Is infinity 'larger' than 1? [closed]

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.

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.