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Is infinity merely a symbol?

I see this type of statement crop up a lot everywhere.

\[\text{"Infinity is not a quantity, nor a number. It is a symbol."}\]

I used to agree wholeheartedly with the above sentiment; however, in recent times, I have come to disagree. There is a way to formalise \(\infty\) as a number; many ways, actually.

You may have heard of the hyperreals \(\mathbb{R}^{*}\). A particular subset of \(\mathbb{R}^{*}\) contains those numbers strictly greater than all numbers of the form \(1, 1+1, 1+1+1, \ldots\). That is certainly one way to bring the concept of infinity into a formal context.

Another way is to define it as \(1/0\). Used purely in the context of limits and allowing for substitution (ie \(x \to \pm \infty \equiv 1/x \to 0^{\pm}\)), this is also a legitimate way to rigourise infinity. I even saw this in a respected book on analysis.

There are still many other ways to define infinity; some are compatible, others not so. We have cardinals, ordinals, surreal numbers: a plethora of axioms to choose from. I simply picked the above two examples for the sake of brevity and intuition.

The problem most people have comes in when you have indeterminate forms, like or . The two seem at first to be and respectively, from rules of algebra, and then you quickly jump into the sobriety of rigour and note that rules of algebra only apply to fields compatible with those rules. The monster group acts very much differently from the additive integers.

So you dismiss it as nothing more than a symbol, placeholder, idea.

Sure it is. But it's really so much more.

The reason we reckon them is simply because - I quote, if memory serves me right, Mr Brian Charlesworth - not all infinities created equal. In all formalisations of infinity, we don't treat every infinity the same. So, \(\displaystyle \lim_{(x,y) \to (\infty , \infty)} \frac{x}{y}\) has no real answer because you have many different paths of \(x \to \infty\) and \(y \to \infty\) to take. (That's what I meant by different infinities.) Or rather, it is not defined since it can be as much equal to \(1\) as \(4.27-e^{\pi}\) as the cat you saw on the street this morning. (You didn't see a cat, you say? Exactly.)

Stop treating infinity like it's just a concept. It's not. As a number, it has real (pardon the pun) value and is very important in maths and science.

Hair tussled (inspired) by Ramiel To-ong's solution on this, which sounds very wishy-washy, swishy-swashy. But eh, I don't know.

Note by Jake Lai
2 years ago

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Would you call a matrix a "number"? At least a matrix is an object, and so is infinity--infinity is an object. See wiki

Mathematical Object

When one says that "infinity is a number", that's already proposing an axiom about it. Do we need to do that, and why? By axiomizing that infinity "is a number", you could be precluding any other mathematical meaning it may have. What if infinity could take the form of some kind of a matrix in some multi-dimensional abstract space? We are already aware that there can be more than one kind of infinity, with properties that they do not all share.

Mathematicians even today are still in dispute about the nature of infinity. See Scientific American article

Dispute over Infinity

The very fact that even after centuries of mathematical thought about infinity there's still dispute about this indicates that it's premature to start axiomizing that infinity is just "a number", and that there's only one kind of infinity. Let's not shut ourselves from other forms infinity can take. At minimum, we should assign specific symbols for different kinds of infinities, and one for the type of infinity that we can associate with reals--along with a cargo ship load of definitions and axioms to go with it. See Cantor.

Let me offer a kind of an analogy---when you marry someone, you're really marrying that person's family and culture. Mathematical objects do not exist in isolation, they exist by definitions and axioms that describe and support it. The concept of infinity is such an example. Michael Mendrin · 2 years ago

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Yes, there are ways to formalize \( \infty \) in the extended reals. However, that doesn't mean that the general populace would call it a number.

If you are interested, I would encourage you to read up on order cardinal arithmetic, and explain why:
1. \( \aleph_0 < 2^{ \aleph_0 } \).
2. \( 1 + \aleph_0 \neq \aleph_0 + 1 \).
3. Does \( \aleph_0 + \aleph_0 = \aleph_0 \times \aleph_0 \)? Why? Calvin Lin Staff · 2 years ago

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Is Jake Lai merely a name? Is http://Brilliant.Org merely an URL? Agnishom Chattopadhyay · 2 years ago

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if we define infinity as 1/0 then we may as well write it as n/0( n belongs to any real number) then if we find 0*infinity ,each time you get the real number itself.This occurred to me quite a time ago it is yet a fancy.Any views? Raven Herd · 2 years ago

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