"Infinity is not a number: that is my point. It is a concept."

In that we agree.

"You can’t add a number to something that is not actually a number. If it were then we would know what one less or one more might be. This is impossible by the definition of infinity — which is bigger than any number you can define or even imagine."

Here you are treating infinity as a number which is bigger than any other number.

Let me ask a question: is 2 centimeters longer than 1 centimeter? Rephrasing it: is the interval [0,2] longer than the interval [0,1]?

In both questions we have a continuous infinite number of points. Jet we instinctively compare those 2 sets and almost anyone will answer that 2 centimeters is longer than 1 centimeter.

Recap: we can make assertions about sets with infinite number of elements, like, for example cardinality (https://en.wikipedia.org/wiki/Cardinality) and even compare 2 sets of infinite number of elements considering some common property of these 2 sets.

Since we cannot treat the infinity ∞ as a number, you cannot state that it is not possible to compare the size of sets with infinite number of elements. That's because you are treating infinite like a number and use the the transitive property of equality.

You state:

- number of Integers = ∞

- number or Reals = ∞

and conclude: number of Integers = ∞ = numbers of Reals => number of Integers = number of Reals = ∞.

ERGO it is false that number of Reals > number of integers. They are both infinity and stop.

That is your flow: you are treating infinity as a number and conclude that you cannot assert anything about the size of sets with an infinite number of elements because this size is ∞ and so they are all equal in size.