- Sep 2015
There is a mathematical object called the surreal numbers. They're a little obscure in the sense that mainstream math doesn't use them much as far as I know. But they're a very interesting curiosity.
Note however that while the surreals do contain the usual ordinals and cardinals, the ordinals and cardinals of set theory come with certain operations which are not the same in surreal number theory. For example, in ordinals and cardinals we have $1 + \omega =\omega$. But in surreal numbers this is no longer true since we demand it to be an ordered field.There is a mathematical object called the surreal numbers. They're a little obscure in the sense that mainstream math doesn't use them much as far as I know. But they're a very interesting curiosity.
They allow all the arithmetic operations, addition, subtraction, multiplication, and division (except by 0), but they include infinite and infinitesimal numbers. We would call them a field, but a field is defined as being a set; and the surreal numbers are too big to be a set. They're a proper class. So they're called a Field, with a capital-F to denote the fact that they are a proper class that satisfies the field axioms.
I wish I knew more about them, in particular whether they include the usual transfinite numbers of set theory or some other kinds of infinite numbers (as in the hyperreals).
The surreals are the absolute largest class of things that can be called numbers, that satisfy the axioms for an ordered field. I think that's even a theorem if I recall. They're provably the largest mathematical object satisfying the ordered field axioms.
You know that division by any real number is defined.Please allow me to restate my question:
Is there a greatest (or least) number which may act as a divisor?