May I query this explanation?
Does “You are not adding to the set. You add to your position within the set” mean something like the following:
Infinite set represented as: {…, x, y, z, …}
Add 1 at position y:
New set looks like: {… x, y+1, z, …}
??
I would like to query the meaning of a series being “infinite”. Skip to the last paragraph if at any point you get frustrated with the following, but coming back to it afterwards will make more sense if you do:
I see no issue in using the synonym “endless” or “boundless” to aid in my explanations, since these are literally what infinite means by derivation as well as definition. Substitute “infinite” back in when I use these terms if you wish, the meaning won’t be changed if you do.
Consider the example (1+1+1+…+1), is it agreed that the “…” represents an endless string of "1+1"s?
If the example of (1+1+1+…+1) is intended to represent an endless string of "1+1"s, any given “1+1” has no specific position relative to any start or end, because an endless string has no start or end. All positions are therefore undefined and cannot be pinpointed as specific, and are therefore arbitrarily interchangeable.
Picking any one non-specific position of a “1+1” and adding a 1 such that it is now “1+1+1”, is the string still endless?
Put another way, picking any non-specific position of a “1+1+1” and removing a 1 such that it is now “1+1”, is the string still endless?
I would say it is endless both before and after in both cases.
I would also say that infinite (endless/boundless) does not communicate quantity by definition - I would say that it does the exact opposite of conveying quantity, which is what infinite means: undefined or quantity-less are synonymous as well.
With or without the “1+1+1” or “1+1” the endlessness is indistinguishable, the endless set is endless before and after the addition or subtraction, the positions picked are equally arbitrary and non-specific, they cannot be counted because there is no “end” from which to begin counting their position.
One (…+1+1+1+…) appears no different from any other, and this representation confers endlessness better than (1+1+1+…+1), which suggests there’s a beginning end and a finishing end, which contradicts the notion of infinity being literally endless. Squeeze a 1 in or take 1 away, the “quality” of endlessness applies either way. The appearance remains the same, the tracability of any change is equally impossible as there is no end from which to check it relative to. It’s both impossible to confirm any change after it has occurred and impossible to equate the series before and after as there is no specific (end) point of reference to use to do so. It’s only possible to present the change as happening as it happens, because it is in finite terms at that point. Even adding in another (…+1+1+1+…) to any given point in the previous one presents the added series as a specific series, representing the infinite with specificity treats it as it was a finite series to inject. Yet afterwards, we see no difference, because it wasn’t finite - it was only represented that way.
Now.
I think I have been hearing protestation this whole time by some that the infinite series of (…+1+1+1+…) was already full, saturated perhaps. If this is the case then I ask what “ends” are resisting any further addition? If all the positions are all already occupied, and there are no empty positions left to fill, how are you judging the beginning and end of these positions? Do these positions have finite ends within this infinite series? Does this suggest that infinite series have finite bounds internally? Finite positions would appear to be a feature of representing infinite series with finite terms like the “1+1+1” in (…+1+1+1+…), or however you wish to represent the set. However, does this “accurately” represent an infinite series? If “infinite” is consistent with itself, including internal consistency, then each “1” would not have a finite bound to its position, it would be a (…+1+1+1+…) in itself, inception style.