Is ∞ considered defined?

Question

$\infty$ (Infinity) is not a number, but infinity is considered to be defined, right?

There are expressions in mathematics such as: $\frac x0,0^0, \frac\infty\infty,$ which are not defined because they do not have a certain place on any domain, and sometimes because there is no single value that satisfies all functions yielding such expression in a limit.

How can infinity be defined?

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Maybe the argument would hinge on how 'defined' is defined?

Perhaps this requires a circular(infinite) argument?

Or is my assumption that infinity is considered to be defined; false?

Answer 1

$\infty$ is just a symbol to denote such quantities which are bigger than any real quantity and nothing else.

Answer 2

We know what is something finite. In the context of the real numbers it simply means that the value is "a real number"; in the context of sets it means "There is a natural number $n$ such that the set has a bijection with $\{0,\ldots,n-1\}$."

Something is infinite if it is not finite. There is circularity, once we know what it means to be finite we simply define infinite as its negation.

If numbers measure some sort of quantity, and we can identify the real numbers with measuring "length" then an infinity is something longer than any finite length. In the context of set theory there are several notions of infinities (ordinals and cardinals) which extend things which are measured by the natural numbers. However where in real analysis there is pretty much one notion of infinity, in set theory we have a whole class of numbers which represent infinite objects.

Answer 3

I assume that you're talking about the $\infty$ that appears in analysis and calculus, when calculating limits. In general, $\infty$ is just a symbol, however it intuitively looks like to the countable infinite $\omega$ cardinal. For instance, when you think about Riemann integrals and you tends a partition $P$ to infinity, $P$ is always finite , then it's at most countable. When you take a limit walking on the real line and it goes to $\infty$, then it just means that it's unlimited and since you're computating it by finite approximations, then it intuitively "looks like" $\omega$ (but, of course, it's not $\omega$).

Another way of thinking about infinity in analysis is to think about it as the element that makes the real line compact (the Alexandroff compactification). If you're thinking about limits over $\mathbb{C}$, then I think that the Riemann sphere is the more intuive way of seeing $\infty$.