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:

- If you use Cantor's set theory to explore numbers, you get to the uncomfortable result that there are different sizes of infinity.
- The smallest infinite cardinal number is called ℵ
_{0},

and it's the size of the set of natural numbers. - There are cardinal numbers larger than ℵ
_{0}. The first

one larger than ℵ_{0}is ℵ_{1}. - We know that the set of real numbers is the size of the powerset

of the natural numbers - 2^{ℵ0}- is larger than the set of the naturals. - The question that the continuum hypothesis tries to answer is: is the size

of the set of real numbers equal to ℵ_{1}? That is, is there

a cardinal number between ℵ_{0}and |2^{ℵ0}|?

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:

x

_{1}= 1/2^{1}= 1/2 = 0.5x

_{2}= 1/2^{1}+ 1/2^{2}= 1/2 + 1/4 = 0.75x

_{3}= 1/2^{1}+ 1/2^{2}+ 1/2^{3}= 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.

x

_{n}= 1.0If 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:

x

_{n-1}= 0.999…x

_{n-1}= 1/2^{1}+ 1/2^{2}+ 1/2^{3}+ 1/2^{4}+ … + 1/2^{n-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 ℵ_{0}, 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 ℵ_{0} 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)*, where^{ℵ}= 1 + x*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}= ℵ - If you take the mean of of all of the possible factorings and reductions, you get a third result, that (κ+1)
^{ℵ}= 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.