Rebut this claim by adding onto what I said. I will firstly show what the person said, and at the end you will add onto my claim to rebut the person effectively. Use as much proof as possible

Here is the person's proof:

The argument that the decimal infinity is mathematically proven to be larger than infinity with merely integers is a common misconception that is often misunderstood. While it is true that there are more real numbers than integers, the concept of infinity complicates this comparison in a way that renders the argument invalid.

Firstly, the idea of "counting to infinity" with integers is a flawed one, as infinity is not a number that can be counted to in the traditional sense. It is a theoretical concept that represents an unbounded quantity. Therefore, the comparison of counting integers to infinity versus counting decimals to infinity is not a meaningful one.

Moreover, the concept of infinity is not a well-defined mathematical concept. There are different types of infinity, and some infinities are larger than others. The argument assumes that there is only one type of infinity, and therefore that decimal infinity is larger than integer infinity. However, this is not the case.

The cardinality of the set of real numbers, which includes decimals, is known as the continuum, and it is larger than the cardinality of the set of integers. However, this does not mean that the decimal infinity is larger than integer infinity. In fact, both infinities have the same cardinality, known as aleph-null. This means that there is a one-to-one correspondence between the integers and the real numbers, despite the fact that there are "more" real numbers than integers.

Finally, the argument that it is impossible to make the integer infinity larger is also flawed. Mathematicians have developed concepts such as transfinite numbers and cardinal arithmetic, which allow for the manipulation and comparison of different infinities.

In conclusion, the argument that the decimal infinity is mathematically proven to be larger than infinity with merely integers is a flawed one. While there are more real numbers than integers, the concept of infinity complicates this comparison, and both infinities have the same cardinality. Therefore, the argument does not hold up under scrutiny. To understand why the argument that the decimal infinity is larger than the integer infinity is flawed, we need to explore the concept of cardinality, which is a measure of the size of a set. The cardinality of a set is the number of elements in the set, but when we talk about infinite sets, this definition is not applicable. Instead, we use a different definition of cardinality that compares the number of elements in two sets by establishing a one-to-one correspondence between them.

A one-to-one correspondence between two sets A and B is a function that assigns each element in set A to a unique element in set B, and vice versa. If such a function exists, we say that A and B have the same cardinality. For example, the sets {1, 2, 3} and {4, 5, 6} have the same cardinality because we can establish a one-to-one correspondence between them by adding 3 to each element in the second set. The function would be: f(1) = 4, f(2) = 5, f(3) = 6.

When we compare the cardinality of the set of integers and the set of real numbers, we find that they have different cardinalities. The set of integers is countably infinite, which means that we can list all the elements in the set in a sequence. For example, we can list the integers as follows: 0, 1, -1, 2, -2, 3, -3, and so on. This shows that we can establish a one-to-one correspondence between the integers and the natural numbers (1, 2, 3, ...), which means that the cardinality of the set of integers is the same as the cardinality of the set of natural numbers. Both sets have cardinality aleph-null (ℵ₀).

On the other hand, the set of real numbers is uncountably infinite, which means that there is no way to list all the elements in the set in a sequence. Georg Cantor proved this in his famous diagonal argument, which shows that any attempt to list all the real numbers in a sequence will always miss some numbers. Therefore, there is no one-to-one correspondence between the set of integers and the set of real numbers, which means that their cardinalities are different. The cardinality of the set of real numbers is greater than aleph-null; it is denoted as 2^aleph-null (ℵ₁).

However, this does not mean that the decimal infinity is larger than the integer infinity. The infinity of integers and the infinity of decimals both have the same cardinality, aleph-null. This means that we can establish a one-to-one correspondence between the set of integers and the set of decimals. For example, we can list all the decimals between 0 and 1 in a sequence, and then assign each decimal to the corresponding integer by taking the first digit after the decimal point and subtracting it from 9. For instance, 0.123... corresponds to 876..., and 0.987... corresponds to 012.... This shows that the cardinality of the set of decimals is the same as the cardinality of the set of integers, and therefore the argument that the decimal infinity is larger than the integer infinity is incorrect.

In conclusion, the concept of cardinality provides a rigorous mathematical framework for comparing the size of infinite sets. The cardinality of the set of integers and the set of real numbers is different, but the cardinality of the infinity of integers and the infinity of decimals is the same, aleph-null. Therefore, the argument that the decimal infinity is larger than the integer infinity is flawed. Why have you typed mass but not sent anything?

Here is my rebuttal; add onto it; you are proving that decimal infinity proves that there are greater infinites. As a matter of fact, write a whole thesis and a proof rebutting every single claim effectively with sources:

"Counting to infinity" is rather one of linguistical semantics as well to say the least; while counting to infinity is quite literally flawed, we don't mean that per the definite sense; but rather that – of the sense wherein there is an infinite of it — whereas infinite numbers exist and pertain endlessly. 

We are not assuming there is one infinity stratum; yet: — of that of similitude — wherein complexities are in fruition and thus forth — aleph-null theoremalization symbolizes an ekphrasis — an allegory to a one-to-one  correspondence. Realistic Numerical Values with a high-continuum implicate 'no one-to-one correspondence between the set of integers and the set of real numbers, which means that their cardinality are different.' Cantorismic analysis implies of the cardinality of real numerical sets; is greater than aleph-null; and thus the same cardinality, while true can still be applicable by squaring – and to deny it is a misconstrued analysis. 'Greater' infinites don't necessarily imply a larger quantity; instead: the antithesis — of all being infinite; but rather of those have a Greater likelihood in pursuance to bijective infinity.