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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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.

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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.

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Too Crazy to Be Fun: Pi Crackpottery

Nov 04 2010 Published by under Bad Geometry, Bad Math

I always appreciate it when readers send me links to good crackpottery. But one of the big problems with a lot of the links that I get is that a lot of them are just too crazy. When you've got someone going off on a time-cube style rant, there's just no good way to make fun of them - the stuff just doesn't make enough sense to make fun of.

For example, someone sent me a really... interesting link recently, to a book by a guy who claims to have proved that \(pi=3.125\). Let me quote the beginning of his book, to give you an idea of what I mean. I've attempted to reproduce the formatting as well as I can, but it's frankly worse that I can figure out how to reproduce with HTML.

CONCEPTIONS OF π

One conception of π is the value 3.141... that is used for calculations, involving geometrical figures containing circles.

Another conception is that the number 3.141... is only an approximation. I interpret

π in this book as the relationship between a circle and its diameter, and not as the irrational number 3.141...

I have attempted to find a value that will result in exact calculations of circles.

SQUARING

The word “squaring” is used for the following:

A. The square with side of 4 u.l. so-called square squaring form

B. A circle with the diameter of 4 u.l., the circle squaring form

C. The only cylinder that has been produced by a square and two circles, from which come the cylinder squaring form

I identify the characteristics found in figures that I call square squaring, circle squaring and cylinder squaring and the principles behind these figures. I refer to three figures:

1. Square

2. Circle

3. Cylinder

It's not particularly easy to make fun of that, because it's so utterly and bizarrely nonsensical.

It's pretty hard to get through his drek... But he's got this way of characterizing different kinds of squares, and then different kinds of circles based on the different kinds of squares. The ways of characterizing the squares are based on screwing up units. There are three kinds of squares: squares where the number of length units in the perimeter are larger than the number of area units in the area; squares where the number of length units in the perimeter are smaller than the number of area units in the area; and squares where they're equal.

That last group contains only one element: the square who's sides have length 4. He concludes that this is a profoundly important square, and says that a square whose side-length is four of some unit is the "square squaring form" of the square. This is a really important idea to him: he goes out of his way to write a special note in extra large font:

N.B.

Squares with sides of 4 u.l. have a perimeter of 16 u.l. and an area of 16 u.a. Perimeter = 16 u.l. and area = 16 u.a. What I immediately observed was the common number for the perimeter and the area.

As you can see, we're dealing with a real genius here.

From there, he launches into a description of circles. According to him, every circle is defined by a square, where the circle is inscribed in the square. It makes no sense at all; this section, I can't even attempt to mock. It's just so damned incoherent that it's not even funny. The conclusion is that for magical reasons to be explained later, the circle with diameter 4 is special.

Then we get to the heart of the matter: what he calls "the circle squaring form". This continues to make no sense. But it's got some interesting typography. It starts with:

ln

of

the logarithm e

For no apparent reason. Then he goes on to start presenting the notation he's going to use... And to call it insane is kind. In includes two distinct definitions: "Logarithm e = log e" and "Logarithm ln of e = log ln". I have no clue what this is supposed to mean.

From there, he goes through a bunch of definitions, leading up to a set of purported equations describing the special circle related to the special square whose sides are 4 units long. What are the equations going to show us?

The formulae will define a circle that shows relation to;

  • Its diameter to its circumference and area.
  • Circles relation to its square.
  • Its relation of the shaded area that is not covered by the
    circle.
  • Finally, how many per cent a circle cover its square’s
    area and perimeter.
  • Also relations to the cylinder.

So he gets to the equations, which are defined in terms of "ln of logarithm e". His first equation, presented without explanation, is:

\(Q = (ln sqrt{(e^{ln s})^2}/ln e^{ln s})^2/2\)

What in the hell that's supposed to mean, I don't know. He doesn't define Q. s is the length of the side of a square. Where eln s comes from, I have no idea... but he gets rid of it, replacing it with s. Apparently, this is supposed to be a meaningful step - we're supposed to learn something really important from it! He goes through a bunch of steps, ending up with "Relevant Formula: ⇒ \(4Q = ( ln sqrt{s^2 *2}/ln s)^2*2\)", which supposedly defines "the relationship between area, circumference, and diameter of a circle".

I'll stop here. I think by now you can see my problem. How can you make fun of this in an entertaining way? There's just nothing that I can say about this stuff beyond "huh? what in the bloody hell is he trying to say here?"

He offers a cash prize to anyone who can prove him wrong. I think he's pretty safe in not needing to worry about paying that prize out; you can't prove that something nonsensical is wrong. Yeah, sure, π=3.2 or whatever in his universe: after all, for any statement S, \(bot Rightarrow S\). Hell, \(4Q = ( ln sqrt{s^2 *2}/ln s)^2*2\), therefore the moon is made of green cheese!

What kills me about this is how utterly, insanely, ridiculously wrong it is... My daughter, who is in fifth grade, did experiments last year in math class where they roll a circle along a piece of paper to get its diameter, and then compare that to its length. A bunch of fourth graders can easily do this accurately enough to show that the ratio of the circumference to the diameter is around 22/7. Any attempt to actually verify his number totally fails. But it would seem that in his world, when reality conflicts with theory, reality is the one that's wrong.

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Turing Crackpottery!

Sep 08 2010 Published by under Bad Math, Bad Software

One of the long-time cranks who's commented on this blog is a bozo who goes by the name "Vorlath". Vorlath is a hard-core Cantor crank, and so he usually shows up to rant whenever the subject of Cantor comes up. But last week, while I was busy dealing with the Scientopia site hosting trouble, a reader sent me a link to a piece Vorlath wrote about the Halting problem. Apparently, he doesn't like that proof either.

Personally, the proof that the halting problem is unsolvable is one of my all-time favorite mathematical proofs. It's incredibly simple - just a couple of steps. It's very concrete - the key to the proof is a program that you can actually write, easily, in a couple of lines of code in a scripting language. And best of all, it's incredibly profound - it proves something very similar to Gödel's incompleteness theorem. It's wonderful.

To show you how simple it is, I'm going to walk you through it - in all of its technical details.

Continue Reading »

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