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.
> 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...)
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.
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.