Archive for the 'Cantor Crankery' category

Infinite Cantor Crankery

Jul 29 2013 Published by under Bad Math, Cantor Crankery

I recently got yet another email from a Cantor crank.

Sadly, it's not a particularly interesting letter. It contains an argument that I've seen more times than I can count. But I realized that I don't think I've ever written about this particular boneheaded nonsense!

I'm going to paraphrase the argument: the original is written in broken english and is hard to follow.

  • Cantor's diagonalization creates a magical number ("Cantor's number") based on an infinitely long table.
  • Each digit of Cantor's number is taken from one row of the table: the Nth digit is produced by the Nth row of the table.
  • This means that the Nth digit only exists after processing N rows of the table.
  • Suppose it takes time t to get the value of a digit from a row of the table.
  • Therefore, for any natural number N, it takes N*t time to get the first N digits of Cantor's number.
  • Any finite prefix of Cantor's number is a rational number, which is clearly in the table.
  • The full Cantor's number doesn't exist until an infinite number of steps has been completed, at time &infinity;*t.
  • Therefore Cantor's number never exists. Only finite prefixes of it exist, and they are all rational numbers.

The problem with this is quite simple: Cantor's proof doesn't create a number; it identifies a number.

It might take an infinite amount of time to figure out which number we're talking about - but that doesn't matter. The number, like all numbers, exists, independent of
our ability to compute it. Once you accept the rules of real numbers as a mathematical framework, then all of the numbers, every possible one, whether we can identify it, or describe it, or write it down - they all exist. What a mechanism like Cantor's diagonalization does is just give us a way of identifying a particular number that we're interested in. But that number exists, whether we describe it or identify it.

The easiest way to show the problem here is to think of other irrational numbers. No irrational number can ever be written down completely. We know that there's got to be some number which, multiplied by itself, equals 2. But we can't actually write down all of the digits of that number. We can write down progressively better approximations, but we'll never actually write the square root of two. By the argument above against Cantor's number, we can show that the square root of two doesn't exist. If we need to create the number by writing down all af its digits,s then the square root of two will never get created! Nor will any other irrational number. If you insist on writing numbers down in decimal form, then neither will many fractions. But in math, we don't create numbers: we describe numbers that already exist.

But we could weasel around that, and create an alternative formulation of mathematics in which all numbers must be writeable in some finite form. We wouldn't need to say that we can create numbers, but we could constrain our definitions to get rid of the nasty numbers that make things confusing. We could make a reasonable argument that those problematic real numbers don't really exist - that they're an artifact of a flaw in our logical definition of real numbers. (In fact, some mathematicians like Greg Chaitin have actually made that argument semi-seriously.)

By doing that, irrational numbers could be defined out of existence, because they
can't be written down. In essence, that's what my correspondant is proposing: that the definition of real numbers is broken, and that the problem with Cantor's proof is that it's based on that faulty definition. (I don't think that he'd agree that that's what he's arguing - but either numbers exist that can't be written in a finite amount of time, or they don't. If they do, then his argument is worthless.)

You certainly can argue that the only numbers that should exist are numbers that can be written down. If you do that, there are two main paths. There's the theory of computable numbers (which allows you to keep π and the square roots), and there's the theory of rational numbers (which discards everything that can't be written as a finite fraction). There are interesting theories that build on either of those two approaches. In both, Cantor's argument doesn't apply, because in both, you've restricted the set of numbers to be a countable set.

But that doesn't say anything about the theory of real numbers, which is what Cantor's proof is talking about. In the real numbers, numbers that can't be written down in any form do exist. Numbers like the number produced by Cantor's diagonalization definitely do. The infinite time argument is a load of rubbish because it's based on the faulty concept that Cantor's number doesn't exist until we create it.

The interesting thing about this argument to be, is its selectivity. To my correspondant, the existence of an infinitely long table isn't a problem. He doesn't think that there's anything wrong with the idea of an infinite process creating an infinite table containing a mapping between the natural numbers and the real numbers. He just has a problem with the infinite process of traversing that table. Which is really pretty silly when you think about it.

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May 05 2013 Published by under Bad Math, Cantor Crankery

A fun game to play with cranks is: how long does it take for the crank to contradict themselves?

When you're looking at a good example of crankery, it's full of errors. But for this game, it's not enough to just find an error. What we want is for them to say something so wrong that one sentence just totally tears them down and demonstrates that what they're doing makes no sense.

"The color of a clear sky is green" is, most of the time, wrong. If a crank makes some kind of argument based on the alleged fact that the color of a clear daytime sky is green, the argument is wrong. But as a statement, it's not nonsensical. It' just wrong.

On th other hand, "The color of a clear sky is steak frite with bernaise sauce and a nice side of roasted asparagus", well... it's not even wrong. It's just nonsense.

Today's crank is a great example of this. If, that is, it's legit. I'm not sure that this guy is serious. I think this might be someone playing games, pretending to be a crank. But even if it is, it's still fun.

About a week ago, I got en mail titled "I am a Cantor crank" from a guy named Chris Cuellar. The contents were:


Haha, ok, not exactly. But you really seem to be interested in this stuff. And so am I. But I think I've nailed Cantor for good this time. Not only have I come up with algorithms to count some of these "uncountable" things, but I have also addressed the proofs directly. The diagonalization argument ends up failing spectacularly, and I believe I have a good explanation for why the whole thing ends up being invalid in the first place.

And then I also get to the power set of natural numbers... I really hope my arguments can be followed. The thing I have to emphasize is that I am working on a different system that does NOT roll up cardinality and countability into one thing! As it will turn out, rational numbers are bigger than integers, integers are bigger than natural numbers... but they are ALL countable, nonetheless!

Anyway, I had started a little blog of my own a while ago on these subjects. The first post is here:

Have fun... BWAHAHAHA

So. We've got one paragraph of intro. And then everything crashes and burns in an instant.

"Rational numbers are bigger than integers, integers are bigger than natural numbers, but they are all countable". This is self-evident rubbish. The definition of "countable" say that an infinite set I is countable if, and only if, you can create a one-to-one mapping between the members of I and the natural numbers. The definition of cardinality says that if you can create a one-to-one mapping between two sets, the sets are the same size.

When Mr. Cuellar says that the set of rational numbers is bigger that the set of natural numbers, but that they are still countable... he's saying that there is not a one-to-one mapping between the two sets, but that there is a one-to-one mapping between the two sets.

Look - you don't get to redefine terms, and then pretend that your redefined terms mean the same thing as the original terms.

If you claim to be refuting Cantor's proof that the cardinality of the real numbers is bigger than the cardinality of the natural numbers, then you have to use Cantor's definition of cardinality.

You can change the definition of the size of a set - or, more precisely, you can propose an alternative metric for how to compare the sizes of sets. But any conclusions that you draw about your new metric are conclusions about your new metric - they're not conclusions about Cantor's cardinality. You can define a new notion of set size in which all infinite sets are the same size. It's entirely possible to do that, and to do that in a consistent way. But it will say nothing about Cantor's cardinality. Cantor's proof will still work.

What my correspondant is doing is, basically, what I did above in saying that the color of the sky is steak frites. I'm using terms in a completely inconsistent meaningless way. Steak frites with bernaise sauce isn't a color. And what Mr. Cuellar does is similar: he's using the word "cardinality", but whatever he means by it, it's not what Cantor meant, and it's not what Cantor's proof meant. You can draw whatever conclusions you want from your new definition, but it has no bearing on whether or not Cantor is correct. I don't even need to visit his site: he's demonstrated, in record time, that he has no idea what he's doing.

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Genius Continuum Crackpottery

Mar 21 2013 Published by under Bad Algebra, Bad Logic, Bad Math, Cantor Crankery

There's a lot of mathematical crackpottery out there. Most of it is just pointless and dull. People making the same stupid mistakes over and over again, like the endless repetitions of the same-old supposed refutations of Cantor's diagonalization.

After you eliminate that, you get reams of insanity - stuff which
is simply so incoherent that it doesn't make any sense. This kind of thing is usually word salad - words strung together in ways that don't make sense.

After you eliminate that, sometimes, if you're really lucky, you'll come accross something truly special. Crackpottery as utter genius. Not genius in a good way, like they're an outsider genius who discovered something amazing, but genius in the worst possible way, where someone has created something so bizarre, so overwrought, so utterly ridiculous that it's a masterpiece of insane, delusional foolishness.

Today, we have an example of that: Existics!. This is a body of work by a high school dropout named Gavin Wince with truly immense delusions of grandeur. Pomposity on a truly epic scale!

I'll walk you through just a tiny sample of Mr. Wince's genius. You can go look at his site to get more, and develop a true appreciation for this. He doesn't limit himself to mere mathematics: math, physics, biology, cosmology - you name it, Mr. Wince has mastered it and written about it!

The best of his mathematical crackpottery is something called C3: the Canonized Cardinal Continuum. Mr. Wince has created an algebraic solution to the continuum hypothesis, and along the way, has revolutionized number theory, algebra, calculus, real analysis, and god only knows what else!

Since Mr. Wince believes that he has solved the continuum hypothesis. Let me remind you of what that is:

  1. If you use Cantor's set theory to explore numbers, you get to the uncomfortable result that there are different sizes of infinity.
  2. The smallest infinite cardinal number is called ℵ0,
    and it's the size of the set of natural numbers.
  3. There are cardinal numbers larger than ℵ0. The first
    one larger than ℵ0 is ℵ1.
  4. We know that the set of real numbers is the size of the powerset
    of the natural numbers - 20 - is larger than the set of the naturals.
  5. The question that the continuum hypothesis tries to answer is: is the size
    of the set of real numbers equal to ℵ1? That is, is there
    a cardinal number between ℵ0 and |20|?

The continuum hypothesis was "solved" in 1963. In 1940, Gödel showed that you couldn't disprove the continuum hypothesis using ZFC. In 1963,
another mathematician named Paul Cohen, showed that it couldn't be proven using ZFC. So - a hypothesis which is about set theory can be neither proven nor disproven using set theory. It's independent of the axioms of set theory. You can choose to take the continuum hypothesis as an axiom, or you can choose to take the negation of the continuum hypothesis as an axiom: either choice is consistent and valid!

It's not a happy solution. But it's solved in the sense that we've got a solid proof that you can't prove it's true, and another solid proof that you can't prove it's false. That means that given ZFC set theory as a basis, there is no proof either way that doesn't set it as an axiom.

But... Mr. Wince knows better.

The set of errors that Wince makes is really astonishing. This is really seriously epic crackpottery.

He makes it through one page without saying anything egregious. But then he makes up for it on page 2, by making multiple errors.

First, he pulls an Escultura:

x1 = 1/21 = 1/2 = 0.5

x2 = 1/21 + 1/22 = 1/2 + 1/4 = 0.75

x3 = 1/21 + 1/22 + 1/23 = 1/2 + 1/4 + 1/8 = 0.875


At the end or limit of the infinite sequence, the final term of the sequence is 1.0


In this example we can see that as the number of finite sums of the sequence approaches the limit infinity, the last term of the sequence equals one.

xn = 1.0

If we are going to assume that the last term of the sequence equals one, it can be deduced that, prior to the last term in the sequence, some finite sum in the series occurs where:

xn-1 = 0.999…

xn-1 = 1/21 + 1/22 + 1/23 + 1/24 + … + 1/2n-1 = 0.999…

Therefore, at the limit, the last term of the series of the last term of the sequence would be the term, which, when added to the sum 0.999… equals 1.0.

There is no such thing as the last term of an infinite sequence. Even if there were, the number 0.999.... is exactly the same as 1. It's a notational artifact, not a distinct number.

But this is the least of his errors. For example, the first paragraph on the next page:

The set of all countable numbers, or natural numbers, is a subset of the continuum. Since the set of all natural numbers is a subset of the continuum, it is reasonable to assume that the set of all natural numbers is less in degree of infinity than the set containing the continuum.

We didn't need to go through the difficult of Cantor's diagonalization! We could have just blindly asserted that it's obvious!

or actually... The fact that there are multiple degrees of infinity is anything but obvious. I don't know anyone who wasn't surprised the first time they saw Cantor's proof. It's a really strange idea that there's something bigger than infinity.

Moving on... the real heart of his stuff is built around some extremely strange notions about infinite and infinitessimal values.

Before we even look at what he says, there's an important error here
which is worth mentioning. What Mr. Wince is trying to do is talk about the
continuum hypothesis. The continuum hypothesis is a question about the cardinality of the set of real numbers and the set of natural numbers.
Neither infinites nor infinitessimals are part of either set.

Infinite values come into play in Cantor's work: the cardinality of the natural numbers and the cardinality of the reals are clearly infinite cardinal numbers. But ℵ0, the smallest infinite cardinal, is not a member of either set.

Infinitessimals are fascinating. You can reconstruct differential and integral calculus without using limits by building in terms of infinitessimals. There's some great stuff in surreal numbers playing with infinitessimals. But infinitessimals are not real numbers. You can't reason about them as if they were members of the set of real numbers, because they aren't.

Many of his mistakes are based on this idea.

For example, he's got a very strange idea that infinites and infinitessimals don't have fixed values, but that their values cover a range. The way that he gets to that idea is by asserting the existence
of infinity as a specific, numeric value, and then using it in algebraic manipulations, like taking the "infinityth root" of a real number.

For example, on his way to "proving" that infinitessimals have this range property that he calls "perambulation", he defines a value that he calls κ:

\[ sqrt[infty]{infty} = 1 + kappa\]

In terms of the theory of numbers, this is nonsense. There is no such thing as an infinityth root. You can define an Nth root, where N is a real number, just like you can define an Nth power - exponents and roots are mirror images of the same concept. But roots and exponents aren't defined for infinity, because infinity isn't a number. There is no infinityth root.

You could, if you really wanted to, come up with a definition of exponents that that allowed you to define an infinityth root. But it wouldn't be very interesting. If you followed the usual pattern for these things, it would be a limit: \(sqrt[infty]{x} lim_{nrightarrowinfty} sqrt[n]{x}\). That's clearly 1. Not 1 plus something: just exactly 1.

But Mr. Cringe doesn't let himself be limited by silly notions of consistency. No, he defines things his own way, and runs with it. As a result, he gets a notion that he calls perambulation. How?

Take the definition of κ:

\[ sqrt[infty]{infty} = 1 + kappa\]

Now, you can, obviously, raise both sides to the power of infinity:

\[infty = (1 + kappa)^{infty}\]

Now, you can substitute ℵ0 for \(infty\). (Why? Don't ask why. You just can.) Then you can factor it. His factoring makes no rational sense, so I won't even try to explain it. But he concludes that:

  • Factored and simplified one way, you end up with (κ+1) = 1 + x, where x is some infinitessimal number larger than κ. (Why? Why the heck not?)
  • Factored and simplified another way, you end up with (κ+1) = ℵ
  • If you take the mean of of all of the possible factorings and reductions, you get a third result, that (κ+1) = 2.

He goes on, and on, and on like this. From perambulation to perambulating reciprocals, to subambulation, to ambulation. Then un-ordinals, un-sets... this is really an absolute masterwork of utter insane crackpottery.

Do download it and take a look. It's a masterpiece.

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For every natural number N, there's a Cantor Crank C(n)

Nov 10 2012 Published by under Bad Logic, Bad Math, Cantor Crankery

More crankery? of course! What kind? What else? Cantor crankery!

It's amazing that so many people are so obsessed with Cantor. Cantor just gets under peoples' skin, because it feels wrong. How can there be more than one infinity? How can it possibly make sense?

As usual in math, it all comes down to the axioms. In most math, we're working from a form of set theory - and the result of the axioms of set theory are quite clear: the way that we define numbers, the way that we define sizes, this is the way it is.

Today's crackpot doesn't understand this. But interestingly, the focus of his problem with Cantor isn't the diagonalization. He thinks Cantor went wrong way before that: Cantor showed that the set of even natural numbers and the set of all natural numbers are the same size!

Unfortunately, his original piece is written in Portuguese, and I don't speak Portuguese, so I'm going from a translation, here.

The Brazilian philosopher Olavo de Carvalho has written a philosophical “refutation” of Cantor’s theorem in his book “O Jardim das Aflições” (“The Garden of Afflictions”). Since the book has only been published in Portuguese, I’m translating the main points here. The enunciation of his thesis is:

Georg Cantor believed to have been able to refute Euclid’s fifth common notion (that the whole is greater than its parts). To achieve this, he uses the argument that the set of even numbers can be arranged in biunivocal correspondence with the set of integers, so that both sets would have the same number of elements and, thus, the part would be equal to the whole.

And his main arguments are:

It is true that if we represent the integers each by a different sign (or figure), we will have a (infinite) set of signs; and if, in that set, we wish to highlight with special signs, the numbers that represent evens, then we will have a “second” set that will be part of the first; and, being infinite, both sets will have the same number of elements, confirming Cantor’s argument. But he is confusing numbers with their mere signs, making an unjustifiable abstraction of mathematical properties that define and differentiate the numbers from each other.

The series of even numbers is composed of evens only because it is counted in twos, i.e., skipping one unit every two numbers; if that series were not counted this way, the numbers would not be considered even. It is hopeless here to appeal to the artifice of saying that Cantor is just referring to the “set” and not to the “ordered series”; for the set of even numbers would not be comprised of evens if its elements could not be ordered in twos in an increasing series that progresses by increments of 2, never of 1; and no number would be considered even if it could be freely swapped in the series of integeres.

He makes two arguments, but they both ultimately come down to: "Cantor contradicts Euclid, and his argument just can't possibly make sense, so it must be wrong".

The problem here is: Euclid, in "The Elements", wrote severaldifferent collections of axioms as a part of his axioms. One of them was the following five rules:

  1. Things which are equal to the same thing are also equal to one another.
  2. If equals be added to equals, the wholes are equal.
  3. If equals be subtracted from equals, the remainders are equal.
  4. Things which coincide with one another are equal to one another.
  5. The whole is greater that the part.

The problem that our subject has is that Euclid's axiom isn't an axiom of mathematics. Euclid proposed it, but it doesn't work in number theory as we formulate it. When we do math, the axioms that we start with do not include this axiom of Euclid.

In fact, Euclid's axioms aren't what modern math considers axioms at all. These aren't really primitive ground statements. Most of them are statements that are provable from the actual axioms of math. For example, the second and third axioms are provable using the axioms of Peano arithmetic. The fourth one doesn't appear to be a statement about numbers at all; it's a statement about geometry. And in modern terms, the fifth one is either a statement about geometry, or a statement about measure theory.

The first argument is based on some strange notion of signs distinct from numbers. I can't help but wonder if this is an error in translation, because the argument is so ridiculously shallow. Basically, it concedes that Cantor is right if we're considering the representations of numbers, but then goes on to draw a distinction between representations ("signs") and the numbers themselves, and argues that for the numbers, the argument doesn't work. That's the beginning of an interesting argument: numbers and the representations of numbers are different things. It's definitely possible to make profound mistakes by confusing the two. You can prove things about representations of numbers that aren't true about the numbers themselves. Only he doesn't actually bother to make an argument beyond simply asserting that Cantor's proof only works for the representations.

That's particularly silly because Cantor's proof that the even naturals and the naturals have the same cardinality doesn't talk about representation at all. It shows that there's a 1 to 1 mapping between the even naturals and the naturals. Period. No "signs", no representations.

The second argument is, if anything, even worse. It's almost the rhetorical equivalent of sticking his fingers in his ears and shouting "la la la la la". Basically - he says that when you're producing the set of even naturals, you're skipping things. And if you're skipping things, those things can't possible be in the set that doesn't include the skipped things. And if there are things that got skipped and left out, well that means that it's ridiculous to say that the set that included the left out stuff is the same size as the set that omitted the left out stuff, because, well, stuff got left out!!!.

Here's the point. Math isn't about intuition. The properties of infinitely large sets don't make intuitive sense. That doesn't mean that they're wrong. Things in math are about formal reasoning: starting with a valid inference system and a set of axioms, and then using the inference to reason. If we look at set theory, we use the axioms of ZFC. And using the axioms of ZFC, we define the size (or, technically, the cardinality) of sets. Using that definition, two sets have the same cardinality if and only if there is a one-to-one mapping between the elements of the two sets. If there is, then they're the same size. Period. End of discussion. That's what the math says.

Cantor showed, quite simply, that there is such a mapping:

\[{ (i rightarrow itimes 2) | i in N }\]

There it is. It exists. It's simple. It works, by the axioms of Peano arithmetic and the axiom of comprehension from ZFC. It doesn't matter whether it fits your notion of "the whole is greater than the part". The entire proof is that set comprehension. It exists. Therefore the two sets have the same size.

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There's always more Cantor crackpottery!

Aug 13 2012 Published by under Bad Math, Cantor Crankery

I'm not the only one who gets mail from crackpots!

A kind reader forwarded me yet another bit of Cantor crackpottery. It never ceases to amaze me how many people virulently object to Cantor, and how many of them just spew out the same, exact, rubbish, somehow thinking that they're different than all the others who made the same argument.

This one is yet another in the representation scheme. That is, it's an argument that I can write out all of the real numbers whose decimal forms have one digit after the decimal point; then all of the reals with two digits; then all of them with 3 digits; etc. This will produce an enumeration, therefore, there's a one-to-one mapping from the naturals to the reals. Presto, Cantor goes out the window.

Or not.

As usual, the crank starts off with a bit of pomposity:

Dear Colleague,

My mathematic researshes lead me to resolve the continuum theory of Cantor, subject of controversy since a long time.

This mail is made to inform the mathematical community from this work, and share the conclusions.

You will find in attachment extracts from my book "Théorie critique fondamentale des ensembles de Cantor",

Inviting you to contact me,

Francis Collot,
Member of the American mathematical society
Membre de la société mathématique de France
Member of the Bulletin of symbolic logic
Director of éditions européennes

As a quick aside, I love how he signs he email "Member of the AMS", as if that were something meaningful. The AMS is a great organization - but anyone can be a member. All you need to do is fill out a form, and write them a check. It's not something that anyone sane or reasonable brags about, because it doesn't mean anything.

Anyway, let's move on. Here's the entirety of his proof. I've reproduced the formatting as well as I could; the original document sent to me was a PDF, so the tables don't cut-and-paste.

The well-order on the set of real numbers result from this remark that it is possible to build, after the comma, a set where each subset has the same number of ordered elements (as is ordered the subset 2 : 10 …13 … 99).

Each successive integer is able to be followed after the comma (in french the real numbers have one comma after the integer) by an increasing number of figures.

0,0 0,10 0,100
0,1 0,11 0,101
0,2 0,12 0,102
0,9 0,99 0,999

It is the same thing for each successive interger before the comma.

1 2 3

So it is the 2 infinite of real number.

For this we use the binary notation.

But Cantor and his disciples never obtained this simple result.

After that, the theory displays that the infinity is the asymptote of the two branches of the hyperbole thanks to an introduction of trigonometry notions.

The successive numbers which are on cotg (as 1/2, 1/3, 1/4, 1/5) never attain 0 because it would be necessary to write instead (1/2, 1/3, 1/4, 1/4 ).

The 0 of the cotg is also the origin of the asymptote, that is to say infinite.

The beginning is, pretty much, a typical example of the representational crankery. It's roughly a restatement of, for example, John Gabriel and his decimal trees. The problem with it is simple: this kind of enumeration will enumerate all of the real numbers with finite length representations. Which means that the total set of values enumerated by this won't even include all of the rational numbers, much less all of the real numbers.

(As an interesting aside: you can see a beautiful example of what Mr. Collot missed by looking at Conway's introduction to the surreal numbers, On Numbers and Games, which I wrote about here. He specifically deals with this problem in terms of "birthdays" and the requirement to include numbers who have an infinite birthday, and thus an infinite representation in the surreal numbers.)

After the enumeration stuff, he really goes off the rails. I have no idea what that asymptote nonsense is supposed to mean. I think part of the problem is that mr. Collot isn't very good at english, but the larger part of it is that he's an incoherent crackpot.

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Yet Another Cantor Crank

Nov 05 2011 Published by under Bad Math, Cantor Crankery

I get a fair bit of mail from crackpots. The category that I find most annoying is the Cantor cranks. Over and over and over again, these losers send me their "proofs".

What bugs me so much about this is how shallowly wrong they are.

What Cantor did was remarkably elegant. He showed that given anything that is claimed to be a one-to-one mapping between the set of integers and the set of real numbers (also sometimes described as an enumeration of the real numbers - the two terms are functionally equivalent), then here's a simple procedure which will produce a real number that isn't in included in that mapping - which shows that the mapping isn't one-to-one.

The problem with the run-of-the-mill Cantor crank is that they never even try to actually address Cantor's proof. They just say "look, here's a mapping that works!"

So the entire disproof of their "refutation" of Cantor's proof is... Cantor's proof. They completely ignore the thing that they're claiming to disprove.

I got another one of these this morning. It's particularly annoying because he makes the same mistake as just about every other Cantor crank - but he also specifically points to one of my old posts where I rant about people who make exactly the same mistake as him.

To add insult to injury, the twit insisted on sending me PDF - and not just a PDF, but a bitmapped PDF - meaning that I can't even copy text out of it. So I can't give you a link; I'm not going to waste Scientopia's bandwidth by putting it here for download; and I'm not going to re-type his complete text. But I'll explain, in my own compact form, what he did.

It's an old trick; for example, it's ultimately not that different from what John Gabriel did. The only real novelty is that he does it in binary - which isn't much of a novelty. This author calls it the "mirror method". The idea is, in one column, write a list of the integers greater than 0. In the opposite column, write the mirror of that number, with the decimal (or, technically, binary) point in front of it:

Integer Real
0 0.0
1 0.1
10 0.01
11 0.11
100 0.001
101 0.101
110 0.011
111 0.111
1000 0.0001
... ...

Extend that out to infinity, and, according to the author, the second column it's a sequence of every possible real number, and the table is a complete mapping.

The problem is, it doesn't work, for a remarkably simple reason.

There is no such thing as an integer whose representation requires an infinite number of digits. For every possible integer, its representation in binary has a fixed number of bits: for any integer N, it's representation is no longer that \(lceil log_2(n) rceil\). That's always a finite integer.

But... we know that the set of real numbers includes numbers whose representation is infinitely long. so this enumeration won't include them. Where does the square root of two fall in this list? It doesn't: it can't be written as a finite string in binary. Where is π? It's nowhere; there's no finite representation of π in binary.

The author claims that the novel property of his method is:

Cantor proved the impossibility of both our enumerations as follows: for any given enumeration like ours Cantor proposed his famous diagonal method to build the contra-sample, i.e., an element which is quasi omitted in this enumeration. Before now, everyone agreed that this element was really omitted as he couldn't tell the ordinal number of this element in the give enumeration: now he can. So Cantor's contra-sample doesn't work.

This is, to put it mildly, bullshit.

First of all - he pretends that he's actually addressing Cantor's proof - only he really isn't. Remember - what Cantor's proof did was show you that, given any purported enumeration of the real numbers, that you could construct a real number that isn't in that enumeration. So what our intrepid author did was say "Yeah, so, if you do Cantor's procedure, and produce a number which isn't in my enumeration, then I'll tell you where that number actually occurred in our mapping. So Cantor is wrong."

But that doesn't actually address Cantor. Cantor's construction specifically shows that the number it constructs can't be in the enumeration - because the procedure specifically guarantees that it differs from every number in the enumeration in at least one digit. So it can't be in the enumeration. If you can't show a logical problem with Cantor's construction, then any argument like the authors is, simply, a priori rubbish. It's just handwaving.

But as I mentioned earlier, there's an even deeper problem. Cantor's method produces a number which has an infinitely long representation. So the earlier problem - that all integers have a finite representation - means that you don't even need to resort to anything as complicated as Cantor to defeat this. If your enumeration doesn't include any infinitely long fractional values, then it's absolutely trivial to produce values that aren't included: 1/3, 1/7, 1/9.

In short: stupid, dull, pointless; absolutely typical Cantor crankery.

113 responses so far

Representational Crankery: the New Reals and the Dark Number

Jan 06 2011 Published by under Bad Logic, Bad Math, Cantor Crankery, Numbers

There's one kind of crank that I haven't really paid much attention to on this blog, and that's the real number cranks. I've touched on real number crankery in my little encounter with John Gabriel, and back in the old 0.999...=1 post, but I've never really given them the attention that they deserve.

There are a huge number of people who hate the logical implications of our definitions real numbers, and who insist that those unpleasant complications mean that our concept of real numbers is based on a faulty definition, or even that the whole concept of real numbers is ill-defined.

This is an underlying theme of a lot of Cantor crankery, but it goes well beyond that. And the basic problem underlies a lot of bad mathematical arguments. The root of this particular problem comes from a confusion between the representation of a number, and that number itself. "\(frac{1}{2}\)" isn't a number: it's a notation that we understand refers to the number that you get by dividing one by two.

There's a similar form of looniness that you get from people who dislike the set-theoretic construction of numbers. In classic set theory, you can construct the set of integers by starting with the empty set, which is used as the representation of 0. Then the set containing the empty set is the value 1 - so 1 is represented as { 0 }. Then 2 is represented as { 1, 0 }; 3 as { 2, 1, 0}; and so on. (There are several variations of this, but this is the basic idea.) You'll see arguments from people who dislike this saying things like "This isn't a construction of the natural numbers, because you can take the intersection of 8 and 3, and set intersection is meaningless on numbers." The problem with that is the same as the problem with the notational crankery: the set theoretic construction doesn't say "the empty set is the value 0", it says "in a set theoretic construction, the empty set can be used as a representation of the number 0.

The particular version of this crankery that I'm going to focus on today is somewhat related to the inverse-19 loonies. If you recall their monument, the plaque talks about how their work was praised by a math professor by the name of Edgar Escultura. Well, it turns out that Escultura himself is a bit of a crank.

The specify manifestation of his crankery is this representational issue. But the root of it is really related to the discomfort that many people feel at some of the conclusions of modern math.

A lot of what we learned about math has turned out to be non-intuitive. There's Cantor, and Gödel, of course: there are lots of different sizes of infinities; and there are mathematical statements that are neither true nor false. And there are all sorts of related things - for example, the whole ideaof undescribable numbers. Undescribable numbers drive people nuts. An undescribable number is a number which has the property that there's absolutely no way that you can write it down, ever. Not that you can't write it in, say, base-10 decimals, but that you can't ever write down anything, in any form that uniquely describes it. And, it turns out, that the vast majority of numbers are undescribable.

This leads to the representational issue. Many people insist that if you can't represent a number, that number doesn't really exist. It's nothing but an artifact of an flawed definition. Therefore, by this argument, those numbers don't exist; the only reason that we think that they do is because the real numbers are ill-defined.

This kind of crackpottery isn't limited to stupid people. Professor Escultura isn't a moron - but he is a crackpot. What he's done is take the representational argument, and run with it. According to him, the only real numbers are numbers that are representable. What he proposes is very nearly a theory of computable numbers - but he tangles it up in the representational issue. And in a fascinatingly ironic turn-around, he takes the artifacts of representational limitations, and insists that they represent real mathematical phenomena - resulting in an ill-defined number theory as a way of correcting what he alleges is an ill-defined number theory.

His system is called the New Real Numbers.

In the New Real Numbers, which he notates as \(R^*\), the decimal notation is fundamental. The set of new real numbers consists exactly of the set of numbers with finite representations in decimal form. This leads to some astonishingly bizarre things. From his paper:

3) Then the inverse operation to multiplication called division; the result of dividing a decimal by another if it exists is called quotient provided the divisor is not zero. Only when the integral part of the devisor is not prime other than 2 or 5 is the quotient well defined. For example, 2/7 is ill defined because the quotient is not a terminating decimal (we interpret a fraction as division).

So 2/7ths is not a new real number: it's ill-defined. 1/3 isn't a real number: it's ill-defined.

4) Since a decimal is determined or well-defined by its digits, nonterminating decimals are ambiguous or ill-defined. Consequently, the notion irrational is ill-defined since we cannot cheeckd all its digits and verify if the digits of a nonterminaing decimal are periodic or nonperiodic.

After that last one, this isn't too surprising. But it's still absolutely amazing. The square root of two? Ill-defined: it doesn't really exist. e? Ill-defined, it doesn't exist. \(pi\)? Ill-defined, it doesn't really exist. All of those triangles, circles, everything that depends on e? They're all bullshit according to Escultura. Because if he can't write them down in a piece of paper in decimal notation in a finite amount of time, they don't exist.

Of course, this is entirely too ridiculous, so he backtracks a bit, and defines a non-terminating decimal number. His definition is quite peculiar. I can't say that I really follow it. I think this may be a language issue - Escultura isn't a native english speaker. I'm not sure which parts of this are crackpottery, which are linguistic struggles, and which are notational difficulties in reading math rendered as plain text.

5) Consider the sequence of decimals,

(d)^na_1a_2...a_k, n = 1, 2, ..., (1)

where d is any of the decimals, 0.1, 0.2, 0.3, ..., 0.9, a_1, ..., a_k, basic integers (not all 0 simultaneously). We call the nonstandard sequence (1) d-sequence and its nth term nth d-term. For fixed combination of d and the a_j's, j = 1, ..., k, in (1) the nth term is a terminating decimal and as n increases indefinitely it traces the tail digits of some nonterminating decimal and becomes smaller and smaller until we cannot see it anymore and indistinguishable from the tail digits of the other decimals (note that the nth d-term recedes to the right with increasing n by one decimal digit at a time). The sequence (1) is called nonstandard d-sequence since the nth term is not standard g-term; while it has standard limit (in the standard norm) which is 0 it is not a g-limit since it is not a decimal but it exists because it is well-defined by its nonstandard d-sequence. We call its nonstandard g-limit dark number and denote by d. Then we call its norm d-norm (standard distance from 0) which is d > 0. Moreover, while the nth term becomes smaller and smaller with indefinitely increasing n it is greater than 0 no matter how large n is so that if x is a decimal, 0 < d < x.

I think that what he's trying to say there is that a non-terminating decimal is a sequence of finite representations that approach a limit. So there's still no real infinite representations - instead, you've got an infinite sequence of finite representations, where each finite representation in the sequence can be generated from the previous one. This bit is why I said that this is nearly a theory of the computable numbers. Obviously, undescribable numbers can't exist in this theory, because you can't generate this sequence.

Where this really goes totally off the rails is that throughout this, he's working on the assumption that there's a one-to-one relationship between representations and numbers. That's what that "dark number" stuff is about. You see, in Escultura's system, 0.999999... is not equal to one. It's not a representational artifact. In Escultura's system, there are no representational artifacts: the representations are the numbers. The "dark number", which he notates as \(d^*\), is (1-0.99999999...) and \(d^* > 0\). In fact, \(d^*\) is the smallest number greater than 0. And you can generate a complete ordered enumeration of all of the new real numbers, \({0,
d^*, 2d^*, 3d^*, ..., n-2d^*, n-d^*, n, n+d^*, ...}\)

Reading Escultura, every once in a while, you might think he's joking. For example, he claims to have disproven Fermat's last theorem. Fermat's theorem says that for n>2, there are no integer solutions for the equation \(x^n + y^n = z^n\). Escultura says he's disproven this:

The exact solutions of Fermat's equation, which are the counterexamples to FLT, are given by the triples (x,y,z) = ((0.99...)10^T,d*,10^T), T = 1, 2, ..., that clearly satisfies Fermat's equation,

x^n + y^n = z^n, (4)

for n = NT > 2. Moreover, for k = 1, 2, ..., the triple (kx,ky,kz) also satisfies Fermat's equation. They are the countably infinite counterexamples to FLT that prove the conjecture false. One counterexample is, of course, sufficient to disprove a conjecture.

Even if you accept the reality of the notational artifact \(d^*\), this makes no sense: the point of Fermat's last theorem is that there are no integer solutions; \(d^*\) is not an integer; \((1-d^*)10\) is not an integer. Surely he's not that stupid. Surely he can't possibly believe that he's disproven Fermat using non-integer solutions? I mean, how is this different from just claiming that you can use (2, 3, 351/3) as a counterexample for n=3?

But... he's serious. He's serious enough that he's published published a real paper making the claim (albeit in crackpot journals, which are the only places that would accept this rubbish).

Anyway, jumping back for a moment... You can create a theory of numbers around this \(d^*\) rubbish. The problem is, it's not a particularly useful theory. Why? Because it breaks some of the fundamental properties that we expect numbers to have. The real numbers define a structure called a field, and a huge amount of what we really do with numbers is built on the fundamental properties of the field structure. One of the necessary properties of a field is that it has unique identity elements for addition and multiplication. If you don't have unique identities, then everything collapses.

So... Take \(frac{1}{9}\). That's the multiplicative inverse of 9. So, by definition, \(frac{1}{9}*9 = 1\) - the multiplicative identity.

In Escultura's theory, \(frac{1}{9}\) is a shorthand for the number that has a representation of 0.1111.... So, \(frac{1}{9}*9 = 0.1111....*9 = 0.9999... = (1-d^*)\). So \((1-d^*)\) is also a multiplicative identity. By a similar process, you can show that \(d^*\) itself must be the additive identity. So either \(d^* == 0\), or else you've lost the field structure, and with it, pretty much all of real number theory.

125 responses so far

Metaphorical Crankery: a bad metaphor is like a steaming pile of ...

Jun 17 2010 Published by under Cantor Crankery

So, another bit of Cantor stuff. This time, it really isn't Cantor
crankery, so much as it is just Cantor muddling. The post
that provoked this
is not, I think, crankery of any kind - but it
demonstrates a common problem that drives me crazy; to steal a nifty phrase
from, people who can't count to meta-three really shouldn't try
to use metaphors.

The problem is: You use a metaphor to describe some concept. The metaphor
isn't the thing you describe - it's just a tool that you use. But
someone takes the metaphor, and runs with it, making arguments that are built
entirely on metaphor, but which bear no relation to the real underlying
concept. And they believe that whatever conclusions they draw from the
metaphor must, therefore, apply to the original concept.

In the context of Cantor, I've seen this a lot of times. The post that
inspired me to write this isn't, I think, really making this mistake. I think
that the author is actually trying to argue that this is a lousy metaphor to
use for Cantor, and proposing an alternative. But I've seen exactly this
reasoning used, many times, by Cantor cranks as a purported disproof. The
cranky claim is: Cantor's proof is wrong, because it cheats.

Of course, if you look at Cantor's proof as a mathematical construct, it's
a perfectly valid, logical, and even beautiful proof by contradiction. There's
no cheating. So where do the "cheat" claims come from?

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50 responses so far

Grandiose Crankery: Cantor, Godel, Church, Turing, ... Morons!

Mar 09 2010 Published by under Cantor Crankery

A bunch of people have been asking me to take a look at yet another piece of Cantor crankery recently posted to Arxiv. In general, I'm sick and tired of Cantor crankery - it's been occupying much too much space on this blog lately. But this one is a real prize. It's an approach that I've never seen before: instead of the usual weaseling around, this one goes straight for Cantor's proof.

But it does much, much more than that. In terms of ambition, this thing really takes the cake. According to the author, one J. A. Perez, he doesn't just refute Cantor. No, that would be trivial! Every run-of-the-mill crackpot claims to refute cantor! Perez claims to refute Cantor, Gödel, Church, and Turing. Among others. He claims to reform the axiom of infinity in set theory to remove the problems that it supposedly causes. He claims to be able to use his reformed axiom of infinity together with his refutation of Cantor to get rid of the continuum hypothesis, and to eliminate any strange results proved by the axiom of choice.

Yes, Mr. (Dr? Professor? J. Random Schmuck?) Perez is nothing if not a true mastermind, a mathematical genius of utterly epic proportions! The man who single-handedly refutes pretty much all of 20th century mathematics! The man who has determined that now we must throw away Cantor and Gödel, and reinstate Hilbert's program. The perfect mathematics is at hand, if we will only listen to his utter brilliance!

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70 responses so far

A Crank among Cranks: Debating John Gabriel

Feb 04 2010 Published by under Cantor Crankery

So, remember back in December, I wrote a post about a Cantor crank who had a Knol page supposedly refuting Cantor's diagonalization?

This week, I foolishly let myself get drawn into an extended conversation with him in comments. Since it's a comment thread on an old post that had been inactive for close to two months before this started, I assume most people haven't followed it. In an attempt to salvage something from the time I wasted with him, I'm going to share the discussion with you in this new post. It's entertaining, in a pathetic sort of way; and it's enlightening, in that it's one of the most perfect demonstrations of the behavior of a crank that I've yet encountered. Enjoy!

I'm going to edit for formatting purposes, and I'll interject a few comments, but the text of the messages is absolutely untouched - which you can verify, if you want, by checking the comment thread on the original post. The actual discussion starts with this comment, although there's a bit of content-free back and forth in the dozen or so comments before that.

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466 responses so far

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