As it turns out, visual_spatial reasoning is an excellent way to show that Doron's cardinality claims are based on confusion. Ironically, these confusions come from relying too much on verbal_symbolic representations.
If you think visual_spatially, the set {1,2,3...} is one thing. Visualize it as a collection of objects stretching on forever. It is one collection of particular objects (numbers). It does not matter how many times the name of that set is written down or how many different names we give it; it remains one thing. Thus, when we have a function from {1,2,3...} into {1,2,3...}, we are still talking about that one thing. It has not split or reproduced itself into a domain and a separate codomain. So keep that visualization of the one thing in your mind.
Thinking this way we can see that a set cannot be bigger than itself. After all, there it is. It is just itself. A function from the set to itself does not alter the set, it just tells us that if we start with one member and consider it as an input, the function tells us which other member is defined as the output. The set is one thing that is not changed by the function, the function is just a way of looking at different members. However, if we lose track of the set and focus entirely on the ways we represent it symbolically, we can run into trouble. One can get confused into thinking the set is over here and also over there, then wondering which is bigger. This is a mistake. In particular, the fact that we can decide that a function defines nothing as the output when 1 is input does not show us that the one set {1,2,3...} has more members than itself. It is only if we start to focus too much on the fact that we write down the domain and codomain as two separate entries, or that we have two sides in our doron-arrow representation, that it could even be remotely tempting to see it that way. Once we think visual_spatially as well as verbal_symbolically, it is easy to spot the error.
This shows again that doron-cardinality is not a legitimate notion of the size of a set. It also shows that it is possible to communicate about visual_spatial reasoning. Doron is content merely to say "it's visual_spatial" as if that completes communication. But it is possible to guide visualization. I can tell you to imagine a triangle, or a line segment, or a single set of elements. The verbal expression does not exhaust the thought, but given our similar cognitive skills, I can convey the visualization I have in mind. So, Doron, if visual_spatial reasoning will help your case, tell us how.
