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Wednesday, May 31, 2006

Contra Cantor

Posted by Adam Roberts on 05/31/06 at 10:59 AM

And next week’s ‘contra’ will be called ‘Contra Con. Art’ and the one after that ‘Contra Contra’.  And then, unless I can think of any more anagrams for ‘contra’, I may have to give the series a rest.

And all that remains now is to find some way of filling a slot by using my high-school maths to tangle with Georg Cantor, one of the very greatest mathematicians of the nineteenth century.  Yeah.  That’s going to work.

Permit me to slide up to this sideways, not so much contra Cantor, of whom you have heard; but contra a couple of more modern thinkers of whom you may not have done, but who have written a very interesting book about the cosmos, infinity, the existence of God and other things.  So let’s talk about proofs for the existence of God.  That’s always fun.

One of the oldest arguments for the existence of God was rehearsed a few years back by William Lane Craig and Quentin Smith, who claim that the physics of Stephen Hawking and Roger Penrose, theories of the Big Bang singularity and modern science in general support it. They put the argument (you’ve certainly already heard of it) in the following terms:

(1) Everything that begins to exist has a cause of its existence.
(2) The universe began to exist.
(3) Therefore the universe has a cause of its existence.  [Craig, William Lane, and Quentin Smith, Theism, Atheism and Big Bang Cosmology (Oxford: Clarendon Press 1993), p.4]

The key term here, they say, is the second one.  If the universe is a finite entity, if it began to exist at some point in the past (let’s say at the ‘big bang’ posited by physics) then something must have caused the universe.  For theists like Craig, or Smith, or Pamela Huby or David Conway or plenty of others, this means that there must be a God.  Which clearly would have serious ramifications.  On the other hand, if the universe is infinite, then it never did ‘begin to exist’, and there is no need to posit a creator.

This is the nub: if the universe is infinite … Craig and Smith’s argument depends upon their assertion that the universe cannot be infinite, because the universe is a real thing, an actual fact, and an actual infinitude cannot exist.

We need to be sure what we’re saying here.  When theorising the infinite a distinction is often made between ‘potential’ infinite and ‘actual’ infinite, a distinction that goes back to the great mathematician Georg Cantor.  Before Cantor it was assumed that ‘infinity’ could only be a potential thing.  A line may potentially be divided into an infinity of points, but nobody could actually do this.  The series of numbers 1, 2, 3 … (and so on for ever) reaches potentially to infinity, but nobody could actually count that far, because it would take an infinite amount of time.  This, it seems to me, has a certain common sense smack of rightness to it.

Cantor accepted this notion of the potential infinite and assigned to it the symbol “∞”.  But he also insisted that there could be an actual infinite; he called this ‘aleph zero’; it takes as its symbol a Hebrew aleph (which ought to be ‘font face="Symbol">ℵ/font’, with a subscript zero after it, but I can’t seem to get this symbol to regster with the Valve’s font software), and it represents the number of all the numbers in the world.  Here is David Hilbert’s definition of these two terms.

Someone who wished to characterize briefly the new conception of the infinite which Cantor introduced might say that in analysis we deal with the infinitely large and the infinitely small only as limiting concepts, as something becoming, happening, i.e. with the potential infinite.  But this is not the true infinite.  We meet the true infinite when we regard the totality of numbers 1, 2, 3, 4… itself as a completed unity, or when we regard the points of an interval as a totality of things which exist all at once.  This kind of infinity is known as actual infinity. [Hilbert, David, ‘On the Infinite’, in Paul Benacerraf and Hilary Putnam (eds), Philosophy of Mathematics (Prentice-Hall 1964), p.139]

Now, Cantor’s transfinite number theory has proved, we’re told, immensely useful to mathematicians.  Which is clearly a very good thing.  But according to Craig and Smith it generates key problems in the real world, the one in which we actually live.  They insist that ‘it is intuitively obvious that such a system [of actual infinity] could not possibly exist in the real world’ [Craig and Smith, p.12].  The life of the universe cannot be infinite, and so therefore it must have been created at some point; and Craig and Smith find in contemporary physics and its theory of a big bang ‘a truly remarkable confirmation of the conclusion who which philosophical argument alone has led us’ that the universe was created ‘about 15 billion years ago’ [Craig and Smith, p.56].

But I want to step back in his proof for a moment, and examine whether they are right in claiming that an actual infinity could never exist in reality.  Their whole argument depends upon this belief; if they are right, then we find ourselves, it seems to me, pushed towards a belief in God.  If they are wrong then there is no need for such a belief.  The intellectual stakes are high.

Craig and Smith insist that an actual infinite, if it existed, would lead to paradox and contradiction.  They produce two thought-experiments to support this theory, both of them famous test cases.  One is known to philosophers as ‘Hilbert’s Hotel’ after the thinker who first formulated it.  Imagine a hotel with an infinite number of rooms, all occupied.  A new guest arrives.  In a finite hotel, if the rooms are all full the manager must turn away the new custom; but in the infinite hotel the proprietor can put the newcomer in room 1, and shift the current occupant of room 1 into room 2, the person in room 2 into room 3, and so on.  In this way a seemingly full hotel can accommodate the new guest, and in fact could accommodate an infinite number of new guests.  Craig and Smith insist that this would work only for a potential infinite (‘such that new rooms are created to absorb the influx of guests’, Craig and Smith p.14); but that in an actual infinite hotel all the rooms are already full, and there is no room for a new guest, even though there is an infinite number of rooms.  Hence, they say, paradox.

Their second case consists of imagining an infinite library, in which every book has a number printed on its spine.  They then posit adding a new book to the library.  What number should go on this new book’s spine?

Because the collection is actually infinite, this means that every possible natural number is printed on some book.  Therefore it would be impossible to add another book to this library.  For what would be the number of the new book?  Clearly there is no number available to assign to it.  Every possible number already has a counterpart in reality, for corresponding to every natural number there is an already existent book.  Therefore, there would be no number for the new book.  But this is absurd, since entities that exist in reality can be numbered.  [Craig and Smith p.12-3]

They conclude that ‘therefore an actual infinite cannot exist in the real world’.

There is however a flaw in their logic, or so it seems to me.  The problem comes at the point when they suggest adding ‘a new book’ to the library.  This is a false step.  If the library is infinite and real, as they say, then there can be no new books to be added:  there is be no place outside the library from which a book could be retrieved.  In this scenario, all books are already in the library and it is meaningless to talk of a new book.  The same is true of Hilbert’s Hotel; no new guest can arrive to trouble the proprietor because every single entity in the universe is already inside the hotel.  Craig and Smith insist that ‘it is obviously possible to add to, say, a collection of books; just take one page from each of the first hundred books, add a title-page and put it on the shelf’ [Craig and Smith, p.14].  But this book, whatever it might be, already exists somewhere in the collection (if the collection is actually infinite).  There is no need for a new number; the book that Craig and Smith hypothesise already has a number, somewhere in the collection.

‘Suppose,’ Craig and Smith continue, ‘we decide to loan out some of the books.’ Suppose, as they suggest, that we loan out every book with an odd number: ‘the collection has been depleted of an infinite number of books, and yet we are told that the number of books remains constant.  The cumulative gap created by the missing books would be in infinite distance, yet if push the books together to close the gaps, all of the infinite shelves will remain full (this is Hilbert’s hotel in reverse)’ [Craig and Smith p.15].  Once again, the fallacy here is imagining that there is somewhere outside the library into which books can be moved.  But if the library is both real and genuinely infinite, there can be nowhere outside it.  Items could be shuffled round inside such a library, but nothing could be removed or added.

Craig and Smith’s fallacy, I think, takes the following form.  They say, in effect, ‘first imagine an infinite library, containing an infinite number of books.  Now, let us add one book to this collection…’ This has the same structure as the following:  ‘let us say that you have one apple.  You only have one apple; there are no other apples in your possession.  Now, take your second apple …’ All the many logical contradictions they outline stem from this initial false step, not from the impossibility of an actual infinite.

In fact the logic leads not to a denial of the existence of an actual infinite, but rather to the necessity of its existence.  We could say that not only can an actual infinite exist, it does exist, and that we are living in it.  We call it ‘the universe’.  What follows?

I’m suggesting that the universe is Hibbert’s hotel, in a manner of speaking; and that it’s a category error to apply the logic of a notional Cantorian infinity of infinities to it, as Craig and Smith do in order to generate their paradoxes.  Which is not to assert a contra to Cantor as such (‘nobody shall expel us from the paradise Cantor has created for us,’ Bertrand Russell once said; and who am I to disagree with Bertrand Russell?), but only to assert that all that’s needful is to expel Cantor from our real universe in order to sidestep the mystical uncaused-causer, or caused-uncauser, or whatever the mumbo jumbo phrase is, that incautious application of his thinking can generate.  So I’m arguing contra-Cantor in only a specialised sense; not to challenge his maths as maths (a field in which my adequacy is microscopic), but rather to protest against doing what Craig and Smith are doing: mixing and matching pure and applied, Cantor’s speculative set-theory aleph-o and the real world’s potential infinity.  Cantorians can continue to frolic and canter in the green fields of their imaginary pure-maths realm.  But my argument is that for us, in the applied maths cosmos, maybe our big bang was the end point of a previous shrinking universe as well as the start point of a new one.  Maybe the cosmos is actually an infinite and uncaused thing.  A bas Cantor, vive the godless cosmos.


If the library is infinite and real, as they say, then there can be no new books to be added:  there is be no place outside the library from which a book could be retrieved.

That appears to assume that an infinite space would necessarily take up the whole universe. Not so; consider a half-plane, an infinite space that leaves room for another infinite space outside it. Indeed if any number of straight lines intersect at a point, the sectors between them are all infinite; thus any infinite space can be divided into an arbitrary number of infinite spaces.

By on 05/31/06 at 12:30 PM | Permanent link to this comment

Contra {rat,tar} con; Rant Co.; Coat, RN; to narc.

By ben wolfson on 05/31/06 at 01:46 PM | Permanent link to this comment

Contra Carton: was it really a far, far better thing?

By on 05/31/06 at 02:47 PM | Permanent link to this comment

This form of argument:

(1) Everything that begins to exist has a cause of its existence.
(2) The universe began to exist.
(3) Therefore the universe has a cause of its existence.

Is a question-begging form of the older argument:

(1) Everything that exists has a cause of its existence.
(2) The universe exists.
(3) Therefore the universe has a cause of its existence.

Argument B generated an infinite regress, which was ended by arbitrarily positing the existence of a Creator who was himself an exception to premise B1. Of course, if B1 is not necessarily true, then there is no reason for a Creator. If he can exist without a cause, then the universe can do alright without one also.

Argument A tries to fix this by changing “exists” to “begins to exist” in A1. Well, that is what is under debate, isn’t it? Did the universe begin to exist? Did it suddenly pop into existence one day ex-nihilo? Even if one accepts the Big Bang Theory, it does not mark the beginning of the universe. It just explains its current shape, as opposed to what it looked like before it blew up.

In the arguments regarding infinity, there is a confusion between infinite and eternal. Time is a measurement of relative motion. Motion does not exist without things which move. Hence, one cannot talk about a time before the existence of the universe. The existence of the universe is necessarily eternal. One can maintain without contradiction that the universe must be finite in space and contain a finite number of existents, yet also be eternal.

But all the arguments about infinity are a distraction from the main point, and in fact, only serve to undermine the theists’ final position: that the cause of the universe is itself an uncaused, eternal, omniscient, omnipotent, not-limited-in-any-fashion-whatsoever, Deity. No one who has a problem with an infinity actually existing (and I do) should be arguing for the existence of God.

By Ardsgaine on 05/31/06 at 03:08 PM | Permanent link to this comment

As an SF reader, I always figured that once technological civilization became far enough advanced somewhere in the universe, some entity would go back in time and cause the Big Bang, thus giving everything a preceeding cause even if the universe was infinite in some sense.

By on 05/31/06 at 03:10 PM | Permanent link to this comment

When theorising the infinite a distinction is often made between ‘potential’ infinite and ‘actual’ infinite, a distinction that goes back to the great mathematician Georg Cantor.

The very Greek notion that the infinite exists only as a potential, rather than in actuality, goes back at least as far as Aristotle’s Physics (Book III in particular). Overcoming Greek-induced diffidence about the infinite took lots of giants of mathematics standing on the shoulders of lots of giants of mathematics topped off by Cantor. It’s peculiar to see Craig and Smith rearguing Artistotle’s point about physical infinities (Aristotle sidestepped the argument against mathematical infinities, presumably because the margins of his book were not wide enough to accomodate it).

By on 05/31/06 at 04:39 PM | Permanent link to this comment

I share your ignorance of Cantor, although apparently the forthcoming Badiouian revolution is going to require us all to learn it or risk being totally irrelevant—but one aspect of this proof that strikes me as problematic is the status of this causative entity.  Does it “exist”?  What sense can existence “outside” the universe have?  Does something “exist” in the same sense if it never began to “exist” (as would presumably be the case with God)?  In a certain sense, isn’t this a proof that God does not “exist,” that God has to be irreducibly outside the universe (since the universe cannot support an infinite being, such as God would be required to be)?

I’ve never been a fan of proofs of the existence of God, by the way.  They never seem to be doing the work people expect them to do (or fear that they’re doing)—they skip a lot of steps between “unmoved mover” and “the God of Abraham, Isaac, and Jacob.” It seems that one could posit the Big Bang itself as this “uncaused cause,” at least with a little conceptual work—there’s no need to say that the “uncaused cause” is some kind of personal agency.

By Adam Kotsko on 05/31/06 at 06:06 PM | Permanent link to this comment

The reason Cantor is really important is not that he posited an actual infinity - it is that he proved that there are different infinities. Specifically, he proved that the totality of natural numbers and the totality of real numbers are of different size in an essential way, so that if one is to treat such totalities as complete objects in their own right, as actual infinities, they will be different, separate infinities.

If there was only one possible kind of actual infinity, you wouldn’t really need actual infinity, and could get by with the potential one.  But it turns out that the infinity of all natural numbers (as with the rooms in Hilbert’s hotel) is only one, the smallest one, of an infinite hierarchy of ever-growing infinities, and any step beyond the first one really requires you to treat the first one, and others, as complete objects in their own right, as actual infinities.

Craig and Smith’s arguments appear to be somewhat confused and unhelpful. They merely reiterate and illustrate unintuitive aspects of infinity well-known to mathematicians. Unuintuitiveness of infinity goes back to Zeno. There’s nothing more profound in their bewilderment about Hilbert’s hotel than there is in wondering how on earth would Achilles ever catch up with the tortoise.

Take the example with the library. If we have an infinite library with every natural number already printed on some book, there is indeed no free number to be assigned to a new book. But this is not “absurd, since entities that exist in reality can be numbered” (by itself a vague and unhelpful statement which ignores the possibility of real-world existence of larger infinities which, in fact, cannot be numbered). The totality of the existing books in the library plus the new book creates a new totality of books which can be numbered - it’s just that some existing books will have to have their numbers changed. One would assign to the new book some number, and shift all the existing ones from that number on to a number 1 higher than before, just as in Hilbert’s hotel (which works extremely well with the actual infinite as well as potential, pace Craig & Smith).

This illustrates a basic fact in cardinal arithmetic ("cardinal numbers” are a technical name mathematicians use for all finite numbers plus all possible sizes of actual infinities): aleph_0 + 1 = aleph_0. An infinity “swallows” infinities less or equal to it in size without swelling. Unintuitive, yes. But not contradictiory. There is no compelling argument here that actual infinity couldn’t exist in the real world.

By on 05/31/06 at 09:12 PM | Permanent link to this comment

Am I crazy or have Craig and Smith, on the one hand, and Ardsgaine, on the other, each taken a side of Kant’s First Antinomy? Their arguments about the eternity of the universe, anyway, strike me as very similar to the ones suggested by Kant.

Oh, and contra Carnot, the safety valve alone knows the worst truth about the engine.

By on 06/01/06 at 03:39 AM | Permanent link to this comment

I believe it was David Hilbert who said, “nobody shall expel us from the paradise Cantor has created for us.”

By Lynn on 06/01/06 at 10:43 AM | Permanent link to this comment

Anatoly: “it turns out that the infinity of all natural numbers (as with the rooms in Hilbert’s hotel) is only one, the smallest one, of an infinite hierarchy of ever-growing infinities, and any step beyond the first one really requires you to treat the first one, and others, as complete objects in their own right, as actual infinities.”

This is very useful; I think I understand Cantor a little more now.

“Craig and Smith’s arguments appear to be somewhat confused and unhelpful. They merely reiterate and illustrate unintuitive aspects of infinity well-known to mathematicians. Unuintuitiveness of infinity goes back to Zeno. There’s nothing more profound in their bewilderment about Hilbert’s hotel than there is in wondering how on earth would Achilles ever catch up with the tortoise.”

My understanding (is it wrong?) is that the missing element in Zeno’s understanding, the thing that unknots his paradox, is the realisation that converging infinite series can be summed: that ‘a half + a quarter + an eighth + sixteenth …’ actually does equal a finite sum (here ‘one’), even though the series itself goes on forever.  That’s the basic premise being calculus … isn’t it?

The point is that, yes, there are lots of infinites; but that only two manifest in the real, actual universe in which we live (converging and non-converging infinite series).  Cardinal numbers don’t exist in and of themselves, after all, however useful they are as abstractions.

I think I’m arguing that if we take the universe itself, in space and time, as an ‘actual infinity’ of entities, then it doesn’t leave us with any external entity that can then be added to the infinity to generate paradoxes (although I take the force of what you say about the limitations in Craig and Smith’s reasoning on this).  I wonder if the problem isn’t the way some people think of infinity as in some sense ‘really really big’.  Thinking in terms of ‘hotels’ and ‘libraries’ confuses the debate rather than elucidating it, since we can’t help but think of such things as definite in some sense.

I was also thinking of Borges’s ‘Book of Sand’, a regular-sized book between two covers that contains an infinite number of pages.  (Let it fall open on a random page; then close it again.  What are the odds that the book will fall open on the same page?  They are zero).  But I’m not sure that can work.  My thinking there, just to be confusing there for a moment, derives from Chris Priest’s Inverted World, a science fiction novel from the 1970s.  Rich is clearly correct that everything in the universe can be explained with reference to SF.

Priest’s inversion is that he thought it would be cool, rather that set a book on a finite world in an infinite universe (which is where we all currently live) to set a book in an infinite world that exists in a finite universe.  It’s a superb book, and I don’t want to give away any spoilers, but my understanding from reading picky critics is that such a world wouldn’t work in reality, because it would intersect itself at every point.

But, see, I get myself tied up here.  Because does this mean that time and space are actually separable, and not an inwoven spacetime after all?  Because, after all, it’s perfectly possible that we live in Infinite Universe A [finite spatially but existing for an infinite timspan), or Infinite Universe B [finite in duration, but infinitely large].  Isn’t it?  And now my head’s hurting.

By Adam Roberts on 06/01/06 at 12:06 PM | Permanent link to this comment

Not to be contrary, but I find myself questioning the viability of some of the proffered Contra titles.  My meagre imagination fails at the thought of writing contra ‘{rat,tar} con’; writing ‘Contra Rant Co’ would require me to discover a company devoted to ranting, something I’d be happy to write against if I chanced upon it.  As for asking me to write ‘Contra Coat, RN’, why, it would require me to pick on student nurses or heroic wearers of Royal Navy coats. And why would I want to do that?

Dell’s suggestions are more plausible; tho’ it would surely be breaking the butterfly on a wheel to Contraify Sidney Carton, and a post entitled ‘Contra Carnot’ would be as wrong as a post titled ‘Contra Bergerac’ (since his surname was indeed the full and splendid ‘Cyrano de Bergerac’, his first name being Savinien); and a post entitled ‘Contra Sadi Carnot’ has the wrong feel to it, somehow.

By Adam Roberts on 06/01/06 at 12:21 PM | Permanent link to this comment


“the thing that unknots [Zeno’s] paradox, is the realisation that converging infinite series can be summed...”

We know that both space and time converge in an infinite sum, in the case of Achilles and the tortoise. Achilles is 1 foot away from it, then 1/2, then 1/4, etc.; he spend 1 second chasing it, then 1/2, 1/4 etc. If we sum up, we get something finite. But does that really take away the unintuitiveness that the paradox displays? After all, we can still imagine each period of time (or space) as a separate entity, and they seem to go on forever, if not in real time, then in our mind’s time. It’s hard for us to accept that the sequence 1, 1/2, 1/4, 1/8 etc. will ever be “finished” in real terms, because in our mind’s eye we look and look and see no end to it - and that is the crux of Zeno’s paradox. The summing-up is a technical way of explaining it away which is true, but doesn’t make it natural and intuitive. “Converging infinite series” is not an intuitive notion for us, unlike that of a sequence that goes on and on and seems to never end.

In another version of the paradox, one has a lamp which is switched on for 1 sec., then off for 1/2 sec., then on again for 1/4 sec., then off for 1/8 sec. etc. Since the time series add up to a finite sum of 2 seconds, we can ask the question: which state will the lamp be on in exactly 2 seconds - but that question doesn’t have a meaningful answer. In technical terms, we have here an infinite sequence which doesn’t converge. In layman terms, we have here something weird that defies our intuition - and it has to do with infinity, once again.

The infinity in Zeno’s paradox and its variants is the infinite divisibility of time and space. If time is not infinitely divisible to ever smaller quanta, there is no Zeno’s paradox - at some point Achilles will make a leap that will not be subdivided further; at some point we will not be able to switch the lamp from on to off for such a short period of time that it doesn’t actually exist. Similarly for space (in case of Achilles and the tortoise). Are time and space infinitely divisible, or are they discrete at some level? No one knows, but the point I’m trying to make is that this kind of infinity leads to paradoxes or unintuitive situation just as well as the “big” infinity of never-ending hotel rooms or infitely wide universe, or infinite extent in time. What underlies paradoxes and failure of our intuition is the notion of infinity itself, not the fact that it extends in space or time indefinitely. Borges’s book, which captures infinity inside a bound volume of quite finite size, is another good example.

After Cantor, we know that it’s not enough that infinity by itself is contra-intuitive to us; for our best mathematical models of the universe to have any hope of being “actually really true” requires real existence of infinities larger than the infinity of natural numbers. This is because, for reasons we do not understand at all (nor do we understand that there must be “reasons"), all the mathematicall apparatus we have been able to come up with for modelling the real world involves calculus, in one form or another. And calculus always requires real numbers, that is, rational and irrational (not to mention complex, but let’s not go there). And Cantor prove that there’s infinitely more irrational numbers than rational. Consider how unreasonable the existence of, say, square root of 2, the canonical irrational number, is. If you look at all rational numbers (i.e. fractions) between 0 and 1, there already is an infinite number of them. Moreover, between any two of them there already is an infinite number of others - they’re already like Borges’s book, with infinite number of pages between any two pages. There seems to be no place for any “holes” in such a tight, infinite arrangement. Yet the square root of 2 sits there, squarely, occupying a “hole” between rational numbers that intution says should not be there at all. And not only that; after Cantor, we know that the ‘holes’ between fractions are infinitely more in quantity than fractions themselves. And you need all that huge extravagance of higher infinities in order to have calculus, in order for the notion of derivative, integral, Newton mechanics, electromagnetic fields, relativity theory, for all that to make sense. They can only be exactly true, and not mere approximations, if the real world is really like that. That - that is truly weird, I think.

Sorry, I got carried away, it seems; but I hope the above explains why I find Craig & Smith’s arguments unpersuasive. It is a very interesting question, no doubt about it, whether the universe is infinite in size, or whether it has been infinite in time. But there’s nothing especially weird in thinking about such infinities that there isn’t already in thinking about infinities that has been done for millennia. Our minds struggle to understand how anything can be infinite, and arguably we don’t really understand that (but instead substitute symbols, equations, and meta-understanding that comes with them). That’s how we are. Is the universe really finite, either in width, or time, or in terms of quanta of time/space that cannot be subdivided further? I don’t know, no one does, and I think arguments about how this or that option defies our intuition will not add to our understanding of these questions.

By Anatoly on 06/02/06 at 10:02 AM | Permanent link to this comment

I just discovered this website today at one o’clock in the morning.  It almost made me cry with joy to see the things that I have debated in my own head for years written in such a coherent manner.  Kudos to all who contribute here.  Although many people will never understand a word out of your mouth, those who do appreciate it.

Mike Bower (18)
Greenville, WI

By on 06/03/06 at 02:09 AM | Permanent link to this comment

Hilbert’s hotel has just been completed with an infinite number of rooms. A guest checkes in. How many rooms are still available? Infinite minus one is still infinite. So, even when guests check in in droves, there will always be an infinite number of rooms that can be filled.
Thus, in my opinion, ‘a hotel with an infinite number of rooms, all occupied’ is not possible, and no conclusions can be drawn from this statement.
(Certainly not that a God exists, who wants us to sing for him on sundays or things like that)

By on 08/02/06 at 02:08 PM | Permanent link to this comment

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