Is infinity a real number? [duplicate]

Is infinity a real number?

If not, why not?

I want some very good arguments.

Thanks.

$$\rightarrow\leftarrow\Huge\Huge\Huge\boldsymbol\infty$$

Solution 1:

No. If you look up the definition of the real numbers, you will not find any of its elements called "infinity".

However, the extended real numbers has two numbers called $+\infty$ and $-\infty$, which become the endpoints of the number line in the extended reals.

There are other structures that have an element or places named "infinity" that are similar but often different.

There are other situations where there are "infinite" objects, although we would never use the noun "infinity" to refer to them; e.g. the infinite cardinal numbers and ordinal numbers, or the unlimited hyperreal numbers.

Solution 2:

Infinity isn't a real number by definition. This definition is sensible because adding $\infty$ to $\mathbb{R}$ would break its field structure, and the fact that $\mathbb{R}$ is a field brings a lot of nice properties to real numbers