@Nihilist Virus , I don't want to wear you out with endless questions, but if it is easy to explain and you have the inclination I was curious how the irrational numbers are defined? I can guess that the rational numbers follow the example of the integers except that division is used instead of subtraction. But the irrational numbers seem that they would be trickier.
There are two ways. √2 can be defined as the positive solution to x²=2. Also, √2 can be defined as the limit of the infinite sequence of rational numbers: 1, 1.4, 1.41, 1.414, ...
But then there are things like π. π is transcendental, which means it cannot be the solution to a polynomial with rational coefficients. We can still define it as the limit of an infinite sequence of rational numbers: 3, 3.1, 3.14, 3.141, ... The only other way to define a transcendental number like π is simply to discover it, like how we discovered it is the ratio of the circumference of a circle to its diameter.
I totally understand if it is too difficult to explain to somebody without the background.
You seem to be doing great. You've got a math background.
It's interesting how the integers are defined relative to the natural numbers, and I suppose that allows mathematicians to say things about the size of the set of natural numbers relative to the size of the set of integers (squaring and then accounting for the equivalence sets).
Well...
N and
Z are actually the same size, even though
N is a proper subset of
Z. So even though {0,1,2,3,...} is fully contained in {...,-3,-2,-1,0,1,2,3,...}, there exists a bijection between them (a one-to-one mapping which goes both ways). For example, take f:
N→
Z such that f(0)=0, f(2n)=n, and f(2n-1)=-n for n>0.
The nested sets with nil for defining the natural numbers is interesting, but the reasons aren't clear to me - other than nil and a set being a simple starting point for defining things.
But I probably am wearing you out, and I don't want to do that.
Don't worry, you're not wearing me out. I do math all day every day, lol.
The dirty truth of mathematics is that it is nothing but assumptions, definitions, and the conclusions that follow. The assumptions are typically referred to as axioms or postulates. This relates to the
Münchhausen trilemma and nihilism.
Why the empty set? We used to define math in terms of line segments, but then Cantor modernized mathematics with his set theory. Mathematics is defined in terms of the empty set. We can define rules all day long but at some point we must assert the existence of something via an axiom, and the empty set is the most simple and basic thing to assume exists. From there, we build sets which contain the empty set as an element.
Speaking of elements, here's a joke for you. What is an element? It is a member of a set. What is a set? It is a collection of elements. Mathematicians don't like to grasp this horn of the Münchhausen trilemma and we've actually decided to leave the set formally undefined. So while mathematics is built upon the innocent notion that you are allowed to group things together into sets, the formal language has no definition for a set. An element is a member of a set, and a set has no definition.