A summary of our discussion to the best of my understanding of what points were made continued:
Obsrvr: …you either have a redundant declaration where you say that A contains itself and you mean that A has in it whatever A has in it. The only thing wrong with that is that it is tautological or redundant or just pointless to say…The other option in saying that A contains itself meaning that an identical but separate instance of A is contained as a member within the boundary of the first mentioned A as a subset member. That is what leads to absurdities…Given the original in question, where is any space for an exact copy of the exact same size and nature - plus 3 more items?
CR: I’m saying A is in A. You’re saying this is either redundant, or contradictory, without accounting for the paradox I pointed out to you.…I cannot say my computer does not have a folder that contains all folders on the computer when the premises I have are folder A is in folder A ad infinitum. If not ad infinitum, then yes, folder A definitely does not truly/fully contain folder A within itself. I know you said “It is going to tell you when you first try to move it into itself that it cannot do that.” And I wholly agree with this. This is in line with not being able to count to infinity. But if I came across a computer that claimed to have all folders on it in one folder with that folder being called A, and it was truly the case that A was in A ad infinitum, then what? Of course I can never empirically verify this because I can never count to infinity (even if I was immortal). But rationally speaking, given the premises at hand, does A contain A?
Obsrvr: You can see that A cannot contain A because that would mean that A has a lesser hierarchy than A…A set that contains itself as merely one of its own members would obviously be greater than itself. Can there be a square that has a larger square inside it?
CR: I agree that a universal set cannot have any duplication (copies of itself), just as I agree that there cannot be more than one set of all sets…I can clearly see that A cannot have a lesser hierarchy than A. A cannot be any different to A otherwise A does not truly contain A. But I don’t see how A containing A necessarily amounts to A having a lesser hierarchy than A. If A was finite, then yes, I clearly see your point.
Obsrvr: What you seem to be saying is that the following two things are identical in every way -
A{}
A{A{A{A{A{A{A{A…}}}}}}}
I added the colors only so we could reference which we are talking about (having no intent to distinguish otherwise).
The red ones are INSIDE the blue ones. That makes them a “subset” of the “superior” blue set. And that makes them SEPARATE instances of the same thing and increases the number of items explosively.
CR: there is only A containing/encompassing itself whilst being equal to itself and not separate from itself. The reference point used is separated, but A itself is not separate from A in any way shape or form…You say there can be no set of all sets because no set can truly contain itself.
I say Infinity/Existence is such that the set of all sets exists. It contains/encompasses Itself fully in the sense that the Infinitesimal is Infinite, which means that the Infinite and the Infinitesimal are the same. We call Existence which is internal to us Existence/Infinitesimal (as opposed to nothing/non-existence) and Existence which is external to us Infinite/Existence (as opposed to nothing). Again, we are in Existence and Existence is in us. So Existence is in Existence as well as outside of Existence. This is because nothingness/non-existence does not exist. There is no beyond Existence.
Obsrvr: So how can a set that is infinite (A{}) have the exact same quantity as a set that is infinite plus two more sets? We just agreed those are two different infinite sizes.
How can a set be larger than itself? That denies the logic of “A = A”.
CR: Consider the folder A in A. You open the A in A, and it has all other folders in it plus A. You go up an A, and it has all folders in it plus the A you just went up from. This is the case whichever direction you go. Where is there a problem with this? How does this lead to a set that is infinite that contains a set that is infinite of the same size, plus two more sets?
Obsrvr: Each one of your folders is smaller than the one it resides in because there are two more folders inside with it. And each one of your folders is larger than every folder inside it because there are multiple folders within. The problem is that one of the folders inside each folder is supposedly that exact same folder plus more.
CR: A contains (1, 2, 3, A). I’m in A. I click 1, and I get what 1 contains. I click A, and I get what A contains (1, 2, 3, A). So here, how are there two more folders inside it? How are there two more folders than (1, 2, 3, A)? If I go up a folder in an attempt to get to the root folder, I get to A. Again, I see (1, 2, 3, A). So again, there are no two additional folders inside this A are there? There is (1, 2, 3, A). Where are you getting these additional folders from?..The A in A contains (1, 2, 3, A) and the A encompassing A contains (1, 2, 3, A). Where is there necessarily an additional folder? Is it in the A in A, or the A encompassing A? If it’s in neither, then where are you getting your additional folders from?
Obsrvr: Regardless of which infinite size ∞ represents it is a different size than ∞ + 3. Every single A{} will have more than A{} inside regardless of how big A{} is…Look closer - you just said that A contains A plus 3 more items – “A = A+3”
CR: (A, 1, 2, 3) is the case…You click 1 you get pictures of cats. You go up a root folder from cats you get (A, 1, 2, 3). Clearly, 1 does not contain itself. You click A you get (A, 1, 2, 3). You go up one folder trying to get to the root folder you get (A, 1, 2, 3). Whether you click A or go up from (A, 1, 2, 3), you always get (A, 1, 2, 3)
You are saying A is itself plus others. This is contradictory. A is not itself plus others because A is A and others are others. See??..Clearly A (which contains A, 1, 2, 3) actually contains A, 1, 2, 3. It contains itself as well as others. So what if it contains all other folders on the computer in addition to itself? This does not mean that A = A+3. It means that A contains A, 1, 2, 3. A = A.
Obsrvr: No! You can NEVER be at that place. That place can’t exist. No matter where to “start” you ALWAYS have another above you. You cannot start at the top. There is no top (and that is exactly what Cantor was saying). Yet you keep thinking that you are there when you start. You cannot ever be there…If you are in that place of ALL folders already - how can you “go up”?
My additional reply to Obsrvr’s last point: If that place can’t exist, then that’s like saying Existence (the place where all existing things exist) can’t exist. The idea that no matter where I start I always have A above me, or A in me, is literally descriptive of the truly Infinite. The Infinite encompasses me, and the Infinitesimal is in me. The Infinitesimal Is Infinite. Infinitesimal = Infinite. A = A. You do not go up from Existence. You go up in Existence via Existence as opposed to non-existence. I agree that my folder example can look as though you exit A and go into another separate A. BUT this is not what I’m implying at all. I’m saying you can never exit A because A encompasses A…which means A is in A. Again, this does not mean A is separate to A or that A is a copy of A. A encompassing A without being a copy of A or separate to A, is just the way A is. A is in me and I am in A. If A is in me and I am in A, then clearly we cannot avoid saying that A is in A. Existence is in me and I am in Existence. I am not in non-existence and there is no non-existence in me.