Not a professional math person here, but I studied Cantor in the past. I think one thing he would point out is that the analogy is imperfect. In the sense that numbers are counting and characterizing other numbers, the telescopes would only have to be able to see and be seen by one other telescope (or not). It is the existence (or not) of a 1 to 1 correspondence between telescopes that matters, not the quality of the images they produce. In fact, as long as the telescopes can be paired in some way, one never need look through them at all. In the same way, numbers do not do the heavy lifting in determining cardinalities. That job is left up to "functions" (in the mathematical sense).
The article started to explain this on the "even numbers" situation, but didn't illustrate the importance of functions. Remember, what is needed is a 1 to 1 correspondence between the numbers. As I remember, cardinality is tested (or defined) in the following manner: If a 1 to 1 correspondence exists between two sets of numbers (in this case 2 infinite sets), then those sets have the same number of members, and hence the same cardinality. If no such correspondence exists, then one set has more members, and a higher cardinality. So what is (one possible) 1 to 1 correspondence between the natural numbers (x) and the even numbers (y)?
y = 2x
For any natural number (x), there is one and only one possible value for (y) that satisfies the equation. Equally important, for any even number (y), there is one and only one (x) that satisfies the equation. Since both of those conditions are true, a 1-1 corresp. has been found to exist, meaning there are the same number of naturals as evens: so both sets of numbers are characterized as having the same cardinality.
It is a bit trickier (and space consuming) to show the 1-1 corresp. between the natural numbers and the rational numbers. It's even worse to show no such correspondence can exist between the natural and irrational numbers. If intrigued by this article, there are plenty of resources out there on it - for all levels of interest and aptitude. Just be warned: Cantor went much deeper than I or the article could ever hope to...and for his troubles, he also went insane.
Not a professional math person here, but I studied Cantor in the past. I think one thing he would point out is that the analogy is imperfect. In the sense that numbers are counting and characterizing other numbers, the telescopes would only have to be able to see and be seen by one other telescope (or not). It is the existence (or not) of a 1 to 1 correspondence between telescopes that matters, not the quality of the images they produce. In fact, as long as the telescopes can be paired in some way, one never need look through them at all. In the same way, numbers do not do the heavy lifting in determining cardinalities. That job is left up to "functions" (in the mathematical sense).
The article started to explain this on the "even numbers" situation, but didn't illustrate the importance of functions. Remember, what is needed is a 1 to 1 correspondence between the numbers. As I remember, cardinality is tested (or defined) in the following manner: If a 1 to 1 correspondence exists between two sets of numbers (in this case 2 infinite sets), then those sets have the same number of members, and hence the same cardinality. If no such correspondence exists, then one set has more members, and a higher cardinality. So what is (one possible) 1 to 1 correspondence between the natural numbers (x) and the even numbers (y)?
y = 2x
For any natural number (x), there is one and only one possible value for (y) that satisfies the equation. Equally important, for any even number (y), there is one and only one (x) that satisfies the equation. Since both of those conditions are true, a 1-1 corresp. has been found to exist, meaning there are the same number of naturals as evens: so both sets of numbers are characterized as having the same cardinality.
It is a bit trickier (and space consuming) to show the 1-1 corresp. between the natural numbers and the rational numbers. It's even worse to show no such correspondence can exist between the natural and irrational numbers. If intrigued by this article, there are plenty of resources out there on it - for all levels of interest and aptitude. Just be warned: Cantor went much deeper than I or the article could ever hope to...and for his troubles, he also went insane.