>> We have a universe of discourse that has a bunch
>> of atomic things. A "set" is then a collection of things
> In set theory, they define numbers (and addition)
> starting from a set and element-of operator. I cannot
> imagine a bunch of atomic things without imagining
> numbers/counting first.
The point is that you can start with nothing, and define the set that has nothing in it. That's {}. Now we have one thing. We can define a set containing that, so we have { {} }. Now we have two things, and we can definea set containing both of them, and so on.
This is what happens when people want to construct a model of numbers and arithmetic using set theory as the basis. It proves that we can useset theory as the foundation. It doesn't mean it's a sensible thing to do in real life- it's a lot like programming in machine code.
> So it seems to me that unless set and element-of are left
> undefined (as others here have suggested), natural numbers
> are more fundamental than sets.
What are numbers? What is your model for numbers? What is "723"?
>> 0,9999... = 1
> What I am getting is that the "limit" is implicit in the
> statement above even though often unstated. The
> ellipsis is what signifies the limit there, being otherwise
> not mathematically defined.
So what is your question? You seem to be saying that when you write down "0.99999..." that is intended to represent the limit of the sequence 0.9, 0.99, 0.999, ... Define what you mean by limit. Once you make a careful definition of "limit" wou find that the limit of the above sequence is 1.
>> Call that collection A
> I get the precise definition of sack B. I am missing it for sack A.
Sack A has a cube with the number 1 written on it. And it has a cube with the number 2 written on it. And it has a cube with the number 3 written on it. And it has a cube with the number 4 written on it. And it has a cube with the number 5 written on it. And so on.
> I understand what happens on a given cube,
I assume you mean a cube in sack B.
> ... and that cubes in sack A are numbered 1 though infinity.
You can't say that with precision, because it is infinity you are struggling with. What you can say is that for every number n there is a cube with n on it, and that every cube has exactly one number written on it.
This level of detail matters.
> Is there anything to say about the numbers written on
> two different cubes in sack A?
Yes - they're different.
> Basically what you are saying is that sac A cannot have
> all possible combinations like sack B is defined to have.
No. I'm saying I have two collections of objects. I have very carefully defined what these objects are. And I'm saying that you cannot pair them off, one-to-one, without having things from sack B left over.
> I am not sure why that is.
I haven't proved it yet, so I haven't explained why this is so. I have merely claimed it is true to see where your understanding fails so far.
>> You can't say that with precision, because it is infinity you are struggling with. What you can say is that for every number n there is a cube with n on it, and that every cube has exactly one number written on it.
This may be nailing it -- struggling with infinity.
When you say "every number n there is a cube with n on it", what does "every" mean. Does every number include infinity? Or should not not consider infinity to be a number? If the latter, this is probably where I went wrong.
> When you say "every number n there is a cube with n on it",
> what does "every" mean.
To be more precise, every finite number.
> Does every number include infinity?
No, in these sorts of discussions infinity is never considered to be a number. You need explicitly to be discussing transfinite arithmetic, and we're not.
> Or should not not consider infinity to be a number?
> If the latter, this is probably where I went wrong.
Absolutely you should not be thinking of infinity as a number.
>> The point is that you can start with nothing, and define the set that has nothing in it. That's {}. Now we have one thing. We can define a set containing that, so we have { {} }. Now we have two things, and we can definea set containing both of them, and so on.
This is what happens when people want to construct a model of numbers and arithmetic using set theory as the basis. It proves that we can useset theory as the foundation. It doesn't mean it's a sensible thing to do in real life- it's a lot like programming in machine code.
What are numbers? What is your model for numbers? What is "723"? So what is your question? You seem to be saying that when you write down "0.99999..." that is intended to represent the limit of the sequence 0.9, 0.99, 0.999, ... Define what you mean by limit. Once you make a careful definition of "limit" wou find that the limit of the above sequence is 1. Sack A has a cube with the number 1 written on it. And it has a cube with the number 2 written on it. And it has a cube with the number 3 written on it. And it has a cube with the number 4 written on it. And it has a cube with the number 5 written on it. And so on. I assume you mean a cube in sack B. You can't say that with precision, because it is infinity you are struggling with. What you can say is that for every number n there is a cube with n on it, and that every cube has exactly one number written on it.This level of detail matters.
Yes - they're different. No. I'm saying I have two collections of objects. I have very carefully defined what these objects are. And I'm saying that you cannot pair them off, one-to-one, without having things from sack B left over. I haven't proved it yet, so I haven't explained why this is so. I have merely claimed it is true to see where your understanding fails so far.