Your statement is this: S_n = {1,2,...,n}, and |S_n| = n. Now, if n goes to infinity, you find it strange that |S_n| goes to infinity, but I think if you view it like this it is not strange at all. The fact that you take the limit to infinity does not mean that the finite cardinality property has to be preserved, so it is not inconsistent at all. But, your mathematical philosophical curiosity is awesome, and you should definitly keep pondering these questions.
Trying to understand an apparent inconsistency (sets and subsets of natural numbers)
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An intuition friendly argument. I love it.
I am not sure what |S_n| = n means. Those look like absolute bars, but I don't think that is what you meant right?
Yeah, thanks, for some reason I really enjoy digging into topics like these.
Yeah, when applied to sets those bars mean cardinality, which means the number of elements in a set
Okay I see.
I think you are saying that I can think of both S_n = {1, 2, 3, ...} as n goes to infinity and |S_n| = n as n goes to infinity. Is that what you meant?
Yes, well, you can (sort of) think of N as a special case of S_n, with n = infinity. So, one obtains S_inf = {1,2,...,inf} = N, and |S_inf| = inf.
I am aware that this notation is not entirely rigorous, e.g. {1,2,...,inf} is incorrect, and should be {1,2,...} = N, but this way of writing it helps you with your philosophical question I think.
I am aware that this notation is not entirely rigorous, e.g. {1,2,...,inf} is incorrect, and should be {1,2,...} = N, but this way of writing it helps you with your philosophical question I think.
Your analogy, S_inf = {1,2,...,inf} = N, and |S_inf| = inf, is exactly my misunderstanding in this topic. From the proper notation of (or what I understand to be proper notation) S_n = {1,2,...,n} = N, and |S_n| = inf , you have swapped 3 n's for 3 inf's. What you have done is exactly what my knowledge and logic is forcing me to believe is true. If your analogy were the actual notation, I would not be on here confused.
Here is my thinking.
There seems to be a crucial difference between "as n goes to infinity" and infinity. The difference being, n is never infinite, even though n goes to infinity, while the set of N is infinite.
Having said that, how it is logical to have some finite n (even though n goes to infinity) from the notation S_n = {1, 2, 3, ...} equate to an infinite number of elements in S_n (the set N)?
What am I saying that is wrong here?
I see what you mean. I think n goes to infinity means that n can become arbitrarily large, and indeed, during that process, n remains finite. Normally, we define S_n as the state of our special subset of N when n has finished growing, but with infinity this does not occur. The question now is, how can one give meaning to S_n when n becomes arbitrarily large, and therefore never stops growing?
We have two options: We can stick with our growing analogy from point A to point B, and forbid sets like N to be defined, or generalize the concept and view it less as a growing process from A to B, but from a point A to possibly point B, or no point at all, or maybe even not as a growing analogy anymore, and give the sets with no point B a cardinality of infinity. As it turns out, option B won the philosophical battle, as mathematicians love to generalize, and I think it is also practical to consider infinite sets.
We have two options: We can stick with our growing analogy from point A to point B, and forbid sets like N to be defined, or generalize the concept and view it less as a growing process from A to B, but from a point A to possibly point B, or no point at all, or maybe even not as a growing analogy anymore, and give the sets with no point B a cardinality of infinity. As it turns out, option B won the philosophical battle, as mathematicians love to generalize, and I think it is also practical to consider infinite sets.
Once again, you are repeating yourself.
S_n = {1, 2, 3, … n} means |S_n| = n
S = {1, 2, 3, … without any limit} means |S| is without any limit
Perfectly parallel.
But no natural number is without limit. Each natural number is its own limit. It is your insistence on not recognizing that a natural number is finite (has a limit) that leads you to the ridiculous notion that infinity is a natural number and therefore infinity is finite. You say that is an inconsistency. It arises because you say infinity is a natural number. The inconsistency never arises if you say infinity is not a natural number.
S_n = {1, 2, 3, … n} means |S_n| = n
S = {1, 2, 3, … without any limit} means |S| is without any limit
Perfectly parallel.
But no natural number is without limit. Each natural number is its own limit. It is your insistence on not recognizing that a natural number is finite (has a limit) that leads you to the ridiculous notion that infinity is a natural number and therefore infinity is finite. You say that is an inconsistency. It arises because you say infinity is a natural number. The inconsistency never arises if you say infinity is not a natural number.
topsquark
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[imath]S_{ab} = \{a, a+1, a+2, \dots , b \}[/imath] is a "section" of the natural numbers, [imath]\mathbb{N}[/imath]. So, [imath]S_n \equiv S_{1n} = \{1, 2, \dots , n \}[/imath] is a section of [imath]\mathbb{N}[/imath].
Now, a, b, and n were used above as elements of [imath]\mathbb{N}[/imath]. Using [imath]\infty[/imath] isn't really allowed because it is not actually an element of [imath]\mathbb{N}[/imath], but we can define the set [imath]S _{\infty} \equiv S_{1 \infty} = \{1, 2, \dots \}[/imath] to be [imath]\mathbb{N}[/imath]. Just be aware that such a notation is not standard and should be written out if you are going to use it.
-Dan
Okay, interesting. I am going to think about this some more. Thanks a lot.
We both agree with this part.
My issue is that |S| must be limited because of the finite property of every n. Every n is finite implies a finite number of elements in S. That's why I don't think it's perfectly parallel.
I am not going to say anymore what I believe it means; that's not important nor the point of all of this. I am only trying to point out what doesn't make sense to me.
If S is limited, then by definition there is a largest element of S. Let’s call it L. But L is a natural number. Therefore L + 1 is a natural number larger than the largest element in S, which is a contradiction.
There is no law of logic requiring that if each element of a set is a finite number, the number of elements in the set must also be finite. You keep saying there is such a logical law even though saying so leads to an immediate contradiction (inconsistency).
There are two ways to go that do not lead to contradiction. One is to say the the set S is numberless and therefore discussing the number of that set is nonsense. That is the finitist position. The other is to say that the number of the set S is a number, but that number is not a natural number (or a real or complex number). That was Cantor and Hilbert’s position. I suppose there may be a third possibility, namely to redefine what is meant by a natural number. But that would require redoing all of arithmetic and analysis (if it is even possible).
The one thing that does cause inconsistency is what in essence you say over and over: namely, because each natural number is itself finite, the number of the set of all natural numbers must be finite too. That is wrong.
Your contradiction does not help explain how my contradiction is invalid.
It is not that I think any set of finite elements must be finite; it's the set of natural numbers in particular.
After thinking about this really hard for a couple days, it all comes down to this. This is exactly where I must be wrong, but I still do not understand how.
I repeat myself because I have yet to see an argument that directly critiques my "logic". Like I said above, giving me a counter proof does not help me understand how I am wrong.
topsquark
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Your flaw has been mentioned several times now. |S| is not limited by the fact that all elements in the set are finite. You are saying this is true and we keep telling you that it is not; it is only true for finite sections of [imath]\mathbb{N}[/imath]. Just because all of the elements of the set S are finite does not mean that the size of the set must be finite.
It is unfortunate, but just because a formula holds for a finite value does not mean that it also holds for infinite values.
-Dan
But all everyone is saying is that I am wrong. Nobody says exactly how. I know that I am wrong (unless I have discovered something which is highly unlikely). I am on here to understand how I am wrong.
If you are right, there is an inconsistency. Hence, you are wrong. It is that simple.
I also think I am wrong for that reason. But I was only hoping to know how.
topsquark
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I told you. The number of elements of a set of finite numbers does not have to be finite. There is no reason that the properties of the elements of the set to have anything to do with the size of the set. Are the number of real numbers between 0 and 1 finite because all the elements are between 0 and 1? If you do not understand that, you need to learn some more Set Theory to get more experience with this material.
-Dan
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