Is $\infty$ undefined? [duplicate]

I am confused at $$\infty$$ in a lot of ways. First, we sometimes say that $${1\over 0}=\infty$$. That gives us these confusing calculations.$${1\over0}=\infty$$ $$1=0*\infty$$ $$2=(0*2)*\infty$$ $$2=0*\infty$$ $$1=2$$ But certainly, this is not true. $$1\ne 2$$. What is happening?
Another confusion: let's say $$x=x+1$$. This is never true. But with $$x=\infty$$, it works.
Third confusion: let's say $$2x=x$$. Subtracting both sides by $$x$$ gives $$x=0$$. But what about $$\infty$$?
Fourth confusion: $$x-x=0$$ is always true, but not for $$\infty$$. $$\infty-\infty$$ is undefined.
What is actually happening??! Any help will be very appreciated.

• $\infty$ is not a number. You have shown that calculating with it leads to contradictions. Division by zero is not allowed. Surely, some users will come with the "extended real line", but this does not make division by zero meaningful. Better avoid it ... Feb 6, 2019 at 18:14
• Yes, that's all very confusing. In fact, you produced a reductio ad absurdum against the very claim that $\frac{1}{0}=\infty$. Feb 6, 2019 at 18:14
• However, to determine limits, setting $1/0=\infty$ can be useful, but never consider this equation to actually hold. Feb 6, 2019 at 18:16
• Related to 1/0 = infinity Feb 6, 2019 at 18:18
• Infinity isn't something you can operate with. For starters, you can't define $\frac10$ to be a number because the operation itself is unperformable. Diving by zero has no meaning, physical or abstract. Infinity isn't a number Feb 6, 2019 at 18:39

You may calculate with $$\infty$$, in a certain sense. For that purpose let $$\infty$$ be an object which is not a real number, e.g., $$\infty:=\{\mathbb R_+\}$$. (You may define $$-\infty$$ as $$\{\mathbb R_-\}$$.) Declare for any real number $$r$$ the set $$N_r(\infty\}:=\{x\in\mathbb R\mid x>r\}$$ as a neighbourhood of $$\infty$$.
Apply the usual definition of a limit to define if as sequence has $$\infty$$ as limit: in each neighbourhood of $$\infty$$ there are almost all elements of the sequence.
Now prove that for the limit of sequences some of the usual rules are valid, for example $$\infty\cdot\infty=\infty$$, which means the product of two sequences which limit is $$\infty$$ has limit $$\infty$$ as well. Also $$\infty+\infty=\infty$$ as well as $$\infty\pm a=\infty$$ for any sequence with limit $$a\in\mathbb R$$ and $$|a|\cdot\infty=\infty$$ for $$a\neq0$$. $$a/\infty=0$$ is another one for non-negative $$a$$.
On the other hand there are so called "indetermined expressions" as $$0\cdot\infty$$ or $$\infty-\infty$$, which can be any number (including $$\infty$$).