Is infinity 'larger' than 1?

Question

If infinity is not a number, how can it be larger than any number? A number is a position on the number line. A larger number is the position further on the number line. Infinity is not on the number line. I understand the definition of the "extended real number system", but it doesn't really answer how infinity can be put in a relation to a number, such as "larger", other than completely arbitrary without sufficient logic. Finally the definition based on the Cauchy sequences is also questionable, as such sequences are seriously challenged by people like Norman Wildberger, a Canadian prof. of math. at the University of New South Wales, Australia.

So what is the consensus on this forum, is infinity larger than 1?

Answer 1

In the extended real numbers, one DEFINES infinity having the order relations $ -\infty < x < \infty $ $\forall x \in \mathbb{R}$

Answer 2

Let me be very clear here, because there is a subtlety you have missed:

Infinity is not a real number.

Infinity is a number, in other contexts. For example, in the Extended Real Numbers, it is a number. This set is of a huge importance for subjects like measure theory and integration theory. In the Ordinals or in the Cardinals (used extensively in set theory), infinity isn't just a number, it is an entire range of numbers.

And yes, in all of these systems, infinity is greater than one.

Answer 3

It's not clear to me what your objection is. The extended real number system is defined to be the real numbers plus the two symbols $+\infty$ and $-\infty$; the relation $<$ on this system is defined so that $-\infty < x$ and $x < \infty$ for real numbers $x$.

Answer 4

I'm going to try to clear up your confusions, as you are not the only one with them.

What is a number?

Surprisingly, you can go through a full mathematics education and not once encounter a definition of "number". What you define is "Set" and "element of a set". These things are defined axiomatically by ZFC (there are alternatives though)

Some sets have common names, for example the natural numbers ($\mathbb{N}$), the real numbers ($\mathbb{R}$), the complex numbers ($\mathbb{C}$), hyperreal numbers ($*\mathbb{R}$), etc. Any element of such a set is commonly called a number. This is not a mathematical definition, just a common name.

However, the set of real numbers is well defined and thus so is the term "real number" (an element of that set). The same is true for the other examples I gave.

What is a relation?

Once we have sets, we put structures on them, extra information about the sets. Order relations are an example of such structure, and so is an operation like addition, or a concept of distance like a metric.

The general definition of a relation can be found here, and as you can see, the idea is the following. If I want to define a relation $R$ on a set $S$, I just have to say which elements of $S$ are in relation with each other, so for each pair $(a,b)$ I choose whether or not the are in relation with each other. If yes, we say $(a,b)\in R$, otherwise we say $(a,b)\notin R$. So in other words, a relation on $S$ is just a subset of $S \times S$

A special case of this concept is a partial order relation. Here we put extra demands on this relation. We demand 3 properties:

• $\forall a \in $S$: (a,a) \in R$
• $(a,b)\in R \text{ and } (b,a)\in R \implies a = b$
• $(a,b)\in R \text{ and } (b,c)\in R \implies (a,c)\in R$

Not all relations have these properties, but some do and we call them partial order relations. A set along with a partial order relation on it is called a partially ordered set or poset. We can verify that $\mathbb{R}$ along with "$\leq$" is a poset. It even makes it a toset which we can intuitively think about as a line.

Now for infinity

There are many sets that contain an element that we call infinity, but I will look at just one example: the extended real numbers $\bar{\mathbb{R}}$. What is this thing?

Well we start with the set $\mathbb{R}$ and another set with 2 elements that aren't in $\mathbb{R}$. These elements have no special role yet, but we will call them $\infty$ and $-\infty$. Now we define the set $\bar{\mathbb{R}}$ to be: $$\bar{\mathbb{R}} = \mathbb{R}\cup \{\infty,-\infty\}$$

Now we put on this set a relation "$\leq^*$". We say that $(a,b)\in \bar{\mathbb{R}}\times \bar{\mathbb{R}}$ is in the relation "$\leq^*$" if and only if: $$(a,b\in\mathbb{R}\text{ and } a\leq b)\text{ or } a = -\infty \text{ or } b = \infty$$ We can again verify that this makes $\bar{\mathbb{R}}$ along with the relation "$\leq^*$" a poset. (again even a toset)

The answer to the question

$\infty$ is not a real number as $\infty \notin \mathbb{R}$, but we can call it a number because it is an element of the extended real numbers $\bar{\mathbb{R}}$.

We can't say it is bigger then any real number using "$\leq$", but we can say that it is bigger than any real number using "$\leq^*$".

So in the end it all boils down to definitions. You might object and say that the concept of infinity already existed before these definitions, and you are right. These definitions just form a mathematical model for it, so that we can be precise about it, so that we know we are all talking about the same thing, and so that we can answer questions about it with certainty.

Answer 5

The "teens" are a set or list of numbers: 13, 14, 15, 16, 17, 18, 19. This set is not a number itself. Furthermore the range of numbers 13-19 (including the non-whole numbers) is not a number. And yet it is entirely natural and correct to say that the teens are greater than 1.

In the same way, Infinity is not a number, but it is greater than 1.

Answer 6

Is infinity larger than 1?

As other answers have said, yes, because infinity isn't a real number. Jens Renders's answer describes this well.

However:

Infinity isn't a number, just a description of a number

Saying that a number is infinite describes it, not defines it, much like saying that it's positive, negative, prime, composite, irrational, etc.. While all numeric assertions are technically just descriptions, descriptions like "$1$" are precise whereas descriptions like "prime" or "infinite" refer to more than just a single value.

In general, $\infty$ should be read as "a number that is infinite", which can concisely stated as "infinity" so long as we're careful to remember that that's shorthand for a class of numbers. It's probably more precise to say that $a{\in}{\infty}$ rather than $a={\infty}$, where the set $\infty$ is usually implicitly defined as the set of all numbers arbitrarily larger than all all non-infinite numbers described in the same context.

When less ambiguity is desired, systems like the hyperreal system provide a framework for more rigorous descriptions. However, in general you can invent your own systems for dealing with infinite values as long as they're consistent. The only real requirement is that all infinite values are, by definition, larger in magnitude than all real values, which are in turn larger in magnitude than all infinitesimal values.

When we're not being that explicit, statements about $\infty$ are limited by the amount of information that they contain. For example, ${\infty}-{\infty}$ is undefined because it basically just says "the difference between two infinitely large values"; and since the difference between two infinitely large values could literally be anything, depending on what the infinitely large values are, then their difference is undefined.

Answer 7

In the context of ultrafinitism, not only doesn't infinity exist, infinite sets such as the set of natural numbers $\mathbb{N}$ don't exist either. Wildberger has emphasized his support for this vision in many of his video lectures. This view is far more radical than finitism. It's not a good idea to on the one hand adopt Wilderger's views and on the basis of that question the way conventional mathematicians work with infinity, as in Wildberger's framework there is no such thing as infinity in the first place. You have to work within a well defined framework, so you have to either accept the conventional view or work within a well defined ultrafinitistic framework. In the former case, there is no problem as is pointed out in the other answers.

Answer 8

The other responses seem to have drawn out the main elements here:

• Infinity is not a real number: $\infty \notin \!R$
• +/- infinity are part of the extended reals system $\pm\infty \in \bar{\!R}$

However, we haven't really discussed the motivation for the construction of the extended reals system. I don't think this was put together to enable the arithmetic handling of infinity - in fact $\bar{\!R}$ is very limited in its algebraic properties when compared to $\!R$ - but to help handling certain limits and topologies, so I'm not sure it really helps with the use of infinity under discussion.

Going back to the OP, I would hesitate to talk about the infinity (concept) as being greater than 1, preferring instead only to use $\infty$ as part of the formal language of limits and the like.