Are there any axiomatic or mathematical paradoxes with assuming that any number divided by zero equals that number multiplied by infinity, if we also assumed concepts such as:
$\infty+\infty$$=2\infty$
$2\infty-\infty=\infty$
$\infty-\infty=0$
$\frac{\infty}{\infty}=1$
ect...
Also, $\frac{1}{0}=\infty$
$-\frac{1}{0}=-\infty$
and
since $\frac{1}{0}=\infty$, $\frac{1}{\infty}=0$ therefore $\infty \cdot 0= \frac{\infty}{\infty}=1$
Because if we did then $\frac{x}{0}=x\cdot \infty$ therefore $\frac{0}{0}= 0\cdot \infty$--> $\frac{0}{0}=1$. Not only do we get rid of the strange {} empty set nonnumber as well as the bizaar $\frac{0}{0}$ omni-number thing, but we just made math a lot cleaner by deciding that $\frac{x}{x}$ ALWAYS equals $1$. Is there any paradoxes with these definitions?
Let's say we play along. We lose a lot of nice properties of multiplication and addition, like associativity and distributivity.
For instance, $2\cdot(0\cdot \infty)=2\cdot1=2$, but $(2\cdot 0)\cdot\infty=0\cdot\infty=1$.
Another example: $(1-1)\cdot\infty=0\cdot\infty$, but multiplying out the brackets we get $\infty-\infty$, which could be anything.
In the end, it's best to let $\infty$ stay where it belongs, which is nowhere near arithmetic.
If you really want to see how to make this work, look into ordinal arithmetic or non-standard analysis. If you're really brave, try to take on the surreal numbers.