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>Cantor's diagonalization is simply demonstrating that same inequality by showing a number in set A is not in set B. If that were true, why go to all the troub

>Cantor's diagonalization is simply demonstrating that same inequality by showing a number in set A is not in set B. If that were true, why go to all the trouble, just show 1/2 which is not a natural number, or sqrt(2) which is not a rational number. Cantor's diagonalization is proving that no mapping exists between the natural numbers and the real numbers in [0, 1]; that no matter what mapping you (try to) come up, there will be a number you would miss. The primes and rationals have the same size (cardinality) as the natural numbers, namely countably infinite. See https://en.wikipedia.org/wiki/Countable_set#Formal_overview_without_details https://en.wikipedia.org/wiki/Countable_set#Formal_overview_...

Retric 10y ago
So, you can show the Real numbers between 0 and 1 is larger than the number of Natural numbers. There are an infant number of points between 0 and 1 and an infinite number of points between 0 and 2. The distance between 0 and 2 is larger. The number of points between 0 and 1 is smaller than the number of points on the unit circle AND they are a different class of infinity.
- pcmonk 10y ago
  "The distance between 0 and 2 is larger" is a question about the metric of the space, not size of sets. There are the same number of points in both sets, since f(x) = 2x is a bijection between them. Try to make arguments from axioms and definitions rather than asserting things from intuition. Intuition is often a useful tool, but (1) it's not an argument, and (2) it's not very helpful once you step into the infinite realm. Incidentally, that's why I went for programming: it's like math, but with no infinity (unless you're using floats, but that's a much easier infinity).
  - Retric 10y ago
    You say same number, you mean same flavor. Either the 'number' is not in R and thus it's not a number or you end up with a host of contradictions. But, I have had my fun poking people who don't really get set theory.
    - JackFr 10y ago
      Best trolling on HN in weeks.

Retric 10y ago

So, you can show the Real numbers between 0 and 1 is larger than the number of Natural numbers. There are an infant number of points between 0 and 1 and an infinite number of points between 0 and 2. The distance between 0 and 2 is larger. The number of points between 0 and 1 is smaller than the number of points on the unit circle AND they are a different class of infinity.

pcmonk 10y ago
"The distance between 0 and 2 is larger" is a question about the metric of the space, not size of sets. There are the same number of points in both sets, since f(x) = 2x is a bijection between them. Try to make arguments from axioms and definitions rather than asserting things from intuition. Intuition is often a useful tool, but (1) it's not an argument, and (2) it's not very helpful once you step into the infinite realm. Incidentally, that's why I went for programming: it's like math, but with no infinity (unless you're using floats, but that's a much easier infinity).
- Retric 10y ago
  You say same number, you mean same flavor. Either the 'number' is not in R and thus it's not a number or you end up with a host of contradictions. But, I have had my fun poking people who don't really get set theory.
  - JackFr 10y ago
    Best trolling on HN in weeks.