NeatNit wrote:If you take an infinite amount of a length infinitely short, what do you get? Just about any length you want...

What does that even mean? There is no such thing as "infinitely short" (well, unless you want to get into the alternative number system of infinitesimals, but that is something else entirely, and doesn't actually solve your problem). Either something has positive length, or it is of length zero.

NeatNit wrote:My **point** is that **points** have absolutely no significance in the way you give them significance.

I think that you are, perhaps, missing the point of an arrow traveling through an infinite number of points. Let me try again.

In order for an arrow to get from the shooter to the target (say the distance is 1 unit), it first has to get to the point 1/2. In order to do so, it must have traveled a

*length* of 1/2 units. Then it has to half of the remaining distance, traveling through the point 3/4. In order to do so, it has to travel a distance of 1/4 units. And so on. At each step of the abstraction, the arrow must travel a positive distance. That is, the arrow must travel an infinite number of positive distances. Yet, somehow, this infinite number of positive distances adds up to a finite number.

Your own intuition, above, was that an unbounded number of positive numbers adds up to infinity. That is most people's intuition, and is the thing that you would want to be true. In general, it is true. Yet here is a model of motion that makes perfect sense, as well. We can divide the interval into an infinite number of sub-intervals of ever decreasing (but always positive) length, and the arrow must travel the distance of each of these intervals. The model and the intuition seem to clash. Hence there is a paradox.

The resolution of this paradox is to

*invent* the notion of a limit. A consequence of inventing limits is the continuum. That is, the continuum is not something that exists

*a priori* (unless you are a Platonist), but a direct result of the rules of logic applied to the invention of a limit. You keep appealing to the existence of a continuum of points in order to resolve the paradox, yet the continuum exists as

*a result of* the resolution of the paradox.

NeatNit wrote:As for the abstraction - it is evidently simple to construct an infinite number of positive values that add up to <anything>.

It's like you aren't even engaging with the ideas... you are just asserting facts on the basis of a Wikipedia article or a math class that you took at some point in the past. You have claimed that 1+1/2+1/4+1/8+... adds up to something finite. How do you know that is true? Can you prove it? What about 1+1/2+1/3+1/4+1/5+...? Does that add up to something finite? If so, what? If I pick a random infinite sequence of positive numbers and add them up, will the result be finite? How does that gell with our intuition? If most infinite series sum to something infinite, what is special about series that sum to something finite? How can we detect these special cases?

NeatNit wrote:Okay. I was not aware of your definition for a paradox, which you posted a few posts back. In that case, it is a paradox that we exist and a paradox that Bruce Willis has been dead throughout the whole movie (a movie which I still have not seen, thank you Scrubs). And a paradox that this sentence ends with a comma,

Perhaps I have done a poor job of explaining what a mathematical paradox is, but none of your examples come even close to qualifying. In mathematics, we have a rigorous structure for determining whether or not a statement is true or false. We start by assuming that some very simple statements are true (in the current context, the ZFC axioms are sufficient---these are nine simple statements that we assume are true), then use the rules of first order logic to determine what those assumptions imply. A paradox occurs when a seemingly false result can be proved to be true, or when we can prove that two seemingly contradictory statements are both true.

xander