Is infinity equal to infinity?

The question "Is infinity equal to infinity?" is more nuanced than it appears at first glance. In mathematics, infinity (∞) is not a number but rather a concept representing something without any limit or end. The idea of equality becomes complex when dealing with infinities.

Different Contexts of Infinity:

- Cardinality: When dealing with sets, infinity can refer to the size or cardinality of a set. Georg Cantor demonstrated that there are different sizes of infinity. For example, the set of natural numbers (1, 2, 3, ...) is countably infinite, denoted by ℵ₀ (aleph-null), while the set of real numbers is uncountably infinite and has a larger cardinality, denoted by 𝔠 (continuum). In this context, different infinities can be unequal.

- Limits: In calculus, infinity often arises as the limit of a function or sequence. Here, we are concerned with what happens as a variable grows without bound. For example, as x approaches infinity in the function f(x)=1/x, the limit is zero. In this context, we talk about "approaching infinity," not infinity as a concrete value.

- Projective Infinity: In some fields like projective geometry or complex analysis, we use a notion of a single "point at infinity." This allows us to work with a more compact way. This concept often merges "positive infinity" and "negative infinity" into a single concept.

When Infinity Is Treated as a Quantity:

- If we consider a simple mathematical operation like adding infinity to infinity (∞ + ∞), the result is still infinity. However, this doesn't mean that all infinities are the same.

- The operation ∞ - ∞ is undefined. This is because if you consider that infinity is just a very large number, two very large numbers might not be the same (infinity is not a number!).

Conclusion:

- In summary, when we speak of "infinity" being equal to "infinity", it often implies the same type of infinity in a specific context. So, if we are both referring to the cardinality of the natural numbers, yes, that is equal to another set of natural numbers. But the cardinality of the real numbers is still a larger infinity.

- However, not all infinities are the same. When considering cardinality of sets or limits, there are different "sizes" or ways of approaching infinity, and it would be incorrect to assume they are all equal.

- Therefore, it is crucial to know the context when comparing infinities. The question "Is infinity equal to infinity?" can be answered both yes and no depending on the specific mathematical usage or concept of infinity being referred to.

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