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

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

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

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

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

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 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.

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).

Every number can divide. For example $\aleph_0$ divides $\aleph_0$. But it doesn't divide $1$.What is the largest set or number which can divide?

Same thing with the number 2, it divides 2 and 4, but not 3.

The largest number which divides every other number is 1.

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

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?

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.

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.