4 ms·

The author specifically states, and I quote: >no matter how close any two distinct real numbers are to each other, there will always be infinitely many real nu

by Kranar 4y ago

The author specifically states, and I quote: >no matter how close any two distinct real numbers are to each other, there will always be infinitely many real numbers in between ... this doesn’t mean that the sets of real numbers and natural numbers have different sizes What is misleading or wrong about that? What specific statement is incorrect?

floxy 4y ago
No matter how close any two distinct rational numbers are to each other, there will always be infinitely many rational numbers in between. But the cardinality of rational numbers and natural numbers is the same.
- Kranar 4y ago
  That's correct and it's precisely what the author states, and I quote: >this doesn’t mean that the sets of real numbers and natural numbers have different sizes
  - housecarpenter 4y ago
    They state that only as a disclaimer, after a bunch of other statements that hint at the opposite. Just adding a disclaimer to misleading writing merely makes it confusing and unclear for those who are paying attention, and still misleading for those who don't notice the disclaimer. I think the writer might have been meaning to imply that there's a spectrum of properties from countably infinite to dense to uncountable, where each property is stronger than the previous one. But this is actually not the case. The Cantor set is uncountable, but nowhere dense. So density is not just something that's in between countability and uncountability---it's an orthogonal thing.
    - Kranar 4y ago
      I would say your criticism is akin to the curse of knowledge [1], wherein you know all of the relevant background information and so you have difficulty putting yourself in the position of someone who does not. The author is writing the article for someone who could very well be learning this for the first time and may think to themselves "Hmm... this is something unusual and counterintuitive.". Since you are not that reader the author's writing could come across as annoying, but it's not wrong or misleading, it's a way for the author to hint to the reader that they are empathizing with them and will address this unintuitive notion further (which the author does). [1] https://en.wikipedia.org/wiki/Curse_of_knowledge https://en.wikipedia.org/wiki/Curse_of_knowledge
WastingMyTime89 4y ago
That’s an interesting quote you got there. Let’s look at the surrounding sentences, shall we? Emphasis mine. > But there’s something unsatisfying about declaring the size of the set of real numbers to be the same “infinity” used to describe the size of the natural numbers. To see why, pick any two numbers, like 3 and 7. Between those two numbers there will always be finitely many natural numbers. > Remarkably enough, no matter how close any two distinct real numbers are to each other, there will always be infinitely many real numbers in between. By itself this doesn’t mean that the sets of real numbers and natural numbers have different sizes, but it does suggest that there is something fundamentally different about these two infinite sets that warrants further investigation. So no, as rightfully pointed by other commenters, the article implies that density somewhat indicates that two cardinalities which is straight wrong as the rational are dense in the natural yet of the same cardinality. They are even dense in R which entirely defeat the point of the analogy. Density and cardinality are not related at all.

by Kranar 4y ago

floxy 4y ago

No matter how close any two distinct rational numbers are to each other, there will always be infinitely many rational numbers in between. But the cardinality of rational numbers and natural numbers is the same.

Kranar 4y ago
That's correct and it's precisely what the author states, and I quote: >this doesn’t mean that the sets of real numbers and natural numbers have different sizes
- housecarpenter 4y ago
  They state that only as a disclaimer, after a bunch of other statements that hint at the opposite. Just adding a disclaimer to misleading writing merely makes it confusing and unclear for those who are paying attention, and still misleading for those who don't notice the disclaimer. I think the writer might have been meaning to imply that there's a spectrum of properties from countably infinite to dense to uncountable, where each property is stronger than the previous one. But this is actually not the case. The Cantor set is uncountable, but nowhere dense. So density is not just something that's in between countability and uncountability---it's an orthogonal thing.
  - Kranar 4y ago
    I would say your criticism is akin to the curse of knowledge [1], wherein you know all of the relevant background information and so you have difficulty putting yourself in the position of someone who does not. The author is writing the article for someone who could very well be learning this for the first time and may think to themselves "Hmm... this is something unusual and counterintuitive.". Since you are not that reader the author's writing could come across as annoying, but it's not wrong or misleading, it's a way for the author to hint to the reader that they are empathizing with them and will address this unintuitive notion further (which the author does). [1] https://en.wikipedia.org/wiki/Curse_of_knowledge https://en.wikipedia.org/wiki/Curse_of_knowledge

WastingMyTime89 4y ago

That’s an interesting quote you got there. Let’s look at the surrounding sentences, shall we? Emphasis mine. > But there’s something unsatisfying about declaring the size of the set of real numbers to be the same “infinity” used to describe the size of the natural numbers. To see why, pick any two numbers, like 3 and 7. Between those two numbers there will always be finitely many natural numbers. > Remarkably enough, no matter how close any two distinct real numbers are to each other, there will always be infinitely many real numbers in between. By itself this doesn’t mean that the sets of real numbers and natural numbers have different sizes, but it does suggest that there is something fundamentally different about these two infinite sets that warrants further investigation. So no, as rightfully pointed by other commenters, the article implies that density somewhat indicates that two cardinalities which is straight wrong as the rational are dense in the natural yet of the same cardinality. They are even dense in R which entirely defeat the point of the analogy. Density and cardinality are not related at all.