Exactly!
Ok, I see. That would make sense considering you don’t agree that (0.\dot9) = 1. But I didn’t know you thought it doesn’t even equal a quantity. I thought, for sure whatever Magnus thinks (0.\dot9) does equal, he thinks it’s a number. But I take it you think that because the infinite sum of 0.9 + 0.09 + 0.009 + … never attains 1 (or just never completes), it doesn’t even attain the status of being a number. Is that right?
I always thought that so long as it falls somewhere on the number line, it’s a number:
… ← -3 – -2 – -1 – 0 – 1 – 2 – 3 → …
…^ ← (0.\dot9) falls here (either 1 or just before 1).
Infinity obviously falls outside the range of the number line (what I mean by “beyond numbers”) and so it does not fall on the number line, and therefore isn’t a number.
It’s true that when a series approaches a limit, that limit can sometimes be attained and sometimes not, and the question of what kind of limit does (0.\dot9) approach is up for debate, but I always thought that the proof given at the wikipedia article settled the matter. If not, what would?
Because arithmetic deals with numbers and only numbers (which are, by definition, finite). It is the practice of calculating numbers. Throw anything else at it–like red, or cows, or shoes–and it makes no sense.
You’re calling the assertion that arithmetic only deals with finite numbers “popular opinion”, huh? I’d call it self-evident, pretty close to axiomatic–that’s just what arithmetic is.
You can ask me to justify my claims ad nausium, but we’re getting pretty close to elementary principles here, which is to say rock bottom, end of the line. Soon it begins to be like asking: why does 1 + 1 = 2? Little children do this. The incessant chain of never-ending whys. Louis C K has a good bit on this:
[youtube]http://www.youtube.com/watch?v=Tf17rFDjMZw[/youtube]
The fact is, no one has an infinite chain of justifications stored up in their minds for all the things they believe (you’re no exception). Press someone to justify their claims long enough, and you’ll eventually get to the end of the line. That doesn’t mean they don’t know what they’re talking about, it just means you’ve reach the fundamental level on which what they believe is based. At this point, it’s just intuitive, self-evident, axiomatic. It’s like trying to explain red to a man color blind from birth. If you find you can’t do it, that doesn’t mean you don’t know what red is, it just means your knowledge of it is fundamental, basic, doesn’t break down any further. At this point, you just know it or you don’t.
So you agree that endlessness is a property of sets, one that sets either have or they don’t. Do you agree that the word “infinite” is just another word for endlessness?
When you say you make an infinite set bigger by adding a new member to it, would you describe this as “more infinite”?
When you say (\infty) + 1 > (\infty), you obviously don’t mean “more endless” since you just agreed that endlessness is a property that a set either has or doesn’t have–no degrees–and I’ll await your reply to see what you say about “more infinite”–but could (\infty) above be replaced with [the size of an infinite set] + 1 > [the size of an infinite set]?
I can see how it might seem intuitive that adding a member to an infinite set would increase the overall number of members in that set, but because the number of members of the set is directly tied into its property of being infinite/endless (in fact, defining it as such), that for me brings into question what things we can say about its size (number of members) and what things we can’t say.
The point of the parallel lines thought experiment I brought up was to show how when you remove a subset of points from one of the lines, making it appear to have changed, rearranging the remaining points can show that nothing has really changed in the line at all. It reacquires its prior state exactly. I know we had a disagreement about whether it was really the same or there was a persistent change, but I explained that it’s not about which points are which but that the structure remains the same. I used the building analogy to solidify this point (you didn’t reply to that which lead me to wonder whether you read or not, so let me know if this doesn’t ring a bell). The overall point is that if we want to say that something about an infinite set changes when we add or remove elements from it–whether that’s its size, length, weight, density, whatever–I’m expecting to be able to detect a difference between the set’s before and after states. But that’s the problem with infinity. Infinite sets turn out to be very resistant to change, at least in terms of their quantity. Assuming uniformity of all its members, the things in it can be rearranged and shuffled around such as to assume exactly the same structure after adding or removing elements as before. And for this reason, I don’t find it as easy as you to simply say the size of the set increases/decreases by adding/removing members.