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 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. And 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 Georg 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. The 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, 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 to be resolved.

The following version shows, to
begin with, that certain possibilities become more and more numerous. Of
course, 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 follows. 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 ‘time-or-super-time’ 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, 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.

Let us begin with a self-evident
truth: these words are distinct from each other. That is self-evident because
that is how we were able to read them. There are, then, numbers of things, e.g.
‘happy’, ‘summer’ and ‘days’ are three words. Now, pairs of those three words –
{‘happy’, ‘summer’}, {‘summer’, ‘days’} and {‘happy’, ‘days’} – are just as
distinct from each other as those words were, because those three pairs differ
in just those three words. And similarly, pairs of those pairs – e.g.
{{‘happy’, ‘summer’}, {‘summer’, ‘days’}} – are just as distinct; as are pairs
of those, and so on. Because of that ‘and so on’ we will have infinitely many,
equally distinct things, if we can indeed count pairs as things. Is there
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*. So such possibilities
are such 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 ‘{‘happy’, ‘summer’}’ to be the name of the combinatorially possible
selection of ‘happy’ and ‘summer’ 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 {‘happy’, ‘summer’} 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 {‘happy’, ‘summer’} is a possible selection is that
‘happy’ and ‘summer’ 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, before choosing to
contemplate creating some particular 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. It makes sense that a being that could create things *ex nihilo *would know so much about them. Standard set theory would therefore be a very good mathematical model of the more
imaginable levels. And 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. Also note that the existence of such a Creator would also
resolve similar paradoxes, such as the Burali-Forti paradox, which concerns
ordinal numbers. A dynamic Creator would be able to construct – and would
probably enjoy constructing – ordinal arithmetic forever.

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.