You continue to somehow believe that the infinite’th point is a discrete point. It isn’t. There exists a infinite + 1 ordinal point and an infinite + infinite ordinal point. They all occupy the same location on the line. Which is why you can’t start at the end of the line and pick one out. You are assuming that you can locate a discrete point for which there are no further points with a greater ordinal. If you can locate this point you have contradicted the definition of infinity, or you have a finite ordinal point. Infinite simply doesn’t work this way. It doesn’t obey the rules that apply to normal numbers. Continuing to try to force it to does get you anywhere.
Since, in “normal math,” there is no such thing as the last digit in either of those two decimal expansions, it’s illegitimate to use a “normal math” subtraction algorithm which requires that you start from the last digit of a decimal expansion.
Hmmm, I don’t know. I would potenitally argue there is a last digit at the infinite’th place.
It is common to think of infinity as never ending, I know. But a line segment can be divided into an infinite number of points. Why Can I not relate those to holding places for my numerals in the decimal expansion of .999… and 1.000…?
Why normal math of subtraction may not apply… perhaps you are right? But I do not see why not here.
And one of the morels of Zeno’s Pardox seems to be we can indeed reach the end of an infinite sequence, so… that seems to be not true that an infinite sequence has no end. Is it somehow there are 2 different infinites here?
You obvioulsy have a stronger math background than me, so I would in all sincereity like your help in understanding the error I have made here, which I do not yet.
So can we start from the beginning of my proof and we’ll go throught it piece by piece and tell me where it goes wrong?
we can simplify this some more, just 1 line - 1 mile long…
As I walk down this line at 1 mph I will imagine we are able to note each half way point, the practicality of this should not really be relivent, agreed?
I am definitely right about the “normal math” subtraction you applied being illegitimate in this case.
Under the rules of “normal math,” the numerical representation 0.999… does not have a final digit, and neither does the numerical representation 1.000… This is just a matter of definition–it is how the ellipses are defined in this context.
But under the rules of “normal math,” the subtraction algorithm you were using requires that you start with the final digit of the numerical representations you are working on.
It requires a final digit–but by definition (see above) there is no final digit. Hence, the algorithm can not be applied.
Well now I think that is how we all taught to think about it, but in reality all it is is an infinite sequence of numbers. Your ordinary plain jane infinity. One kind of assumes you can never find the end… but I don’t know that is stated as a requirement of the number.
So I am wlaking down my line at 1 mph, when I reach 1/2 mile at 30 mins, we note this is the first 0 after the decimal of 1, ie the 1.0 <- zero. Let just ignore the 1 in fornt of the decimal and assume we maked it at the beginning of th eline. I also place my first 9 of the .9 here.
Again the physical practicality of marking is not really an issue, i dont care if we actually mark, it note the place holder maybe sounds better.
Regarding the question of whether an infinite sequence can have an end:
An infinite sequence can have an end. For example, …6, 4, 2, 0.
The question isn’t whether an infinite sequence can have an end, but rather, whether the particular sequence 0, .5, .75, .875, … itself has an end. By definition (see the ellipses) it does not.
You’re asking, how can you reach the end of it if it doesn’t have an end?
Well, there are actually two senses of “end” at play here. One sense is “last member.” The other sense is “limit.” The sequence doesn’t have a last member, and so, you can’t reach the last member. It doesn’t have one. But the sequence does have a limit. And there’s no reason to think you can’t reach the limit. Clearly you can’t reach the limit simply by enumerating the members of the sequence. But that’s not relevant. You can reach the limit by other means. (In this case, by walking the required mile at a normal pace, you’ll eventually reach it. You won’t reach the “final member of the sequence” that way, because there’s no such thing. Rather, you’ll reach the limit.)
Hmmm, interesting at which decimal place would that limit occur? Or in my walk anology what is the count we would be at to reach the limit, assuming we are counting each 1/2 + 1/4 + 1/8 + … + 1/infinity
assuming the cont would go 1, 2, 3, … respectively?
Not to pick on you, since I’ve seen other people make similar comments elsewhere, but it’s important to note that we typically add from right to left because of a specific algorithm that we learn in elementary school. We use it because it works, but there’s nothing exceptional about this algorithm (or the typical algorithms for subtracting, multiplying, dividing…) What I’m lecturing, for example, I typically add from left to right, since that’s the way we read the numbers. I just make sure to scan one place ahead to see if it may become necessary to carry. There are many other algorithms that can be used as well, although some aren’t as effective.
You don’t reach the limit at any step on the walk. You reach the limit after all the steps are completed. (I think! Someone should be along shortly to correct me if I’m wrong.)
Interesting point. I never thought of that. I’m not sure how it relates? One must clearly reach the end for either direction to get an answer though, or no?
No, it’s the same infinity. Both are aleph null–simple, countable infinity.
That’s in terms of cardinality. I now forget how to compare infinities for ordinality. Aleph-null plus one is the same cardinality as aleph null (namely, aleph null) but has a higher ordinality (it is one greater). I forget how to determine the ordinality of an infinite number in general. So I can not say for certain whether the two sequences have the same ordinality.
But they have the same cardinality, and that is almost certainly what’s relevant to this discussion.