Tuesday, September 08, 2009

Something That Keeps Me Up At Night

Since a commenter recently noted that Steve's been writing almost all of the Triablogue posts of late, I figure I can post this one on the T-Blog even though I'm not quite sure there's any practical apologetic use for it. On the other hand, it's stuff that I find "wicked kewl" and therefore is interesting to me. But it'll have a bit of math in it, so if you don't like that, well I'm sure Steve will write something new shortly :-)

One of the questions that cosmologists have pondered is whether the universe is open or closed. An open universe would extend infinitely in all directions, whereas a closed universe would have a "boundary." However, even a closed universe could still be infinite. If space was curved in such a manner that, just like you could always travel East on Earth and return to the point you started from, in the universe you could always pick a direction and travel long enough and you'd return to your starting point. In other words, you could travel infinitely in one direction yet always return to your starting point (this would assume space was curved in the fourth, or higher, dimension that we cannot physically see).

I have to admit that I have a strange attraction to these kinds of loops. I don't know why, but they appear "pleasing" to me. And therefore I find it no surprise that I've discovered one such loop within numbers themselves. In other words, just as we could say that the universe is infinite yet closed because it loops back (assuming that theory is correct, I must add—by far this is not proven!), I say that numbers themselves are infinite and yet closed because they loop back on themselves too.

For a simple proof (simple in that it requires nothing more than algebra), consider the following.

1. The number 1 (one) is that number which has no factors other than 1.

This can be restated as:

1'. If a number n has only 1 as a factor, then n = 1.

This seems fairly straightforward to me, yet by the end of this you'll see why it might be tempting to deny the above.

Now we need to give one other tidbit of information. I'll explain it below (and note that because we are dealing with factors, by definition we're only considering positive values and whole numbers, so all the numbers below are positive integers):

2. Let c be a factor of w.

3. Since c is a factor of w, the next integer greater than w that c can likewise be a factor of is w + c.

Since many people don't like thinking with letters instead of numbers, let me give a concrete example. Let's say that c = 7 and w = 21. 7 is a factor of 21, so (2) above is satisfied. (3) states that if 7 is a factor of 21, the next number greater than 21 that 7 could be a factor of would be 21 + 7, or 28. And this is obvious because 22, 23, 24, 25, 26, and 27 cannot have 7 as a factor. Indeed, (3) is really nothing more than restating the definition of a factor.

Now let my proof begin in earnest:

4. Let x be the product of all positive integers. That is x = 1 x 2 x 3 x 4 x … x infinity.

5. Since x is the product of all positive integers, then x has all positive integers as factors.

6. Let a be a factor of x.

7. The next number greater than a than will be a factor of x is x + a.

8. Consider x + 1.

9. Let a = 1.

10. a is a factor of x + 1 (per (7)).

11. Therefore, 1 is a factor of x + 1.

12. Let a be greater than 1.

13. a cannot be a factor of x + 1 because the next greatest number than x that a could be a factor of is x + a (per (7)), and a > 1 (per (12)).

14. Therefore, 1 is the only factor of x + 1 (per (11)).

15. Therefore, x + 1 = 1 (per (1')).

16. If x + 1 = 1, then x = 0 (algebra).

17. But(!) x = 1 x 2 x 3 x 4 x … x infinity (per (4)).

18. Therefore, 1 x 2 x 3 x 4 x … x infinity = 0.

Now the way that I see it, there are one of two options that mathematicians can take here. Either we can simply rule that when x is 1 x 2 x 3 x 4 x … x infinity, then x + 1 is undefined (similar to the way that division by zero is undefined), or we can say that numbers themselves contain some sort of looping mechanism, wherein by the time you reach infinity (the infinity defined as the product of all positive integers), you "loop back" to zero.

You already know which way I'll go because I like loops. :-) But there is more evidence. I think we can see the "loop back" when looking at a tangent graph. Since I don't want to throw in Greek symbols here, assume that a is an angle: tan(a) = sin(a)/cos(a). So, whenever cos(a) = 0, tan(a) is undefined because of division by zero.

The tangent graph looks like this:

That's with the classic orientation, where the origin (where the arms of the graph cross) is located at (0,0). You can tell that since the right-hand portion of the graph is running up toward infinity and the left-hand portion is running down toward negative infinity why there would be a sudden "jump" in the graph at pi/2 (since cos(pi/2) = 0). If the x value is just slightly less than pi/2, you have positive infinity, but if it's just slightly more than pi/2 you have negative infinity.

Instead of assuming these things just go off to infinity, what happens if we assume that they "connect" at infinity and redraw the graph from that perspective? If I did it correctly (and since it's late at night right now, I am subject to correction), you'd get something that looks like this:

For this graph, we're looking at how it relates to infinity. Basically, what I did was assume that the graph "rolls over" at infinity, and made the horizontal axis the point where positive infinity and negative infinity intersect. In essence, you move the lower left to the upper right on the tangent graph and vice-versa. Naturally, the graph is horrifically distorted since it's representing two infinities on the vertical axis—the lines would actually appear to be virtually synonymous with the vertical axis for most of the trip, with the hook out at the very end; but I think this is sufficient to at least give a faint picture. (Note: technically, the origin on this graph would still be undefined, since the origin in this view is the point where the division by zero takes place.)

In any case, note that this graph would continue in sequence, just like the tangent graph does. That means that you could print out a row of these figures. The interesting thing about them is that you can then take the top of the graph and "fold" it down so that the 0s appear on the same line (the graph would now be on a donut-shaped paper rather than a 2D screen). At this point, the line graphed would look continuous (bearing in mind that at the origin of each cross point (multiples of pi/2) the graph would still be undefined).

This would imply that the graph, represented flat on a 2D surface, takes on the characteristics of a bent 3D object. Though only two dimensions are present in the tangent graph, there is an assumed third dimension where the graph "rolls over" from positive to negative infinity. In this curved 3D representation, the graph no longer has an infinite jump from positive to negative infinity, but rather that jump is a mere point, more akin to switching from positive to negative numbers at 0.

In short, it would be a curved space of infinite length, curved in a higher dimension.

This might actually affect physics. If it is true that math on the number line itself assumes a higher dimension of curved "space" then one could question whether that means reality really is curved, or whether it means that our math will always make it appear to be curved regardless of what it really is. In other words, is the fact that the math involved in physics seems to indicate a curved universe the result of the way that the universe actually is, or is it because the only method by which we have of probing the universe on such levels is mathematically, and math itself is curved? To use an analogy, suppose you use a level and see that a board appears warped; is the board warped or is the level warped? If we define the level as being level, then the board is warped; but what if we begin to see evidence that the level itself shows a curve?

1. Hold on.

"4. Let x be the product of all positive integers. That is x = 1 x 2 x 3 x 4 x … x infinity."

When you go on from there, and try factoring x, doesn't that assume that x is an integer? Or at least that this series* has a finite limit? Which is obviously doesn't, so... Doesn't this go wrong at the beginning?

* Sorry, I've forgotten my math terms. A series uses addition, not multiplication. I'm rusty here.

2. This post is meant as satire, right? Especially the "earnest proof".

3. I believe you are correct that a series would use addition. I'm quite sure most mathematicians will say the problem is the x + 1 when x is infinity (because what does it mean to add 1 to infinity?).

And yes it is true that x is infinitely long.

But for the record, I have a graphical way to represent this too that shows the same thing. In other words, if I showed you the section of the graph that was equal to "1", you would not be able to tell if the number before 1 was 0 or infinity because they would graph identically. (In reality, I came up with the graph before I came up with any of the math I showed above.)

In any case, it might be helpful to think of the infinity I set up kind of like the way Cantor set up his diagonal proof (although that dealt with specifically with sets, namely how even an infinite list of sets will be missing an infinite number of other sets, such that there are uncountable sets, etc.). In short, Cantor showed that an infinite list of infinitely long sets would not contain an infinite number of other sets by looking at only a portion of the begining of each set, which is similar to what I did in looking at x as the product of all integers. This means that while we are considering x as an infinite, we are going through it in a step-wise fashion. As an infinite, we'd not be able to conceptualize what's going on, but when we realize how that infinity is formed then we can draw conclusions from it.

In short, if x has 2 as a factor (and it does, because by definition x is created by a product of numbers that includes the number 2), then x + 1 cannot have 2 as a factor, etc.

As for A Helmet--you obviously misunderstood what I meant by "in earnest." That is, the first part of the proof was definitions, and the part where I said it began "in earnest" only meant to set it apart from the definitional "header" of the proof.

And no, it's not satire. At the very least, I think this demonstrates a paradox in math, where you can get 0 = infinity (and mathematical paradoxes are quite common). Usually such a paradox only occurs with hidden division by zero, but this uses only multiplication of positive numbers to accomplish the same result.

4. BTW, an easy example of hidden division by zero yielding irrational results:

1. a = b

2. 2(a - b) = 3(a - b).

3. Factor out (a - b) from both equations.

4. 2 = 3

Of course, the problem is (a - b) = 0, so factoring it out of the equation is dividing by zero.

Oh, and one other thing I could add regarding my earlier proof. Mathematicians could always say that "If a number n has only 1 as a factor, then n = 1" is itself wrong.

Finally, Jugulum, since I mentioned the graphs earlier, I will post them eventually so you can see what I'm talking about too.

5. Peter Pike,

I understood what you meant by earnest, but anyway the notion that the natural numbers loop back is in contradiction to the very definition of the natural numbers, which is basically given by the Paeno axioms. These are as follows:

i) 0 is a natural number

ii) every natural number has a successor

iii) every natural number is successor of at most one natural number

iv) 0 is not successor of a natural number

v) among all sets that satisfy the two conditions:

a) 0 is an element

b)if the number n is contained then its successor n' is contained also

the set of natural numbers is smallest.

----

By these 5 axioms the positive integers (including 0) are defined. Now, with regard to a loop axiom iv is crucial. Zero is not the successor of any natural number, however according to your theory, the number x-1 would be the preceding number of 0, hence 0 be the successor of x-1. This would discard the very foundation of the integers.

Well, the Paeno axioms are not overly used anymore to establish the natural numbers' fundament, rather today Zermelo-Fraenkel set theory provides axioms used to define the numbers. But the ZF axioms to define the natural numbers do not allow for a loop either. However, what is even more remarkable is, that by ZF set theory it was possible not only to describe the concept of natural numbers but also to prove their existence. And the natural numbers thusly defined don't allow for 0 being the successor of some other natural number any more than Paeno axioms permit such a loop. There exists no natural number that loops back to the beginning.

What does this mean? All mathematical proofs are traceable to the ZF-theory. All. Not just sentences about integers. And to discard crystal clear implications of this axiomatic fundament renders all known proofs null and void. In order to make the integers loop, you'd need to redefine axioms to begin with.

But what is especially objectionable is your step (4) that has been criticized by Jugulum. You cannot describe a natural number in such a way. Numbers given by expressions like "...up till infinity" as you are doing by the expresson 1x2x3x4x5.....xinfinity in order to describe x, are wrong. Rather the numbers are defined inductively, that is, by defining how a successor emerges out of a current instance. (An example of an inductive definition is Paeno axiom v above). The ZF set theory uses the inductive infinity axiom as the foundation for the natural numbers. And this doesn't allow for a definition of x as you propose in step 4!

But there are arithmetic reasons why you cannot apply steps 4 and 5 that way.

If you build the faculty of a finite number, lets say

4! = 1x2x3x4 = 24

Then let this result of 24 be x, so x=24

Lets apply step 6 and seek a factor a of x=24 for instance, a=8

Now 8 is greater than the largest number in the product, which is 4.

24/8 = 3

However, we could as well set a=3 and divide 24 by 3:

24/3 = 8

Now, x=24 and the largest number in the product was 4. Consigned to your example, where infinity would be the largest number in the product (here 4) and the product x would be infinite too (here 24), we would in any case receive a number between these two "degrees of infinity" (here 8).

Thus, if a is a factor of x, then there is another factor b such that a times b = x

Either a or b must be infinite. In other words, you could calculate infinity/infinity to receive a or b, though this division is basically undefined (like 0/0). I don't see what this proof is getting at. The underlying error is due to your usage of infinity. You cannot treat it just like finite numbers. And proper number definitions get by without the blatant use of infinity in such a way.

-a helmet

6. A Helmet,

One quick thing I have to point out is that this statement seems to fly in the face of Gödel:

---
All mathematical proofs are traceable to the ZF-theory. All.
---

But of course we know that no mathematical system can be both complete and consistent, so if ZF-theory is complete (as you indicate with your above statement) then it cannot be consistent. In which case you should expect there to be things that "don't fit."

Ultimately, I'm not concerned with whether this specific example "violates" Paeno or ZF because, again, Gödel's inconsistency theorum requires exceptions to either completeness or consistency. Either Paeno/ZF is complete and inconsistent or it is consistent but incomplete.

I will write more later when I can produce some graphs for both you and Jugulum. (Note: The graphs assume one of Euclid's Common Notions: "C.N.4. Things which coincide with one another equal one another.")

7. I'm not sure that a graph can address the hidden (possibly invalid) assumption that you're making: That x is a number.

In other words, I suspect the flaw is that a multiplicative series which diverges cannot be said to "equal" a number.

But we'll see what graphs you mean.

8. I meant all proofs that have been done so far. They're all traceable to the fundaments of ZF set theory. A proof that the postive integers loop back cannot be wrought on this overall accepted basis of mathematics but would be in contradiction to them.

-a helmet

9. When I get home, the graphs will be fairly easy to make.

Jugulum, I wonder what your take on this would be then:

Suppose you have the following set, N, which is supposed to contain all the sets of possible single-digit positive numbers (obviously, duplicates are allowed):

N(1) = 4,3,5,6,2,5,7...
N(2) = 6,9,1,0,5,3,4...
N(3) = 0,3,0,8,4,4,9...
N(4) = 4,7,9,0,1,2,2...
N(5) = 1,3,1,6,7,2,1...
N(6) = 5,5,9,2,2,9,7...
N(7) = 2,6,4,3,9,2,0...
.
.
.

The diagonal (starting at row 1, column 1; then row 2, column 2; etc.) is: D = "4,9,0,0,7,9,0"

It is possible to construct a row, N(R), such that each value of N(R) is the value of the diagonal incremented by 1, using modal math so that 9 + 1 = 0. Hence:

N(R) = "5,0,1,1,8,0,1"

(Note: I'm using != to mean "not equal" since I can't type the equal sign with a slash.)

The first value of N(R) != the first value of N(1); therefore N(R) != N(1).

The second value of N(R) != the second value of N(2); therefore N(R) != N(2).

Etc.

This is a version of the Cantor diagonal. Note that it deals with infinitely long sets. N is itself infinitely long, and each row contains an infinite number of elements. Yet it does not contain the row N(R), which is built off the diagonal.

Further note that N(R) is a derrived infinite. It's value depends on what amount you care to increment the original value by (you could just as easily add 2 or 7; or for that matter you could alternate where the first increments by 1, the second subtracts 3, etc. with the only rule being that you could never add 0 or 10 because that would not change the value of the diagonal number, thus making it possible for there to be a match). But the value of N(R) ALSO depends on the order in which you write the entries into N, etc.

Would you agree that this is a valid way to probe infinites, even though N, being an actual infinite, can never be fully written out, nor can N(R) ever be fully known, etc.?

The reason I ask is because it's similar reasoning to that that I have employed above.

10. Yes, you can probe infinite sets by comparing them element-by-element, systematically.

I'm not sure that's all you're doing.

The issue is where you treat x as an integer--introduced in 4, and then used in 7.

Points 1-3 assume that c and w are integers with definite values, which seems to be an invalid assumption about x. Series (additional or multiplicative) don't have values. You don't evaluate them. You take their limit, which either converges or diverges. And if it diverges, I question whether you can apply points 1-3 to it at all--because the limit won't be a number.

P.S. I'm also wondering about how you used the phrase "value of N(R)". Did you just mean, "The list of elements of the set"? If so, fine--never mind. But if you meant the sum or product of its elements, then that might be a problem.
P.P.S. I'm speaking with a math major in my past, but a very rusty one.

11. And I just realized that's close to what you said here, sort of:

"Either we can simply rule that when x is 1 x 2 x 3 x 4 x … x infinity, then x + 1 is undefined (similar to the way that division by zero is undefined), "

I think my comment might be pointing to why that's undefined. Maybe.

---
I'm also wondering about how you used the phrase "value of N(R)". Did you just mean, "The list of elements of the set"?
---

Yes :-) I was writing fast and...wait a second, this is America. It was Bush's fault!

13. Jugulum said:
---
Points 1-3 assume that c and w are integers with definite values, which seems to be an invalid assumption about x.
---

I agree that it appears x does not have a specific value since it is infinite, but where I quibble is that I'm saying we can know that x has a factor of 2, 3, 4, 5, etc. because that was how we defined x.

So my point would be this. Assume x is not infinite. Or better yet, let's use the letter k so we're looking at something different.

Assume you are given some value k and you are told that 2 is a factor of k. Without knowing what k's actual value is, you can say that k + 1 does not have 2 as a factor. Similarly, if we say that k has 3 as a factor, we know that k + 1 cannot have 3 as a factor. If we then proceed such that we say all values of n, from n = 2 to n = "some extremely large value that is yet finite", are factors of k, then you would agree that k + 1 does not contain any of the values of n as factors, right?

My argument would merely push n into infinity, and I could ask at what point does the finite portion (which I'm certain you would agree with) suddenly change so it's no longer applicable when it becomes infinite?

And, of course, I also offered a different track for my thoughts with the tangent graph, since we know that if you're just slightly to the left of X = pi/2 on the tangent graph you're at positive infinity, but if you're just slightly to the right of X = pi/2 you're at negative infinity--which is sorta like dropping 2 infinities instantaneously at that single point.

14. Hey Paul,

This concept of x=1x2x3x...xinfinity, I think "undefined" is a good way to describe it, "does not exist" is probably better. Proof by contradiction should suffice:

Suppose x=the product of all positive integers exists. By definition, all positive integers are factors of x.

But what about x+1? x+1 would also exist, but since x+1 > x, x+1 cannot be a factor of x (exercise to the reader: if y > x, y is not a factor of x)

Contradiction, thus x does not exist.

The real problem is that "infinity" is not "a number" that can be handled as other concrete numbers.

You might be very interested to study orders of infinities, however. For instance, the "number" of positive integers is the same as the "number" of positive rational numbers (fractions of positive integers).

You'd think that since there are fractions that fit between the integers, there should be more of them. But you can define a 1-1 mapping, so that every positive integer is associated with one and only one fraction, and vice versa, so the sets are said to have the same cardinality, and it is called "aleph-null" (printed hebrew letter, subscript zero) or "countably infinite". aleph-null is the "smallest infinity."

If you never read them before, you might be interested in this Puzzle on my blog (really a series of posts) from a few years ago. All told, it constitutes a non-constructive existence proof (and the existence of a non-constructive proof of existence is a refutation of any atheist claim "in order for me to believe in God, you must show him to me so I can inspect him")

Another thing you might be interested in; does there exist a proof of the irrationality of sqrt(2) which does not somehow involve proof by contradiction? I've never found one, and I doubt there could be one (since the definition of "x is irrational" is "there do not exist integers q,r such that q/r=x", and every proof I have ever seen starts with "suppose integers q/r=sqrt(2)"

I can't recall the name, I'm sure you know, there's a group of philosophers that don't believe in the law of the excluded middle, i.e. not(not(A)) doesn't necessarily imply A. But without this law, proof by contradiction doesn't work, thus it seems to me that these jokers can't even prove that sqrt(2) is irrational.

15. Oh, I meant to say, you either were recently exposed to the standard proof of the infiniteness of the set of primes, or you almost rediscovered it yourself.

Suppose there are a finite number n of primes P1=1, P2=2, P3=3, P4=5, ...PN=?

Let x = P1xP2xP3x...xPN (Since this is a finite product, this is well defined), and consdider the number x+1. As you describe above, 2 doesn't divide x+1 (but it does divide x+2), 3 doesn't divide x+1 (but it does divide x+3), ... PN doesn't divide x+1 (but it does divide x+PN). Since none of the known primes divide x+1 (except for P1=1), this new number x+1 must be prime!

But we previously assumed PN < x+1 was the largest prime. Contradiction, thus there does not exist finite n which is the number of primes. QED.

16. BTW, I ran out of time to get the graph done yesterday. I'll try to finish it up by the end of the week, but I'm not sure if this Triablogue would deserve being subjected to another post :-D If anyone reading this wants to, you can always feel free to e-mail me directly using "yahoo.com" as the end of the address and "petedawg34" at the front, with of course an "@" between them. (Isn't it fun trying to avoid spambots?) I'll definitely post it on my personal blog when it's finished and will try to provide a link to it here too.

17. ...I'm not really into Pokemon.