Parting the Sea

The Set.

What a wonderful construct a set is, simply a collection of something which only has to be distinct - and nevertheless, “It can be viewed as both the foundation upon which nearly all of mathematics can be built and the source from which nearly all mathematics can be derived.”

You don’t like sets? They remind you of school? Annoying, deadly boring?

Think again, you are grown up now.

Just focus. Forget about native or axiomatic theories, forget about paradoxes, forget about cardinality, unions or intersections: focus on the set.

A collection of something distinguishable.

How can such a simple concept be so important to mathematics? Something put together and voila, it’s a set by the touch of a magic wand?

How about a glass of water? Do the molecules build a set? No, they don’t, just because we can’t distinguish one H₂0 from the other.

But if?!? Yes, if we could, it would be a set, really.

For nothing more than separating the water from the rest of the world into a special glass and be able to “call them by their names” – we would have created a set.

Imagine that!

Imagine, how much you know about things, if they form a set, because the fact, that the things in a set have to be unique proves you a very, very important point: you have to be able to measure something, some distinctive value like identity to distinguish one set member from the other.

But that’s just a part of the information you got from sets. Because the simple little word “collection” shows you something else: You must be able to decide, whether something belongs to that collection or not.

Again, you must be able to measure something to decide, whether it’s part of the collection.

Look at our water glass of magically distinguishable H₂0-molecules. “Water in my glass” defines the collection. So air or children or cars don’t fit with the definition – and water? Water has to be in “my glass” to belong to my set. My glass can be defined by its space-time curve or by form or matter, the keys you choose to measure can be anything: They just have to be unique...

and stable.

Remember? It wouldn’t make sense to use changing values as identity, because after the change, how could you track the changes?

That’s the reason, why a set is so important to mathematics – and to each and every information processing system or procedure: it defines identity and properties...

it defines measurability: the precondition for mapping systems.

It’s the “zero-point-information”, the point, where you can stand to structure the whole wide world in two distinct parts: inside the set, outside the set.

And inside the set, you know, there is identity – therefore you can track movings, you can follow events, you can construct relations, you can map and store, retrieve and compare.

A set creates order in a random world.

That’s the magic.