Carleas wrote:James S Saint wrote:You do NOT know its length. You only know it to be infinite.

These two sentences are mutual incompatible. We know its length to be infinite. That is a known length.

Infinite is NOT a length. It is a quality. You already agreed that not all infinites are equal.

Carleas wrote:James S Saint wrote:Carleas wrote:Two infinite sets are the same size if there's a function that uniquely maps every member of one set onto a member of the other set (i.e., it's a bijection).

Well that is true. The problem is that you can't actually do that.

I showed two examples, specifically addressing your claims that 1+infinity > infinity, and that two line segments of different lengths have different numbers of points. If you think my reasoning is flawed, show your work.

A little off topic, so just briefly:

You can specify a particular infinite list, say the integers. And you can specify that you are going to take every other number, which you call "even" to make another list. In so doing, you have already stated that you only took half of the first list to make the second list. QED.

Alternatively you can specify the first list as the integers. And then specify a second list as 2 times each of the first list. In such a case, you have two equal cardinality lists. The problem is that

you cannot multiply 2 times the maximum/last element of the first list, because

there isn't such a maximum/last (NOT, NOT, NOT because you didn't have time to get to it). Thus you can't actually fulfill your second list specification. QED.

In either case, you cannot actually accomplish what you imagined doing.

Carleas wrote:James S Saint wrote:Infinite means that there is no "there"... There is no "at infinity" to be finally reached...Why do you think that you can represent the solution that would be obtained "at infinity" when there is no "at infinity"?

You seem to be using a notion of infinity as somehow uncertain.

Not at all. "Infinity" is certainly not existent.

Carleas wrote: But infinite quantities are certain, they are static, they're just infinite. They aren't growing or changing, we aren't waiting for them to get somewhere, they are just infinite.

Except that there are different infinites. And even if you did know the number of items in the infinite list, you still would not know their sum, only their limit. Different size infinite sets can have the same summation limit. The limit DOES NOT specify the final summation.

Carleas wrote: An infinite string isn't a process. An infinite string of 3s isn't growing, we aren't actually doing long division when refer to it. 1 divided by 3 produces an infinite, repeating decimal expansion .333... That infinite expansion is how 1 divided by 3 is written.

That is an issue with you, not me. As I have said many times, I am not referring to the process unless I state that I am. There are different cardinalities and/or sizes of infinite. Which one do you have? You don't know. What is the sum of all of its elements? You don't know that either. All you know is the digits involved and the limit which is being eternally approached without ever, ever, ever being reached. It is certain that the limit is NOT reached, which is why it is an infinite, never ending string, with an always present remaining difference to 1.0 not summed.

You keep thinking that by stating that it is infinite, you have specified the end point of the string, "infinity", and thus of the summation. There is no end point to specify. And the summation never tallies as anything because it never ends. Even if you have added an infinite quantity of elements, you still have an infinite quantity to go. And you always do. It is "statically" unknown as to how many elements there are. You only know that it is an endless list.

Carleas wrote:We can show that .333... equals 1/3, using a point you made earlier:

.333... = 3/10 + 3/100 +3/100 ...

Multiply each side by 10:

10 (.333...) = 3 + 3/10 + 3/100 + 3/1000 ...

Subtract the first equation from the second:

9(.333...) = 3

.333... = 3/9 = 1/3

Fault. 9 times 0.333... is NOT 3.0.

What is 9 times 3? 27.

Carried infinitely gives you:

2.999....You merely presumed the consequent.

And when you "subtract" the two series, you get:

2.7 + 0.27 + 0.027 + ...You know that every element in the infinite list:

has a 1 at the end. And you know that such is the

entire list of the differences between 1 and ALL partial sums of the 0.999... series. Yet you keep insisting that "the final sum" has zero difference. Where did the 1 disappear to in that "final sum"? It is required to be in the specified list.

A) there is NO FINAL SUMMATION.

B) Even if there was a final summation, there would still be required a difference between it and 1.0