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

[RoadRunner79]

You ask, how can more numbers be added if it is complete, but by definition an infinite amount is never completed.    I think you are switching to sets now, because you can abstractly complete the set just add "..." .  You loosely define it, to include any possible integer.   You want to increase the set... you just loosen the definition.  So you either end up with a contradiction (it is both complete and not complete at the same time in the same way).   Or what you mean by complete is not the same.  Or you mean infinite in another way.

I switched to sets for your sake, not mine. We can go back to Hilbert's Hotel if that's what you really want. Hilbert's Hotel is completely occupied in terms of all rooms representing each integer from 0 to positive infinity. So represented by the infinite set of all whole numbers. But we can also change the example so that the hotel is infinite both ways, and there are rooms representing negative integers. Doesn't matter.

Complete, in the context of infinity, means every element of the concerned infinite set or sequence or whatever you want to call it, is there. And yes, in mathematics, infinite sets like infinite sets of integers are complete already. But what I cannot do is then represent that literally in writing because then it would be me trying to complete a potential infinity (assuming classical concepts of motion and time). What I write or draw is different from what is in the abstract math world.

But even going back to Existence itself, there is nothing logically impossible about actual infinities existing. So long as they are already complete, I need NOT complete anything. And so logically, it can exist in the real world. That's the logic. What you seem to have a problem with is how can a complete actual infinity be a thing? But this is more a problem with your intuition rather than with logic (so incredulity is not enough, you need to demonstrate on paper that it is a logical problem).

No one can perhaps really fully grasp infinity intuitively, but part of what's amazing about it is that it's counterintuitive. There are no contradictions happening with infinity, only apparent paradoxes. This is expected when we're dealing with infinity.

I'm not all that concerned about the hotel or the library either.  My point is that you started talking about sets, because then you can have a concept, but my question is about what that concept represents.  

My problem is that it is easy in concept to just say that it is complete.   This is because nothing ever needs to be complete (which is good, because by definition infinity is never complete)  The problem is how do you have an actual infinity when there is always something more.   In concept, we can use the magic ellipses  1+2+3....+N  to say that the set is complete.  However those ellipses represent something, and that representation keeps going on forever.    In what way is it complete if it keeps going.   What I am asking (and doubting) is that you can ever tie this abstract to something physical.   Even if you are granted any  starting condition you like (infinity always requires more).

Infinity turns out to be the opposite of what people say it is. It is not 'that which has nothing beyond itself' that is infinite, but 'that which always has something beyond itself'.

Logically, how can it be complete, and never complete at the same time?

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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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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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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.

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Also, just to emphasize again, I have no doubt, that you can potentially make up numbers forever in your head (at least hypothetically).  What I doubt is an actual infinity of physical things.  What is it that you are saying that these numbers represent?

What I think that you are saying, is that there are an infinite number of positions (points) which have zero size in and of themselves, that you can envision on that line. You could potentially always envision another point of subdivision for any set of points I could give.   Would you agree that this represents what you are saying?    My contention is that these points are just an abstract.  There are not an infinite number of things on the line, and you don't even really need the line.