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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

by 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
    > Clearly set theorists worldwide find nothing* wrong with the current definition. Are you saying that popularity defines reality? If enough people agree on something that’s the truth and we shouldn’t question thing anymore? About the two points: 1) the definition of what a set is vague, so math has determines that set is an atomic concept that through formality should conform to our expectations of what a set is. So instead of dormally defining what a set is, they formally define a description for a certain set (https://math.stackexchange.com/questions/1452425/what-is-the-definition-of-a-set https://math.stackexchange.com/questions/1452425/what-is-the...) 2) not sure what you mean. All symbols are finite. Think of a symbol as a tag for something else. In the case of infinity, what’s happening here is that the label for infinity is being equated with actual infinity, and then used to determine other stuff. Yes, I want this to mean something real. Because if we are just going to make stuff up, let’s just use a different language instead of co-opting concepts from natural language. Cantors proof doesn’t prove that reals are uncountable, they prove that he couldn’t come up with a system to count them, if he had already counted the naturals and paired them up. But you cannot do that, it just doesn’t make sense. Similarly, saying that the cardinality of the naturals is the label aleph-null, might be useful to perform operations. But whatever you conclude, will be based on the wrong assumption that you can finish counting infinity.
denton-scratch 3y ago
/me not a mathematician! It seems to me that this definition of "countable" is obscure, and furthermore fails to show why the word "countable" is used. Is it wrong to say that a countable set is a set in which it is reasonable to ask which element is the "next" element? That's a natural-language definition, right? You can do that with natural numbers (just add 1) and with rationals (so I understand, but I can't give a procedure). Both sets have the same cardinality: aleph-null. You can't do that with the reals; that set is not countable, and furthermore has grater cardinality than aleph-null. It can also be shown that (a) there is no set with cardinality greater than aleph-null and smaller than the cardinality of the reals; and (b) no set with cardinality greater than aleph-null is countable. I can't do these proofs, but I've read them (in their non-symbolic, natural-language forms), and I was convinced.
- Viliam1234 3y ago
  > You can... (define "next number") ...with rationals (so I understand, but I can't give a procedure). 1/1 1/2, 2/1 1/3, 2/2, 3/1 1/4, 2/3, 3/2, 4/1 1/5, 2/4, 3/3, 4/2, 5/1 ... The first rational number is 0. Next is 1/1, that is 1. Next number to a positive rational number is its negative version, for example 1/1 is followed by -1/1. Next number to a negative positive number can be found by taking its absolute value (i.e. the previous number), locating it in the pyramid, and choosing the next number... skipping those fractions which can be simplified. For example -1/1 is followed by 1/2. So it goes like this: 0, 1/1, -1/1, 1/2, -1/2, 2/1, -2/1, 1/3, -1/3... now we skip 2/2 because that can be simplified to 1/1, and proceed with... 3/1, -3/1, 1/4, -1/4...
- nico 3y ago
  > Is it wrong to say that a countable set is a set in which it is reasonable to ask which element is the "next" element? That's a natural-language definition, right? You can do that with real numbers as well. F(x)=randomreal() At any step you want a next element, you just produce a random real element. You can’t pre-determine how that random element will relate to the previously generated elements, but you can still produce real numbers, one at a time, and count them as you go. And if you do it for infinity, then you will never finish, but at every single step you will have assigned a natural number to a real number. The definitions used for the proof are fiction that you can ever fully finish counting something. The illusion that will fall for is that we think that all natural numbers are somehow pre-computed. If we are going to be generating infinite elements one by one, then you can count/number/order all of them, one at a time, no problem.

by nmilo 3y ago

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
  > Clearly set theorists worldwide find nothing* wrong with the current definition. Are you saying that popularity defines reality? If enough people agree on something that’s the truth and we shouldn’t question thing anymore? About the two points: 1) the definition of what a set is vague, so math has determines that set is an atomic concept that through formality should conform to our expectations of what a set is. So instead of dormally defining what a set is, they formally define a description for a certain set (https://math.stackexchange.com/questions/1452425/what-is-the-definition-of-a-set https://math.stackexchange.com/questions/1452425/what-is-the...) 2) not sure what you mean. All symbols are finite. Think of a symbol as a tag for something else. In the case of infinity, what’s happening here is that the label for infinity is being equated with actual infinity, and then used to determine other stuff. Yes, I want this to mean something real. Because if we are just going to make stuff up, let’s just use a different language instead of co-opting concepts from natural language. Cantors proof doesn’t prove that reals are uncountable, they prove that he couldn’t come up with a system to count them, if he had already counted the naturals and paired them up. But you cannot do that, it just doesn’t make sense. Similarly, saying that the cardinality of the naturals is the label aleph-null, might be useful to perform operations. But whatever you conclude, will be based on the wrong assumption that you can finish counting infinity.

denton-scratch 3y ago

/me not a mathematician! It seems to me that this definition of "countable" is obscure, and furthermore fails to show why the word "countable" is used. Is it wrong to say that a countable set is a set in which it is reasonable to ask which element is the "next" element? That's a natural-language definition, right? You can do that with natural numbers (just add 1) and with rationals (so I understand, but I can't give a procedure). Both sets have the same cardinality: aleph-null. You can't do that with the reals; that set is not countable, and furthermore has grater cardinality than aleph-null. It can also be shown that (a) there is no set with cardinality greater than aleph-null and smaller than the cardinality of the reals; and (b) no set with cardinality greater than aleph-null is countable. I can't do these proofs, but I've read them (in their non-symbolic, natural-language forms), and I was convinced.

Viliam1234 3y ago
> You can... (define "next number") ...with rationals (so I understand, but I can't give a procedure). 1/1 1/2, 2/1 1/3, 2/2, 3/1 1/4, 2/3, 3/2, 4/1 1/5, 2/4, 3/3, 4/2, 5/1 ... The first rational number is 0. Next is 1/1, that is 1. Next number to a positive rational number is its negative version, for example 1/1 is followed by -1/1. Next number to a negative positive number can be found by taking its absolute value (i.e. the previous number), locating it in the pyramid, and choosing the next number... skipping those fractions which can be simplified. For example -1/1 is followed by 1/2. So it goes like this: 0, 1/1, -1/1, 1/2, -1/2, 2/1, -2/1, 1/3, -1/3... now we skip 2/2 because that can be simplified to 1/1, and proceed with... 3/1, -3/1, 1/4, -1/4...
nico 3y ago
> Is it wrong to say that a countable set is a set in which it is reasonable to ask which element is the "next" element? That's a natural-language definition, right? You can do that with real numbers as well. F(x)=randomreal() At any step you want a next element, you just produce a random real element. You can’t pre-determine how that random element will relate to the previously generated elements, but you can still produce real numbers, one at a time, and count them as you go. And if you do it for infinity, then you will never finish, but at every single step you will have assigned a natural number to a real number. The definitions used for the proof are fiction that you can ever fully finish counting something. The illusion that will fall for is that we think that all natural numbers are somehow pre-computed. If we are going to be generating infinite elements one by one, then you can count/number/order all of them, one at a time, no problem.