You do not distinguish between the ever smaller element that exists between any pair of some closer smallest elements.
Wrong, your “ever smaller element” is just your personal ascription that you try to associate with the fact that in a continuous space there is always another point between any two points (and thus smaller line segments between the endpoints of any line segment). While your “smallest elements” are just your personal ascription that you try to associate with the fact that points are zero dimensional and thus have no extents.
Your difficultly remains, Doron, that you just don’t distinguish between your own flippant personal ascriptions and the well established concepts you try to associate them with.
The different names of the smallest elements is possible because of the co-existence of the smallest AND the ever smaller, and it is known by the name "collection".
“different names”? What different names? Again, location is an aspect of ordering not just ‘naming’. Oh and “collection” need not even involve line segments or points, as some simply don’t.
Actually, without the ever smaller element, the smallest elements can't be related to each other.
Sure they can, spaces do not need to be continuous, as some simply aren’t.
So the relative locations of the smallest elements are the result of the ever smaller AND the smallest.
Nope, as noted above and before your “ever smaller” requires a continuous space and as not all spaces are continuous not all locations (relative or absolute) require a continuous space.
You can add relative to the list of the concepts that you (yet) do not comprehend, by your smallest-only reasoning.
Again stop trying to simply posit aspects of your own failed reasoning onto others.
An orange is already a result of the co-existence of the ever smaller element, which exists at once at least at two smallest elements' locations.
No smallest element exists at once at more than one location, which is a property that an ever smaller element has.
So your “ever smaller element” requires at least two of your “smallest elements” for what you consider to be its defining “property”. That should tell you something, if you’re listening to, well, yourself.
This distribution is a result of the co-existence of the ever smaller AND the smallest.
Distributions also don’t need to be continuous, while your “ever smaller” requires a continuous space along with, by your own assertion, at least two of your “smallest elements'”
Yet no smallest element has the property of being at once at both given locations, which a property that the ever smaller element has.
A “property” you claim above requires at least two of your “smallest elements”
You still the not grasp the co-existence of the ever smaller AND the smallest.
You still, apparently deliberately, don’t grasp that your “ever smaller”, explicitly and by your own assertions, requires at least two of your “smallest” for just what you consider its defining “property”. So your “co-existence of the ever smaller
AND the smallest” is just redundant and superfluous nonsense on your part.
That is a challenge that your reasoning simply ignore.
Really? Well evidently you’re the only one failing to meet your own challenge. Perhaps because you just fail to challenge yourself or your own “notions”.
By my own determination only an ever smaller element has the property of being at once at lest at two locations of the smallest elements, where no smallest element has this property.
So once again the defining “property” of your ever smaller element requires at least two of your “smallest element” by your “own determination”.
The continuum is a property of an ever smaller element. No smallest element has this property.
Actually as noted above and before a continuous space is a requirement for what you refer to as your “ever smaller element”.
Your local-only view, which gets only the smallest elements of some collection, ignores the ever smaller elements between them, with actually have the property of the continuum.
Again stop simply trying to posit aspects of your own failed reasoning onto others.
You still to not get the co-existence of the continuum, which is the property of the ever smaller element, and the discrete, which is the property of the smallest element.
You still don’t get that a continuous space is a requirement for what you refer to as your “ever smaller element” as well as , by your own “own determination”, at least to of your “ever smaller element”.
You can wait as much as you like, it does not change the fact that your location-only reasoning does not help you the comprehend the co-existence of the continuum, which is the property of the ever smaller element, and the discrete, which is the property of the smallest element.
Again stop simply trying to posit aspects of your own failed reasoning onto others.
As far as waiting goes, evidently we are going to have a very long wait before you can agree with just yourself or even be bothered to look up what defines a continuous or discrete space.
