…is always light

# Tag Archives: empty set

## Is the Empty Set a set Part II

Let us assume again, that a person has a collection of 100 books. The books are in a shelf in way that between some books there is an empty space. Probably we can say, that the empty space is part of the collection at least visually considering the whole, although that empty space is something different than a book.

In general an empty space can be filled with a book, so that the book is surrounded by an empty space. If no empty space existed, a book couldn’t be taken away from the collection of the books (or in this example put a book to the shelf).

Therefore empty space is part of every collection of books..

## Do we need a new ”set” – going beyond the empty set

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…

## Zero, none, nothing, the empty set and the infinite

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.

## Is the Empty Set a set?

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? 🙂

## About zero and the empty set

If zero (0) is added to any real number a, as a result we get a: a + 0 = a. What now was added to a? Nothing? I would say wrong. In some sense.

Namely from our friend, the set theory, point of view set A = {0} is not empty, there is something, namely number zero. If one would say, that 0 is nothing, in set A weren’t anything. In our case there clearly now is something, element 0.

Somehow philosophically 0 isn’t in same extent ”nothing”, that it would lead from view of set theory as the only element in the set to same state as the empty set ({} or ∅), that is so empty, that there simply is nothing; the empty set is more ”nothing” than 0. As  a number, zero is considered as neutral element in some cases. But it obviously is more… What?

As to empty set, more philosophical question is, does the empty set contain itself – and is it then empty.

Emptyness and nothingness have their differences.

Let us imagine an empty room where there is four walls and a roof. Emptyness gives there space. And also this emptyness, space, has many meanings; if the room has only little space you would probably feel quite uncomfortable there.

In music the fact the there isn’t a note is known as pause. In this case emptyness in the notes gives rhythm to the music, without this non-existence of a note (nothingness from point view of sound?) we would’n have music as we know it.

In speech silence, a pause, can give one some kind of power to the speech itself.

Emptyness and nothingness really are powerful from their beings!

I consider ”nothing” as something that doesn’t exist. Still it does.

Update! (16/5/2016)

In order to express all this more precisely zero is interpreted as an positive integer, but not genuinely positive integer; ”nothing” can’t be a positive integer. As to 0 + a = a, to number a is added an positive integer – something else than ”nothing”. This makes me consider the empty set being ”more nothing” than number zero. In this particular case to a is added a neutral element zero – not ”nothing”, in some sense…

It seems we’re pushing the limits of semantics of ”nothing” to new boundaries… Anyway it is essentially something else than ”emptyness”. See my post on creation.

The question now is: Is the rank of {} zero (0)? Do we need to “divide” zero; do we need a new “zero”?