6 ms·

Countable (roughly) means that you can count from one element to any other in a finite amount of time, it doesn't mean you can count the whole set in a finite a

by nmilo 3y ago

Countable (roughly) means that you can count from one element to any other in a finite amount of time, it doesn't mean you can count the whole set in a finite amount of time.

nico 3y ago
Think about counting as three operations: 1. Produce a number 2. Add it to the set of numbers I’m counting 3. Sort the numbers Usually when you produce a number, you produce it in order (eg. 1, 2, 3, 4). But you could also count like: 4, 1, 3, 2 and then sort the set. As long as you have an algorithm/function to keep producing numbers for “each iteration”, you can always keep counting one at a time. The case you say: “count from one element to any other” is equivalent to a finite operation, it has a very clear beginning and end. But an infinite set doesn’t, unless you stop counting.
- cscurmudgeon 3y ago
  > Think about counting as three operations: > 1. Produce a number 2. Add it to the set of numbers I’m counting 3. Sort the numbers Missing in your definition is the operation has to produce all numbers that you are counting over. If you are counting something, you don't leave out elements.
  - nico 3y ago
    If you are counting a set that is larger than your count, you will always leave elements out. The only way you won’t leave elements out for an infinite set is if you keep counting forever.
    - cscurmudgeon 3y ago
      "If you are counting a set that is larger than your count" What does that mean?
      - nico 3y ago
        If you want to count 4 natural numbers, your count is going to be 4, but the set of natural numbers is infinite. The set is larger than the count.
- nmilo 3y ago
  I think you're mixing up math and natural language. A "countable set" in math, by definition, means what I said it means, (or more precisely means it is 1-to-1 with the natural numbers). You can argue that the word itself is misleading/unclear, but that's a semantic argument and you should make clear that that's your stance.
  - nico 3y ago
    The definition of a countable set is that it has the same cardinality as the natural numbers set, “aleph-null”. This is an ill defined concept for two reasons: 1) a set is supposed to be a “collection of items”, but the natural numbers is not a collection of numbers, it’s a method to generate an infinite number of numbers… which technically you could never count 2) it’s trying to use a symbol (aleph-null), to make infinity appear finite - the cardinality of an infinite set (like the natural numbers), cannot be a finite symbol. So essentially there’s two contradictions in the definition of the cardinality of the set of natural numbers. But that definition is taken as true, and then used as the basis for justifying other concepts, like comparing sizes of unknown infinities.
    - nmilo 3y ago
      1) Who said a set is supposed to be a "collection of items" or any other kind of set in the CS definition? A set is a mathematical object which can be said to contain, or not contain, an element. The natural numbers fall into this definition of a set. 2) There's no such thing as a "finite symbol." What does that even mean? A symbol is a symbol, it symbolizes something. The symbols themselves are finite; they're just letters. But if that something that it symbolizes is an infinite quantity, why stop it? You are living too close to reality and numbers and words. Even if sets were rigidly defined as finite objects, why not define some new thing, call it an "infinite collection," and let all the theorems like countability and aleph-null come out of that? It lets us do useful math, after all, so why let the words we use stop us? Clearly set theorists worldwide find nothing* wrong with the current definition.
      - nico 3y ago

by nmilo 3y ago

Countable (roughly) means that you can count from one element to any other in a finite amount of time, it doesn't mean you can count the whole set in a finite amount of time.

nico 3y ago

Think about counting as three operations: 1. Produce a number 2. Add it to the set of numbers I’m counting 3. Sort the numbers Usually when you produce a number, you produce it in order (eg. 1, 2, 3, 4). But you could also count like: 4, 1, 3, 2 and then sort the set. As long as you have an algorithm/function to keep producing numbers for “each iteration”, you can always keep counting one at a time. The case you say: “count from one element to any other” is equivalent to a finite operation, it has a very clear beginning and end. But an infinite set doesn’t, unless you stop counting.

cscurmudgeon 3y ago
> Think about counting as three operations: > 1. Produce a number 2. Add it to the set of numbers I’m counting 3. Sort the numbers Missing in your definition is the operation has to produce all numbers that you are counting over. If you are counting something, you don't leave out elements.
- nico 3y ago
  If you are counting a set that is larger than your count, you will always leave elements out. The only way you won’t leave elements out for an infinite set is if you keep counting forever.
  - cscurmudgeon 3y ago
    "If you are counting a set that is larger than your count" What does that mean?
    - nico 3y ago
      If you want to count 4 natural numbers, your count is going to be 4, but the set of natural numbers is infinite. The set is larger than the count.
nmilo 3y ago
I think you're mixing up math and natural language. A "countable set" in math, by definition, means what I said it means, (or more precisely means it is 1-to-1 with the natural numbers). You can argue that the word itself is misleading/unclear, but that's a semantic argument and you should make clear that that's your stance.
- nico 3y ago
  The definition of a countable set is that it has the same cardinality as the natural numbers set, “aleph-null”. This is an ill defined concept for two reasons: 1) a set is supposed to be a “collection of items”, but the natural numbers is not a collection of numbers, it’s a method to generate an infinite number of numbers… which technically you could never count 2) it’s trying to use a symbol (aleph-null), to make infinity appear finite - the cardinality of an infinite set (like the natural numbers), cannot be a finite symbol. So essentially there’s two contradictions in the definition of the cardinality of the set of natural numbers. But that definition is taken as true, and then used as the basis for justifying other concepts, like comparing sizes of unknown infinities.
  - nmilo 3y ago
    1) Who said a set is supposed to be a "collection of items" or any other kind of set in the CS definition? A set is a mathematical object which can be said to contain, or not contain, an element. The natural numbers fall into this definition of a set. 2) There's no such thing as a "finite symbol." What does that even mean? A symbol is a symbol, it symbolizes something. The symbols themselves are finite; they're just letters. But if that something that it symbolizes is an infinite quantity, why stop it? You are living too close to reality and numbers and words. Even if sets were rigidly defined as finite objects, why not define some new thing, call it an "infinite collection," and let all the theorems like countability and aleph-null come out of that? It lets us do useful math, after all, so why let the words we use stop us? Clearly set theorists worldwide find nothing* wrong with the current definition.
    - nico 3y ago