>>11581740>the reals are constructedthe reals are constructed IN HINDSIGHT motherfucker. you don't begin with the construction of the fucking reals. you BEGIN with solutions to polynomial equations that force you to use fractional exponents as solutions thereby "creating" irrational numbers and "completing" the rationals.
Sets are created by polynomials. You do not begin with sets. You begin with polynomials. You can, in hindsight, begin with a set, but again, that is always in hindsight, after you have produced the set from the polynomial.
n + 1 > n produces the naturals
n + 1 = n produces the naturals with zero
n + 1 = 0 produces the integers
n*m = 1 produces the rationals
x^2 = 2 produces the irrationals
x^2 = -1 produces the complex numbers
and so on
THEN, once you have produced the sets, you can in HINDSIGHT observe that when you restrict yourself to only using certain solutions to certain polynomials (certain "sets"), you realize under this restriction some polynomials have no solutions for certain operations and that is when you begin creating the generalization of groups, fields, rings, and so on.
You do not begin with a fucking field. That is such a stupidly literal interpretation of mathematics, like you started with abstract algebra and have no idea where the generalizations came from.
>finite fields and the rationals are not generalizations of the real numbersI didn't say this. I said fields are a generalization of the reals. One of the cool things about generalizations is that you can find instances of the generalization that are not the same thing as the original thing you generalized from. You generalize the properties of a field from the reals, and then re-instantiate. Finite fields end up being an instance of a field when considering modular systems.
Do you not understand how abstraction works? You don't begin with a field. It's an abstract concept, it has to be derived from something.
