To be more slightly articulate… when i said, “Why not just make a new list?” I was actually saying that infinities DEMAND it!!!
Look at it vertically…
[1+1+1+1…] + [1+1+1+1…] … now vertically, it’s easier to understand, that the first set of 1’s never ends, so you can’t add the second set of 1’s.
Understand?
Err… I meant horizontally. I have brainfarts too =)
Again, you are confusing the symbols with their meaning, “conflating the map with the terrain”.
“[1+1+1+…]” does NOT represent a process. It represents only the implied sum.
“[1+1+1+…]” is a single symbol representing “infA”.
infA ≡ [1+1+1+…]; a uniquely defined quality that in hyperreal math can be used as a quantity.
And
infA + infA = 2*infA = [1+1+1+…] + [1+1+1+…] = [2+2+2+…]
None of that has anything to do with any process of adding (horizontally nor vertically), but merely a way of representing a total. That exact same total could be represented by “[1+3+1+3+1+…]”, as long as it represents the same length of series.
And that concern also applies to your use of infinity in your proposed paradox resolutions. The time it might take to add an infinite series is totally irrelevant because the infinite series is not a process, but a single symbol to represent an otherwise difficult thing to express.
Oh now we’re having fun =)
The implied sum of 1+1+1+1 … = infinity
The implied sum of 1+3+1+3… = infinity
And yet different degrees of infinite. All infinities are not alike.
Oh come on… these infinities are exactly alike. This is probably why we have the concept of equalities existing in the cosmos.
There are very many ways to prove that all infinities are not alike. It is sad that one would have to do so.
I will offer only one (out of the very many) and if you choose to disagree with it, then you will continue to insist on whatever you wanted to believe regardless of any and all reasoning and rationality.
Given an infinite line, one could begin to count the number of points on the line using the natural number set. One might begin with;
[1,2,3,4…]
But given two infinite lines, side by side, one could begin to count the number of points on both lines simultaneously by counting the pairs, one from each line as so;
[2,4,6,8…]
The progress along each line remains the same as the other throughout the counting procedure. When one gets to count 14 on one line, he is also at count 14 on the other line. And the number of counts from the beginning would be the same whether counting only one line or counting two lines simultaneously. That means that the infinite series representing each line are the “same length”.
And at all times, with every count, the count of the two lines is twice the count of one line. That means that no matter how far one counts down the line, the two line count is always twice the one line count, even if one could count to infinity. Given any arbitrary position and count in the one-line process, the two-line process will always have a count that is twice as large.
Thus if the single line is said to have “an infinite number of points”, the double line must have twice that many, or;
2 * infinite.
And that is of logical necessity. If you lust to deny it, feel free. But I have a very limited desire to argue with those who simply deny logic rather than try to rectify proposed logic.
Remember, infinity is not a number… but you can use numbers to make infinity. This is an important concept to understand, all infinity is equal. What happens in fact, when you reverse engineer pi and say the square root of two, is that they become infinite decimal outputs, and they end up both equalling infinity. When you divide 1/9 you get .1… which I’m sure you know, but when you multiply .1 by 9, you don’t get 1. This is what I mean by reverse engineering infinity.
But you’re contradicting yourself… you just said infinity isn’t a process. It has to be a process in order for a place to be twice as large as the other.
No.
I said that the symbols were not a process, but representative of a sum. In the counting example proof, I began a counting or summing process so as to show where the total would end up. The example proof is a process. But the sum is merely the sum, not a process trying to get to the sum.
[size=85]… and it disturbs me that I sensed that you were going to say that.[/size]
For the children of the Crack generation, the following video explains most of it correctly (although not all of what she says is right);
[youtube]http://www.youtube.com/watch?v=lA6hE7NFIK0[/youtube]
What I’m saying is that when you reverse engineer something to an infinity, you are left with an infinite sequence, and this always equals infinity instead of what you started with. So you start with 1 in the 1/9 scenario and end up with infinity, not .9999… Does that make sense to you? I mean, does it make sense to any of us? But that’s besides the point.
What I meant to clarify it is that it doesn’t leave you with 1 or .9999…, it just leaves you with infinity, and by saying this, I’m saying that you cannot add or multiply infinities…
It’s nonsense to say .1111… * 9 = .9999…
Just like it’s nonsense to say that
.1111…
.1111…
=
.2222…
And i wanted to additionally clarify… I once told you that compared to the counting numbers, even the number 1 is infinitesimal, as are any finite set of counting numbers…
The way i conceive it, is that there are an infinite combination of the counting numbers… so that sometimes 1 becomes the infinitesimal (the last number) and sometimes 1 becomes the first number.
Oooo… since you haven’t replied yet, I can add this…
It’s different with infinities than finites…
As I already stated… an infinity never ends to be able to add it, what exactly is the symbol for 0.111… ?? 1/9th? Is that the infinity that you’re referring to? I already explained that decimal outputs are ALWAYS approximations to infinity, as are algorithms…
The thing about infinities, is that they never end, so you can’t add one row or column to another without dealing with dimensional flooding (having one infinity in front of another infinity (or finite)).
This is why I say it’s absurd to add or multiply them.
Infinities are different than finites… they go on forever, and are like their own numbers in their own right…
0.111… only has one instance… if you do it again or try to add it to itself, you have dimensional flooding… either vertically or horizontally, so what happens is that you get a list, a new list…
List 1: 0.1111…
List 2: 0.1111…
etc…
And in that way, they become numbers… just like an infinite list of zeros added to zeroes becomes numbers!!! That’s what I was trying to explain before.
No. It doesn’t. But it is difficult to articulate the obvious error that you are making.
But first, it isn’t “reverse engineering”. It is “deconstructing”. And any infinite series that you “are left with”, is merely your deconstructed pieces of the former whole, left un-whole, deconstructed.
Those pieces, of the infinite series, can only be “equal to the original” after you reconstruct the pieces back into the whole.
The infinite series might or might not sum to infinity. The fact that it is an infinite series does not make it “equal to infinity”.
And again, you are confusing a process of dividing 1/9 with the end result that is being represented by the symbol, “0.999…”. You seem to be confusing “infinitely” with “infinity”.
Let’s use your deconstruction argument and suppose for a moment that the division sign is the remaining data… why do some divisions only end in finiteness that can be “deconstructed” and “reconstructed”?
Of course it makes it equal to infinity… offer me a series that’s infinite that doesn’t equal infinity.
We already established that
1+1+1+1… = infinity
and that:
1+3+1+3… = infinity
So show me an infinite sequence that doesn’t equal infinity!
ALL series are infinite. But we don’t concern ourselves with those that end with an infinite series of zeros. Some series “end” with an infinite series of zeros. Because they are all zeros throughout infinity, they can all be dropped or ignored as one settles for what was already resolved before the string of zeros appeared.
I have already done that. You (and John) keep ignoring my answers to your objections.
And realize that;
[size=150]2 = 2.0000…[/size]
…an infinite series.
This is where you and I disagree… it equals APPROXIMATELY … 2, it doesn’t actually equal 2.
It CONVERGES at 2. And 2 is no where near infinity.
That means that IF it were to be carried out infinitely, it would become 2. In math, that is called “the limit as the divergence becomes zero”.
As I stated before with the “1 = 0.999…” issue, I agree that the bound number (1) is NOT identical to the unbound number (0.999…).
Oh come-on James, you’re contradicting yourself… I agree with the 1, .999… issue as well, but you can’t see how a convergence to 2 is the same problem? That actually astonishes me. You must have a reason behind the facts why you can’t let that go.