The reason I ask this is that I've discovered that, even though they don't satisfy all field axioms, there are sets called projectively extended real number line and Riemann sphere, which are ℝ∪{∞} and ℂ∪{∞} where division of every nonzero number of the set by 0 is defined as ∞. However, the two sets' arithmetic operations aren't total and some operations are left undefined. These include: ∞+∞, ∞-∞, ∞·0, 0·∞, ∞/∞, and 0/0. My question is that if there is, or could be a field-like set that can also define the results of these operations. And could there be a logical definition for those operations, especially 0/0?

A different approach than that of the wheel structure(s) in E. Joseph's answer is given by considering the linear/vector subspaces of the plane (e.g. $mathbb R^2$), regarded as binary relations and with appropriate operations.

Motivation

One way to look at numbers is as linear operators: The number $r$ corresponds to the function given by $f(x)=rx$. This gives us a new lens through which to examine things, especially multiplication and multiplicative inverses.

Given $f(x)=rx$ and $g(x)=sx$, then $fcirc g$ sends $x$ to $(rs)x$, so multiplication arises as composition of functions. Similarly, for $rne0$, the inverse function of $f(x)=rx$ would be $f^{-1}(x)=left(frac{1}{r}right)x$.

Both of these ideas can be understood in more generality as composition of relations and inverse relations (sometimes called converse relations). So if we select a slightly broader class of relations than "1-d linear operators", we can embed the numbers in a larger structure. By thinking of the graphs of these operators/relations, one "broader class" would be all of the subspaces of the plane.

Setup

Old and New Elements

For each number $r$, there is a corresponding subspace $[r]={(x,y)mid y=rx}$. So the numbers can be viewed as non-vertical lines through the origin with the number as its slope.

There are three other subspaces of the plane:

1. The vertical line ${(x,y)mid x=0}$, which we will denote by $infty$.


2. The origin ${(0,0)}$, which we will denote by $bot$.


3. The whole plane, which we will denote by $top$.

The operations

There are four particularly important operations on numbers: the unary operations for negation (aka minus) and reciprocals, and the binary operations of addition and multiplication.

All of these can be understood in terms of the corresponding subspaces, without appealing to the nonlinear structure directly:

1. The negation of $[r]={(x,y)mid y=rx}$ should be ${(x,y)mid y=(-r)x}$, which is ${(x,y)mid (x,-y)in [r]}$.


2. If $rne0$, the reciprocal of $[r]={(x,y)mid y=rx}$ should be ${(x,y)mid y=(r^{-1})x}$, which is ${(x,y)mid (y,x)in [r]}$.


3. The sum of $[r]={(x,y)mid y=rx}$ and $[s]={(x,y)mid y=sx}$ should be ${(x,y)mid y=(r+s)x}$, which is ${(x,z)mid exists y_1,y_2: (x,y_1)in [r] land (x,y_2)in [s]land y_1+y_2=z}$.


4. The product of $[r]={(x,y)mid y=rx}$ and $[s]={(x,y)mid y=sx}$ should be ${(x,y)mid y=(rs)x}$, which is ${(x,z)mid exists y: (x,y)in [s] land (y,z)in [r]}$ (the order is inspired by composing the linear functions).

But those latter characterizations work just as well for other relations, so we will take those as general definitions:

1. $-R={(x,y)mid (x,-y)in R}$


2. 
   $R^+={(x,y)mid (y,x)in R}$ (we might call this pseudoinverse)


3. $R+S={(x,z)mid exists y_1,y_2: (x,y_1)in R land (x,y_2)in Sland y_1+y_2=z}$


4. $R*S={(x,z)mid exists y: (x,y)in S land (y,z)in R}$

In particular, we can look at $[0]^+$ (even though $frac{1}{0}$ does not define a number) and evaluate any of these operations on the three subspaces that do not correspond to numbers.

0/0?

We have not defined division, but for numbers $r,s$ with $rne0$, $[s/r]=[s]*[r]^+=[r]^+*[s]$. Therefore, $0/0$ could be interpreted as either $[0]*[0]^+$ or $[0]^+*[0]$.

Note that $[0]^+=infty$ (if you reflect a horizontal line about $y=x$ you get a vertical line), so the question reduces to the value(s) of $[0]*infty$ and $infty*[0]$.

For $[0]*infty$, the only input $infty$ allows is $0$ and the only output of $[0]$ is $0$, so this is just the origin: $bot$.

For $infty*[0]$, $[0]$ sends all inputs to $0$, and $infty$ sends $0$ to all outputs, so this is the entire plane: $top$.

Operation Tables

For completeness, we can show all of the results of these operations. Below, $r$ and $s$ represent any nonzero number.

Negation

begin{matrix}X: & [r] & [0] & infty & bot & top\
-X: & [-r] & [0] & infty & bot & top
end{matrix}

Pseudoinverse

begin{matrix}X: & [r] & [0] & infty & bot & top\
X^{+}: & [frac{1}{r}] & infty & 0 & bot & top
end{matrix}

Addition

Note that addition of these subspaces is commutative since addition of numbers is:
begin{array}{r|ccccc}+ & [s] & [0] & infty & bot & top\hline{}[r] & [r+s] & [r] & infty & bot & top\{}[0] & [s] & [0] & infty & bot & top\infty & infty & infty & infty & infty & infty\bot & bot & bot & infty & bot & infty\top & top & top & infty & infty & topend{array}

Multiplication

Since Multiplication is not commutative, $R*S$ will be the entry with the row $R$ and column $S$:
begin{array}{r|ccccc}
* & [s] & [0] & infty & bot & top\hline
[r] & [rs] & [0] & infty & bot & top\{}
[0] & [0] & [0] & bot & bot & [0]\
infty & infty & top & infty & infty & top\
bot & bot & [0] & bot & bot & [0]\
top & top & top & infty & infty & top
end{array}

Source

None of these ideas are my own. I first saw this in the Graphical Linear Algebra blog (though there it's wrapped up with a discussion of, well, graphical linear algebra). The most relevant entry is Keep Calm and Divide by Zero, but the following two entries contain interesting context as well. Given his work on graphical linear algebra, this approach may have been discovered by Paweł Sobociński.

There is a structure called a wheel, whose purpose is to define division by $0$. More specifically, there is $x/0$ for $xne 0$ and $0/0$ in a wheel, and those two elements are not the same.

It is used for formel computations in computers.

You can find more informations there (and how it is formally constructed).