I have to correct some things I’ve written about the empty set.

Let’s start. Let A = {0}. Now A \ {0} = {} = Ø.

That is: We get the empty set. Let’s go beyond that.

I’ve written in this blog earlier ”emptyness can be created, ‘nothing’ can’t be created; nothing is from which the creation begins”.

At the beginning of this post, we created emptyness, we got the empty set as a result. By the means of the set theory we can’t get rid of the empty set. It’s as empty space as we can get. Also, in the poetic thought of mine I’ve written, that emptyness can be created, ‘nothing’ can’t be created.

So, we can’t get to this ‘nothing’, we can just see, that it is a ”state” that is ”before” the empty set, for the sake of perfectness. 🙂

A symbol to this ”state” could be

The order: ‘nothing’ → emptyness → something.

This new “collection of nothing” can’t be considered really a set; it actually doesn’t exist, but still, it is there — at least it was… Perhaps it will be… Somewhere…

In the previous post I wrote, that as far as I can see at the moment, semantically zero is *none*, not nothing. I’ll clarify that a bit. Let’s ask a question: How many? None.

The opposite to “none”, might be everything of some finite amount (in the sense of the set theory). But as I wrote in the previous post, zero might be considered neutral in the way, that zero doesn’t have an opposite.

And I can’t really say, that the opposite to none is everything of some finite amount. If zero is none and the opposite to none is everything of some finite amount, every positive integer could be considered as an opposite to zero, that doesn’t make sense.

Or does it, if we have special cases, where something in some collection or set is some finite amount of elements, for example we have 7 elements in a set. Then, if we have all the elements from the set, we have 7 elements, all of them, instead of having none of the elements from the set.

What about dividing by zero, particularly 0/0? If there is zero amount of something, there isn’t this something at all. So 0/0 could be phrased as “none isn’t divided at all”.

Though I’m using too many negatives in one sentence. Perhaps now it’s better phrased: “None is divided.” Now there’s only one negative in one sentence and it’s better English and we don’t divide anything, particularly not zero. Word “none” prohibits the division.

In case *a*/0, *a ≠ *0, perhaps we could say: “Something is not divided.”

Therefore particularly zero can’t be divided by zero. 🙂

I’ve been thinking about zero, none, nothing and the empty set time to time… And the infinite.

Can the opposite of zero be infinite? No. Why? Zero is a number, infinite is categorically different concept than a number. In the set **N **(all whole numbers) is infinite amount of numbers, but none of those is infinite.

Therefore, the opposite of zero is not infinite. And as far as I can see at the moment, semantically zero is *none*, not nothing.

Does zero really in terms of mathematics have an opposite? Is it neutral in a way, that it doesn’t have an opposite?

As to the empty set, it is an empty collection, one could say ”collection of nothing”. Poetically one could ask: Does the empty set exist? The empty set is ”collection of nothing”. If there is a collection of nothing, a collection that consists of nothing, the collection seems non-existent.

Can non-existent exist? Though, to ask, that does some kind of mathematical concept (the empty set) exist, is quite meaningless…

So, perhaps one can say, that the empty set is some kind of nothing… What’s the opposite to nothing, to something completely non-existent? Everything? Everything of what? Everything of everything that exists.

Can one say that there is an opposite to the empty set? If it would be the set of all sets, there is a problem: Also the empty set would be included in the set of all sets — if the empty set exists in same sense than non-empty sets. If the empty set exists, the set of all sets couldn’t be an opposite to the empty set.

The infinite is difficult concept. I’ve read, that Gauss himself objected at first to bring the *actual *concept of infinite to the mathematics. He would at first wanted to keep it only in philosophy and religion.

As to the infinite, perhaps, to be precise, one really can’t find an opposite to the infinite, not in terms of mathematics nor by the terms of semantics.

This is just humorous thought of mine…

If we have a “collection” where isn’t anything, is it a collection? If someone has got 100 books, the person has a collection of 100 books. But if a person hasn’t got books at all, does the person have a collection of books? No.

So, is the empty set as such a set? A collection where is nothing isn’t a collection.

But is it an empty collection? But is an empty collection a *collection* at all? 🙂

I’ve been reading e-book ”Introduction to Mathematical Philosophy” originally written by Bertrand Russell and published in the year 1901.

Among other interesting thoughts Russell gives thought to the definition of a number. This is something very interesting; I’ve been thinking myself strange things about number zero. Can zero be considered as a whole number? It doesn’t describe anything existing as whole. If the number of something is zero, this something doesn’t exist at all in somewhere, particularly not as whole.

As to definition of number, Russell discusses about classes. From an old Finnish book that discusses university level algebra, I recently learned the definition of zero as a class. In Russell’s book zero is defined as a class in slightly different way: Russell doesn’t say anything about the empty set, instead he mentions ”null-class”. I think I will read this part of the book over and over again.

This is something fascinating…

Hopefully you got interested in this great book:

In my old article I have implementation for Cantor’s set without recursion. Now I implemented the Cantor’s set with recursion in JavaScript. This is better solution.

The definition of the Cantor’s set in the language of set theory is the following:

If this doesn’t mean anything to you, you might want to check the old post. 🙂

Below is a picture from the output of the program of this post:

Below is the JavaScript listing from Notepad++ in full as png-file:

Back to the world of fractals…

One of the famous fractals is the Koch curve, described by Helge von Koch in the early 1900s.

The basic idea of this fractal is as follows:

- Take an equliteral triangle, build another triangle in the middle of each side of the shape, the new triangle having a base length of 1/3 of the length of the side.
- Repeat
*ad infinitum.*

Turtle graphics are very handy in the implementation. The needed functions or methods are *Forward*: Draw a line in given amount of units forwards to the current drawing angle. The second is *Turn*: Turns the drawing angle in given amount of degrees.

With recursion we are now ready to go to build a snowflake based on the Koch curve.

Below is two videos of my implemantation:

..and the same with more iterations: