I’m not entirely sure I understand your comment, but the fact that there are more real numbers than natural numbers can be readily shown using something called cantor’s diagonal argument. It goes something like this:

Suppose the set of real numbers and natural numbers had the same size. Then we could write down an infinite list, where each line represents some real number written in it’s decimal representation. So something like

1: 3.14159265
2: 1.41421356
3: 0.24242424
...

This list goes on forever. We will now construct a new real number r as follows: The first number after the decimal point of r shall be different from the first number after the decimal point of the first number in our list, the second shall be different from the second decimal of the second number on the list, and so on (the name diagonal argument comes from this, we consider the entries on the diagonal from top left towards the bottom right).

The key point now is that this constructed real number is different from every single number on the list: After all, we have made sure it differs from each number on the list in at least one place. Therefore, it is impossible to write down the real numbers in such a way that each real number gets its own natural number: There are simply too few natural numbers for this. In particular, there are at least two different “sizes” of infinities.

Infinity cannot be divided, if it can then it becomes multiple finite objects.

It really depends on what you mean by infinity and division here. The ordinals admit some weaker forms of the division algorithm within ordinal arithmetic (in particular note the part about left division in the link). In fact, even the cardinals have a form of trivial division.

Additionally, infinite sets can often be divided into set theoretic unions of infinite sets fairly easily. For example, the integers (an infinite set) is the union of the set of all integers less than 0 with the set of all integers greater than or equal to 0 (both of these sets are of course infinite). Even in the reals you can divide an arbitrary interval (which is an infinite set in the cardinality sense) into two infinite sets. For example [0,1]=[0,1/2]U[1/2,1].

Therefore there cannot be multiple Infinities.

In the cardinality sense this is objectively untrue by Cantor’s theorem or by considering Cantor’s diagonal argument.

Edit: Realized the other commenter pointed out the diagonal argument to you very nicely also. Sorry for retreading the same stuff here.

Within other areas of math we occasionally deal positive and negative infinities that are distinct in certain extensions of the real numbers also.

If infinity has a size, then it is a finite object.

Again, this is not really true with cardinals as cardinals are in some sense a way to assign sizes to sets.

If you mean in terms of senses of distances between points, in the previous link involving the extended reals, there is a section pointing out that the extended reals are metrizable, informally this means we can define a function (called a metric) that measures distances between points in the extended reals that works roughly as we’d expect (such a function is necessarily well defined if either one or both points are positive or negative infinity).