Sets and Structures
Set Theory is the Machine Language of mathematics. It's a very low level language, close to the machine, so to speak. In set theory with define really primitive operations on even more primitive types data. Set theory in a way isn't as illuminating as high level mathematics but in another way, it is the light that makes all of math possible.When I first came across set theory I found it really quite a strange thing. I couldn't quite wrap my head around why it was there at all, why the symbols looked so weird and what the fuss was about at all. I had no teachers, just some books and the internet at large. It was a mess.
I hope that in the following pages I can do something to reduce the mess. Although I suspect that like everyone else, I will just be adding to it.
What Sets do
There are going to be parts of this site which are rigorous and parts that are not, parts that are more explanatory, especially from the viKid sense of view. So bear with me here.You will find the Zermelo-Frankel Axioms of Set theory below. Which are great, a rigorous foundation that is good to understand but yet, on looking at them again utterly lifeless.
What I want to do here is explain the need for set theory at all. I want to understand why we need such a theory.
The point of set theory, the whole point of set theory in my opinion is that it studies the idea of what 'an idea' is.
It is the ontological basis for all ideas in terms of themselves. It is recursive like most modern theories, but its importance is greater than that.
In set theory. An idea is defined by the contents of the idea and nothing else.
Both parts are important, 1. the contents and 2. nothing else.
Let me repeat. An idea is its contents and nothing else.
So what happens when we have only nothing, that is, no contents.
This is the special case, the root of all set theory. It is called the empty set.
Otherwise represented as {}
What Structures are and what they do
Structures are essentially Sets with Rules. The rules, often called operations allow us to transform (or create) one set into another.For example, the + operator is one such rule.
1 + 1 = 2.
All we have done here is operate on a set (1) and produced another (2).
And yes, in set theory, numbers are sets themselves.
And what are these Rules? Well they are sets too! So in many senses, structures are the study of how sets interact to produce new sets.
Or how ideas interact to produce new ideas. This is genesis, it mathematics.
Set Relations
Source, Links & Reading Lists
- Topic List @ Wikipedia
- A Bunch of Set Theory and a Bit of Logic by Ronald Garcia
- Metamath
- Modus Ponens & Tollens
- Affirming the Consequent @ Wikipedia
| 08dec16 | admin |