Is the universe a tautology? (Explanation inside)

I have read about the premise of The Library Of Babel in a book a few years ago. Then recently I got an e-book of Jorge Luis Borges, in which I found the original short story, and it got me thinking.

(If you are not familiar with the thought experiment/concept, read this!)

So it is established, that the library contains all the books that can be written in all languages. The short story has the statement that "language is a tautology", since even for the seemingly incomprehensible books you can devise a language on which it makes sense. So anything could make sense, which means that language is a tautology.

With me so far? Good [smile]

My thoughts begane to race as I read this, and I have 2 questions to discuss about this concept.

If all the books can be found in the library, is there a book that describes the process of how to build a computer from scratch? The answer is of course yes. So my first question to discuss is:

• 1. Is there a book in the library which describes the working method of how to levitate and another which describes the same method of how to teleport? Or fly? Or gain super strength? Or whatever, the point is that it has to work in Real Life.

Prologue for the second question: It is established in the short story that every book has an opposite book, a book that falsifies all the statements in the original book. So this means that if you read the book about levitation, there will be a book which "debunks" it. Which means that the How to build a computer? book has one too. So my second question to discuss is:

• 2. Is the universe a tautology? If everything is possible and everything is not possible at the same time, what does this mean? Does anything exist at all?

If we stay in the frame of the concept of the library, I think this question makes perfect sense. Of course, in Real Life it might not... Or does it?

"We have done the impossible and that makes us mighty." - Malcolm Reynolds

In the same way that a book could make sense if you assume a language for it, a method of learning to fly could exist if you assume a certain set of natural laws. Thus, while you couldn't find a book that could teach you to fly, you might find a book that contained a completely self-consistent description of how to learn to fly, with its only flaw being that it's utterly wrong.

That's Feo . . . He's a disgusting, mysoginistic, paedophilic asshat who moonlights as a shitty writer—Something Awful

And the same goes for the computer-falsifying book: it'll tell you exactly why a computer described in the computer instruction book won't work or indeed even be possible to build, yet as soon as you build it and turn it on, you'll know that the other book was BS.

If every imaginable book is in there, you'd think that it means that it includes things that explain how impossible things are possible (and vice versa), but if there is no way that the claims of the book could be correct, then they simply aren't correct, no matter what.

Unless you assume magic or a very fluctuating kind of universe in which one's individual perception of reality (which the books one reads would alter) defines the actual reality of the universe in relation to that person - in other words, a universe that is in fact a dream, a delusion or a self-contained one that has the person experiencing it as its God.

In which case I assure you that there are much better books to find that something as simple as "here's how you fly". Look for "here's how to do, become or experience anything you can or can't imagine at will" and fuck 18-dimensional 900-breasted space babes with your 38 dicks. Assuming that that's what floats your boat (which is a natural assumption, considering that you're looking for a book like that, pervert!)

edited 25th May '11 2:08:00 AM by BestOf

Quod gratis asseritur, gratis negatur.

The universe is Shaped Like Itself..

A guy called dvorak is tired. Tired of humanity not wanting to change to improve itself. Quite the sad tale.

The way out would involve L-space, I assume.

Quod gratis asseritur, gratis negatur.

Assuming that such a thing is possible in Real Life, and that the method to do so can be described through a book of sufficient length, then yes.

Do you mean a scenario like the Ultimate Ensemble hypothesis, in which every possible state of things is true somewhere? That's an amusing idea, but the problem of course is that there seems to be no way to verify if this is the case - at least for now.

edited 25th May '11 2:37:32 AM by Carciofus

But they seem to know where they are going, the ones who walk away from Omelas.

OK, so you guys argue that language is an arbitrary construct, while the laws of nature are what they are and limit certaing things.

From this however, I conclude that The Library Of Babel doesn't contain all books, since it doesn't contain the books which describe the working methods of how to break the laws of nature.

Am I wrong?

edited 25th May '11 3:03:21 AM by Sati1984

"We have done the impossible and that makes us mighty." - Malcolm Reynolds

Well, if such books do not exist, this is a given.

It also does not contain a book whose number of words is the greatest prime number, because no greatest prime number exists.

But they seem to know where they are going, the ones who walk away from Omelas.

Is it possible to imagine impossible things? If so, it should be possible to write them down...

The library could contain such number, since there are infinite books, so the number of books which contain the number can also be infinite.

"We have done the impossible and that makes us mighty." - Malcolm Reynolds

To rephrase the part about the greatest prime number: the library could contain the greatest prime number in the sense that it could contain all the prime numbers. And all the prime numbers by definition contain the greatest too.

If you don't limit the books to 410 pages (as in the original short story), it's even possible to have a book with the word count of the greatest prime number, since that should be infinite too.

I'm not good at math, but this is how I interpret it

"We have done the impossible and that makes us mighty." - Malcolm Reynolds

Depends on what you mean with "imagine". You can certainly write such a description as "the greatest prime number", but you cannot write this number, because it just does not exist.*

No, because no such thing as a greatest prime number exists. Give me a prime number, any prime number, and I can calculate another prime number which is bigger than it.*

^{Well, if you give me enough computing power and time, of course...}

It's the same thing as saying that the Library will not contain a number which is even and odd. The fact that the library is infinite, or that it contains all combinations of characters, is irrelevant here: the Library will not contain this number, because all odd numbers are not even and all even numbers are not odd.

edited 25th May '11 3:35:24 AM by Carciofus

But they seem to know where they are going, the ones who walk away from Omelas.

The short story only involves a finite number of books, since the set of strings that would fit into a 410-page book is finite. Are we just assuming the Library contains every book of every finite length?

edited 25th May '11 4:25:35 AM by Yej

Da Rules excuse all the inaccuracy in the world. Listen to them, not me.

I was assuming that this was the case - although, by Cantor's Theorem, this would still imply that there exist real numbers which are not defined in any book of the Library.

Hm. a transfinite version of the Library of Babel would be pretty amusing...

edited 25th May '11 4:29:04 AM by Carciofus

But they seem to know where they are going, the ones who walk away from Omelas.

But how would you have |R| books?*

I'm fairly sure that even the set of all possible strings of all possible alphabets is still only |Z|.

edited 25th May '11 4:33:08 AM by Yej

Da Rules excuse all the inaccuracy in the world. Listen to them, not me.

Well, you need books of infinite length.

First you start by considering books with an infinite (but countable) pages: perhaps, these can be imagined as displays of some sort that let you enter a number n and return you the contents of the n-th page.

Then you put all these books in your library - the practical details are left as an exercise. These are already enough to have |R| books, but why stop here?

If you want to go even higher in the transfinite hierarchy, you treat each one of these books as if they were page numbers instead of books: that is, you develop another kind of "book" which is a display that lets you enter one of the first-level "books" and return the corresponding page.

In a certain sense that can be made precise, there are more "second level books" of this sort than there are "first level books".

Then you repeat the procedure again, and build "third level books", "fourth level books", and so on.

You do this for for all integer levels. Then what can you do? Why, of course, you can build "omega-th level books" that take in input any book of a previous level and return a page. Then you build "omega-plus-one-th" level books, "omega-plus-two-th" ones, and so on, up to infinity. And then you do it again, and again, and again, and do this too infinitely many many times, and then you do this infinitely many times, and so on.

Again, the practical details are left to the reader - what am I, an engineer?

Transfinite arithmetics is freaky.

edited 25th May '11 4:53:07 AM by Carciofus

But they seem to know where they are going, the ones who walk away from Omelas.

But an entire book can be reduced to a single integer, and so your machine that accepts an entire book as an index is equivalent to the one that accepts a single integer.

Da Rules excuse all the inaccuracy in the world. Listen to them, not me.

A function from integers to letters (like an infinite book of the sort I was describing at the first step) is not reducible to an integer. You can see it by checking that all infinite sequences of zeroes and ones (and, hence, all the real numbers between 0 and 1 written in base 2) are expressible as books of this kind.

It is reducible to a real number, though. So the second-level books are basically books with a real number of pages. And there are more books of this sort than there are real numbers, by Cantor's Theorem.

edited 25th May '11 4:50:22 AM by Carciofus

But they seem to know where they are going, the ones who walk away from Omelas.

Oh right, I thought we were starting with a book of arbitrary but finite length. However, all the infinite sequences of 1s and 0s won't cover all real numbers, since that sequence is... well, a sequence and enumerable, and hence can't be larger than |Z|. I think.

Transfinitism is freaky.

edited 25th May '11 4:54:44 AM by Yej

Da Rules excuse all the inaccuracy in the world. Listen to them, not me.

You are right, if you start of books of bounded length it does not work.

Well, it could still do, but you need infinite steps to get started and reach the first true "infinite".

Honestly, I am starting to wonder if Ultrafinitism (basically, the philosophy of mathematics that says that all of this is nonsense and that there exists a "biggest natural number" above which it does not make sense to talk about numbers) might not have a point. Still, thinking about this sort of thing is amusing

edited 25th May '11 5:02:24 AM by Carciofus

But they seem to know where they are going, the ones who walk away from Omelas.

@Carciofus:

But if the library contains an infinite number of prime numbers, your argument becomes sort of mute, since if you give me the next number, I can always show you that yes, the library does have that too. And the next one too. And the next one too...

"We have done the impossible and that makes us mighty." - Malcolm Reynolds

Yes, it contains all prime numbers. But it does not contain the greatest one, because it does not exist. You cannot show me a book of the library and tell me "see, this book here contains the greatest prime number", because then I can reach for another book of the library and tell you "no, it cannot be the greatest one - see, this one is also a prime number and is bigger".

edited 25th May '11 5:05:52 AM by Carciofus

But they seem to know where they are going, the ones who walk away from Omelas.

Actually, it doesn't work even with books of unbounded length.

• Any finite alphabet A is going to be isomorphic to a subset of the natural numbers, specifically {0,1,2,..|A|}.
• Start with a "blank" book composed entirely of 0s, and imagine this book written right-to-left.
• Interpret the text of the book as a natural number in base |A|.
• The text of the next book is defined as adding one to this number.
• This procedure will, eventually, produce every possible book.
• Thus, the books are enumerable.
• Thus, the set of all books is isomorphic to |Z|.

edited 25th May '11 5:07:18 AM by Yej

Da Rules excuse all the inaccuracy in the world. Listen to them, not me.

Your argument meets some trouble right here. What is the code that your mapping will give to the book

111111111... ?

More precisely, let us imagine that your alphabet contains only two letters A and B, just to make things simple (I could use 0 and 1, but I don't want to make confusion between numbers and letters — just map A to 0 and B to 1 if you prefer).

Then what you are attempting to do is basically just listing the books as follows:

Book 0: AAAAAAA....
Book 1: BAAAAAA....
Book 2: ABAAAAA....
Book 3: BBAAAAA...
Book 4: AABAAAA...

and so on. All the books of your list end with a trailing sequence of AAAAA, and therefore the book BBBBBB... will not be in your list.

More in general, for any listing as the above one I can pull a Cantor and write another book which is different from Book 0 in the first character, from Book 1 in the second one, and so on. For the one you gave me, it would be the book BBBBB...., but ABABABABAB.... would also work as a counterexample for that case. The important thing here is that for any listing that you could give me, Cantor's method gives me a counterexample.

edited 25th May '11 5:25:08 AM by Carciofus

But they seem to know where they are going, the ones who walk away from Omelas.

Oh right, yes. (Though what happens if I start with BBB... and CCC... and combine all the results? ) Now what are you going to do with your real-numbered books?

edited 25th May '11 5:26:44 AM by Yej

Da Rules excuse all the inaccuracy in the world. Listen to them, not me.

You won't get BCBCBCBC..., perhaps? It really depends on how you combine them, though.

Ideally? Pick the one that contains the answer for the Halting Problem for all Turing Machines and all inputs (hey, it's going to be there somewhere) and proceed to answer all mathematical problems ever.

edited 25th May '11 5:35:36 AM by Carciofus

But they seem to know where they are going, the ones who walk away from Omelas.