Jesus as Lord - why is this appealing to so many?

[polymath257]

Yes, an infinite number of finite numbers.

I gave the argument above for the collection being infinite. It is a simple argument. As long as you admit that there is a point between any two given ones (no matter what the definition), the result follows.

I think that there a couple of problems with this as a refutation.   I think that you point is an abstraction, which is why you do not want to define it.  I also think that if you do define it, as anything physical and finite, your argument fails.  It's also on this point, that you have an issue, because the contention is that you cannot have an infinite number of things by successful addition (or division or multiplication for that matter).   No one is arguing that you cannot have an infinite set of abstract numbers (which is really all I think your points are).  If you disagree, then give a definition which is not an abstract, and we can test it.  I think that your point is zero (or nothing), or always changing and not talking about the same thing.

Secondarily similar to successive addition, you will never reach an actual infinite by successive division.  At any given step in the process, you only have a potential infinite.  You don't posses and actual infinite at any time, but are extrapolating out the action infinitely.

Wrong again. Look at the entire collection of points. Because they are on a line, there have an order (directionality) to them. If there are only finitely many, say N, we can write them as
x_1 < x_2 <...x_N
(say, from left to right). Remember that this is assumed to be the *complete* list. That is the assumption of finiteness.

But then, there is some point y with x_1 <y<x_2. In other words, y is between those first two points. And so y is NOT in the list you started with. In other words, that list was NOT complete. That contradiction shows there cannot be a complete list of points that is finite. In other words, the complete list of points is infinite.

I'm not doing successive division. I am doing *one* division to show the assumed finite list cannot be complete.

And this isn't a 'potential infinity' (damn, I hate that phrase---very bad philosophy). The whole collection of points is there and completed. And I just showed that it cannot be a finite collection of points. So it must, by definition, be infinite.

You don't have to be able to do the subdivision an infinite number of times. You just need to be able to do it *once* to show that having a finite total number of points is contradictory.

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I don't have to complete anything. That's the thing. It's already complete but boundless.

Can you add one more?  How about subtract one more?    If so, then how is it complete?

Edit:   Or what do you mean by complete?

The complete collection of points on a line segment has every point on that line segment. No point can be added that is in that line segment. Yes, you can take a point away. You can take infinitely many points away. If you do the latter right, you will still have infinitely many points left over.

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About Zeno's paradox and infinite divisibility, a general question for all: would it be reasonable to argue that what works on paper does not necessarily translate effectively to the real because mathematics doesn't suffer the constraints reality (or at least this local universe) tends to have

Math has it;s limits..but in this context that isn;t ne of them.

Consider the following;  I place you in a rubber room with a piece of granite.  You can't actually break it in half.but that doesn't mean it doesn't have two halves, hemispheres, if you will.  Further, your practical inability to break a piece of granite in two in a rubber room does't mean that each of those hemispheres can't be divided into quadrants.  So on and so forth.  Whereas once it was thought that an atom was indivisible, it was always similar to a piece of granite in this regard..and in truth we learned that we could split them practically as well.  Now we're down to the subatomic..and while at some point we have no practical ability to split x..this doesn't make the concept of the hemispheres and quadrants (ad infinitum) of x any less intelligible or valid.

But we can *still* talk about positions and half-way positions, etc. Even if the measured distances are quantized, the whole collection of real number are potential eigenvalues for the position operator.