)) = S²(N), and there is similarly S³(N), and so on. All the things in all those collections are collectively the union of those collections, U, which is bigger than each of those Sⁿ(N), for natural numbers n, because it contains all the things in each S(Sⁿ(N)). Furthermore S(U) is even bigger, and so on; so there is also the union, say V, of all the Sⁿ(U) for natural numbers n. And again, S(V) is even bigger, and so on and so forth. Now, each of the possible selections that such endlessly reiterated selection collections and infinite unions would or could ever show to be there is already there, as a combinatorial possibility that is implicitly distinguished from all the others of that kind, and so they are collectively some collection, C, of all the possible selections of that kind. But by the diagonal argument, their selection collection, S(C), would contain even more things of that kind. That contradiction shows that we went wrong somewhere, but why should even highly evolved primates know where?

Natural Numbers

There are clearly numbers of things in the world, e.g. you and I are two individuals. And to think of us in that way is essentially to think of us as the elements of a pair-set, {you, I}. So let us say that a natural set is any whole number of things. Note that {you, I} is also a subset, there being other people. And note that numbers and sets are themselves things in the sense that there are numbers and sets of them.

Do mathematicians discover facts about such natural numbers as 2? If so, then Georg Cantor’s famous paradox of the 1890s was essentially a mathematical proof that the natural numbers (i.e. 0, 1, 2, …) are ever-growing in number, because what Cantor did was to obtain a contradiction from the assumption that there is a set of them all. I will go through the mathematical steps of the proof below; but to begin with, Cantor’s proof was paradoxical because each whole number is essentially the logical possibility of that many things, and if something is ever going to be possible, then it was never logically impossible.

Cantor’s proof has therefore been taken to be a refutation of the assumption that mathematicians discover facts about such natural numbers: Most twentieth-century mathematicians defined the natural numbers, e.g. 0 is usually defined to be the empty set within an axiomatic set theory (usually ZFC), 1 to be {0} and so on. Axiomatic set theory is used because Cantor’s proof used a natural kind of set that has been blamed for the paradox (and called “naïve”). But such sets exist as clearly as natural numbers do and so, like the natural numbers, they should not be defined, but scientifically described.

That is because there is at least one way in which the natural numbers could, possibly, be ever-growing in number, because logical possibilities can become more fine-grained over time. Suppose that time is not so much like space that future events are already there (at future times), and consider any existing man. He was always possible, but it is only with hindsight that we can describe the logical possibility of his existing with such direct reference to him. Before he existed there was only the possibility of someone just like him, to whom we could not directly refer.

The logical possibility of n things, for each natural number n, could, then, have originally been part of the logical possibility of numbers of things, only becoming the logical possibility of things in that number when that number was created, perhaps as part of the analysis of the concept of a thing by some creative power that exists primarily in a world of spiritual stuff (and which may therefore appear triune to us), thereby creating an abstract realm of sortal natural kinds and their associated numbers prior to the physical objects and incarnate creatures of this world. Note that what matters here is only that that is a logical possibility (not even Richard Dawkins claims that God is impossible, only that He almost certainly does not exist).

Let us, then, assume that there is a natural set of all the natural numbers, N = {0, 1, 2, …}. Clearly the subset {0, 1, 2} is already part of N, as is every other subset; all the subsets of N are there implicitly, and so there is the set of all of them, P(N), the power set of N. Cantor showed that P(N) is cardinally bigger than N (two sets have the same cardinal number of elements when the elements of each set can all be paired up with those of the other) by way of a diagonal argument; and both of those steps generalise: Given any set, there are implicitly all of its subsets, so that there is also its power set, to which the following diagonal argument applies.

Let S be any set, and let P(S) be its power set. If S and P(S) had the same cardinality, there would be one-to-one mappings from S onto all of P(S), so let us assume that they do and let m be one such mapping. Let a subset of S, say D, be specified as follows: For each member of S, if the subset that m maps it to contains it, then D does not contain it, and otherwise D does. Since D differs from every subset that m maps the members of S to, D differs from every subset of S, whereas D is by definition a subset of S. Consequently D is contradictory, and so there is no such m, and so S and P(S) do not have the same cardinality. And since P(S) contains a singleton for each element of S, hence P(S) is cardinally bigger than S.

So as well as N, there is also P(N), and P(P(N)) and so forth, an infinite sequence of power sets. Consequently there is also the set of all the elements of all of those sets, U, their union. U is cardinally bigger than each of those power sets because it contains all the elements of the power set of each of them. And of course, P(U) is cardinally bigger than U. And so on (there is another infinite sequence of power sets, then another union, and eventually an infinite sequence of unions that we can also take the union of, and so on).

There must be a set, T, of all that could possibly be found in that way (via power set and union), because all of it is already there to be found. But if T is a set, then P(T) contains cardinally more of precisely those sorts of elements; and that contradiction means that we went wrong somewhere. And from our assumption of N we made only logical moves, so it must have been that assumption that was false: The natural numbers are certainly ever-growing in number (in time that is not much like space).

Logic and God

Vagueness is a well known problem in logic. Imagine, for example, a rough table-top being gently sanded flatter and flatter. Eventually it will become flat (i.e. flat enough to count as flat in some apposite context). However, since ‘flat’ is not so precisely defined that sanding away a few scratches could be enough to flatten the table-top, hence after each bit of sanding the table-top will still not be flat, from which it follows that it will never be flat. That contradiction is a problem that cannot be solved just by redefining ‘flat’ more precisely, because all the terms of natural languages are, in such ways, at least a little vague, and it is within such languages that we all reason. So, there is a borderline, between the table-top being flat and it not being flat, that is more like a pencil line than a mathematical line – there are borderline cases of flatness – but, there is no region between the table-top being flat and it not being flat where it is neither flat nor not flat, because in such a region the table-top would not be flat and yet would be flat (which the meaning of ‘not’ rules out). Nevertheless, it is logically possible for the table-top to be about as flat as not. At such times it would not so much be false as only about as false as not to say that it was not flat, and similarly, a little later, to say that it was flat (which resolves our logical problem). At such times, we might be more likely to say that the table-top was getting flat, since that would be true. We reason best with descriptions that are either true or else not true (false, in classical logic). Of course, ‘getting flat’ is no less vague than ‘flat’, but its borderlines are in different places; and in general, while we cannot remove all the imprecision from our languages, we can always move the borderlines out of the way of our logical language-use. Our words are defined as precisely as our purposes have required them to be, with the two classical truth-values – ‘true’ and ‘false’ – meeting at a place where descriptions are described as well by ‘not true’ as by ‘true’. We do not have to do much with such descriptions, other than identify them as needing to be replaced with truer descriptions, and so we need only add the following definition to the classical definitions of ‘true’ and ‘false’: To say, of what is about as much the case as not, that it is the case, or that it is not the case, that is to say something that is about as true as not. A description that is much truer than not will be true enough to count as true (by definition of ‘much’), while one that is not much truer than not will be about as true as not (by definition of ‘about’); and if we need to make sharper distinctions than that, then we need to avoid borderline cases and use classical logic. We do not need a formal definition of ‘as true as not’ (in some non-classical logic), because mathematical precision is inapposite when the sharp distinction between something being the case and it not being the case is absent. It would, in particular, be wrong to model the idea that self-referential claims like ‘this claim is not true’ are about as true as not as such claims having truth-values of 0.5, as the fuzzy logicians do.

There are similar resolutions of the other semantic paradoxes (see other posts of mine), but the set-theoretic paradoxes have no such resolutions: Sets are essentially non-variable collections and it makes no sense to think of a collection as being about as variable as not. That distinction, between semantic and set-theoretic paradoxes, originates with Frank Ramsey, who was a mathematical constructivist; and quite a few mathematicians believe that the set-theoretic paradoxes show that there are too many numbers – too many possible sizes of sets – for them all to exist as distinct numbers. But, such constructivism seems to clash with the objectivity of arithmetic: How could 2 exist but not, say, 4? Four is just two twos. So, most mathematicians think that the set-theoretic paradoxes should be showing something else, which may have motivated formalizing the borderline truth-value in a mathematics that would then apply, instead of classical logic, to those paradoxes. But in fact, although the existence of whole numbers, n, is essentially the possibility of sets of n objects, and although such possibilities are intuitively timeless, such possibilities can emerge as distinct possibilities from more general possibilities. To see that, consider how the possibility of you would have been, had you never existed, the possibility of someone just like you: Looking back now, there was always the possibility of you yourself, as well as that more general possibility; but, there could have been no such distinction had you never existed. It is, then, logically possible for distinct numbers to emerge in an unending stream from some more indistinct coexistence – as possibilities inherent in the concept of a thing – and so a coherent story can be told of 1 + 1 = 2 existing – via the concept of another thing of the same kind – and 2 + 1 = 3 existing, along with the question of what 2 + 2 is, and only then 2 + 2 = 2 + 1 + 1 = 3 + 1 = 4 existing. Note that such a story might be more plausible were the small natural numbers replaced by large transfinite numbers. Furthermore, if the concepts involved were divine conceptions, then such arithmetic would be as objective as anything. So the main reason why the set-theoretic paradoxes are paradoxical is the prevailing atheism within science (which is all but a reductio ad absurdum of atheism).