# Cardinality of real #s

#### romsek

Math Team
What is the largest set or number which can divide?
It's going to depend on whether you are talking actual division or finding limits

We divide by infinity all the time when we find limits. Or at least by some variable that tends to infinity.

topsquark

#### Maschke

What is the largest set or number which can divide?
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.

topsquark

#### [email protected]

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.
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 theorem that the surreal numbers in fact contain every other totally ordered field (not with capital f). So the hyperreals are definitely contained in the surreals, but not canonically. The embedding probably depends crucially on the axiom of choice. While the transfinite numbers are very canonically embedded in the surreals (aside from the operations).

#### [email protected]

What is the largest set or number which can divide?
Every number can divide. For example $\aleph_0$ divides $\aleph_0$. But it doesn't divide $1$.
Same thing with the number 2, it divides 2 and 4, but not 3.

The largest number which divides every other number is 1.

Loren and idontknow

#### Loren

The operation 0/0 is called indeterminate. Does its "set of outcomes" include all real numbers?

#### mathman

Forum Staff
What is the largest set or number which can divide?
"largest" is strange in this context. There is no largest number. Division by other than numbers needs to be defined.

#### Loren

Please allow me to restate my question:

Is there a greatest (or least) number which may act as a divisor?

#### romsek

Math Team
Please allow me to restate my question:

Is there a greatest (or least) number which may act as a divisor?
You know that division by any real number is defined.
You also know that the reals increase without bound.

So you know there is no greatest number that may act as a divisor.
Since you can divide by negative numbers as well there is no least number.
Maybe you mean is there a least magnitude number.
No. Basically the set of possible divisors, over the real numbers, is the non-zero reals.
This is an open set. It has no minimum or maximum.

topsquark and Loren

#### Loren

Forgive my reposting, but I feel that this is important:

The operation 0/0 is called indeterminate. Does its "set of outcomes" include all real numbers?

#### romsek

Math Team
It's tough to answer this, as 0/0 isn't really a thing. I would say it has no defined set of outcomes. You can't divide by zero.

Suppose you were to try and use limits to determine what this quotient might be in the form of say

$$\displaystyle \lim \limits_{x,y\to 0} \frac y x$$

In this case, you can set $$\displaystyle y = c x,~c \in \mathbb{R} \Rightarrow \lim \limits_{x,y\to 0} \frac y x = \lim \limits_{x\to 0} \frac{cx}{x} = c$$

So yes, in some non-rigorous sense that I don't even think is legitimate at all, you can make the "set of outcomes" of 0/0 any real number you choose.

Loren and topsquark