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> "Not all infinities are equal" In other words, there are different categories of infinite, and it might be inappropriate to represent infinity with just one

> "Not all infinities are equal" In other words, there are different categories of infinite, and it might be inappropriate to represent infinity with just one symbol! This article is about how many types of infinity might exist. I was taught there is countably and uncountably infinite. Integers are countably infinite because the number of integers between any two numbers if finite. Real numbers are uncountably infinite because there are infinite numbers between any two real numbers.

ceh123 5y ago
>I was taught there is countably and uncountably infinite. Integers are countably infinite because the number of integers between any two numbers if finite. Real numbers are uncountably infinite because there are infinite numbers between any two real numbers. So this isn't exactly right, although I suppose the argument for integers isn't exactly wrong. When we get to the rational numbers however, they are countable but there are an infinite number of rational numbers between any two rationals in the typical way of thinking about "between" numbers. Based on your argument above, this would mean that rationals are uncountable. A better way to think about this is that the set of natural numbers is the first (and smallest) infinite set you can construct. This is the set of all counting numbers so we call it countable. We then say two sets are the same size (cardinality) if you can create a one-to-one mapping between the two that covers both sets (a bijection). This way you exactly pair one element of one set with one element of another. You can do this with the natural numbers and integers by just alternating positive and negative (so 0 -> 0, 1 -> 1, 2 -> -1, 3 -> 2, etc.). All the sets that you can do this sort of mapping with the natural numbers are considered countable. You can't do this sort of mapping from the natural numbers to the reals (see cantor's diagonalization argument) so since you can't "count" the reals, the set is "uncountable."
jmgao 5y ago
> I was taught there is countably and uncountably infinite. Integers are countably infinite because the number of integers between any two numbers if finite. Real numbers are uncountably infinite because there are infinite numbers between any two real numbers. This is unsound: a set is countable if there's a mapping that assigns an integer to each element of that set. There are an infinite number of rational numbers between any pair of rational numbers, but you can pretty easily construct a mapping between integers and rationals. Imagine representing a rational x/y as a point on the plane, and draw a square spiral around the origin. Every rational number lays on that spiral, and there's a unique "shortest arc-length of the spiral from the origin" for each rational.
- gotostatement 5y ago
  I guess you could rephrase OP's argument in a sound way as: a set is countable if there's a way to order it such that between every two numbers there are finitely many numbers
chaoticmass 5y ago
https://www.youtube.com/watch?v=SrU9YDoXE88 https://www.youtube.com/watch?v=SrU9YDoXE88 I really like how VSauce covers this topic as well.
omnicognate 5y ago
The article isn't about how many types - or rather cardinalities (sizes) - of infinity exist, it's about which of those cardinalities describes the real numbers. "Countably" infinite sets have cardinality Aleph_0, the "smallest infinite size". There is an infinity of "larger infinite sizes", all of which are "uncountable", so to refer to a set as "uncountably infinite" doesn't pin down which particular cardinality it has. The article is about what specific cardinality the set of real numbers has, and in particular whether it's Aleph_1 or Aleph_2 as it seems less likely to be any of the infinite other possibilities.
mikepurvis 5y ago
Veritasium did a really good video on this recently, based around Hilbert's Hotel: https://www.youtube.com/watch?v=OxGsU8oIWjY https://www.youtube.com/watch?v=OxGsU8oIWjY

ceh123 5y ago

>I was taught there is countably and uncountably infinite. Integers are countably infinite because the number of integers between any two numbers if finite. Real numbers are uncountably infinite because there are infinite numbers between any two real numbers. So this isn't exactly right, although I suppose the argument for integers isn't exactly wrong. When we get to the rational numbers however, they are countable but there are an infinite number of rational numbers between any two rationals in the typical way of thinking about "between" numbers. Based on your argument above, this would mean that rationals are uncountable. A better way to think about this is that the set of natural numbers is the first (and smallest) infinite set you can construct. This is the set of all counting numbers so we call it countable. We then say two sets are the same size (cardinality) if you can create a one-to-one mapping between the two that covers both sets (a bijection). This way you exactly pair one element of one set with one element of another. You can do this with the natural numbers and integers by just alternating positive and negative (so 0 -> 0, 1 -> 1, 2 -> -1, 3 -> 2, etc.). All the sets that you can do this sort of mapping with the natural numbers are considered countable. You can't do this sort of mapping from the natural numbers to the reals (see cantor's diagonalization argument) so since you can't "count" the reals, the set is "uncountable."

jmgao 5y ago

gotostatement 5y ago
I guess you could rephrase OP's argument in a sound way as: a set is countable if there's a way to order it such that between every two numbers there are finitely many numbers

chaoticmass 5y ago

https://www.youtube.com/watch?v=SrU9YDoXE88 https://www.youtube.com/watch?v=SrU9YDoXE88 I really like how VSauce covers this topic as well.

omnicognate 5y ago

The article isn't about how many types - or rather cardinalities (sizes) - of infinity exist, it's about which of those cardinalities describes the real numbers. "Countably" infinite sets have cardinality Aleph_0, the "smallest infinite size". There is an infinity of "larger infinite sizes", all of which are "uncountable", so to refer to a set as "uncountably infinite" doesn't pin down which particular cardinality it has. The article is about what specific cardinality the set of real numbers has, and in particular whether it's Aleph_1 or Aleph_2 as it seems less likely to be any of the infinite other possibilities.

mikepurvis 5y ago

Veritasium did a really good video on this recently, based around Hilbert's Hotel: https://www.youtube.com/watch?v=OxGsU8oIWjY https://www.youtube.com/watch?v=OxGsU8oIWjY