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On Balzano's definition of greater and less can be based a definition
of Multitude. That is the multitude of a collection will be defined as consisting in
its being greater than those collections that is greater than and being less than those it is
less than. This definition is at least a practicable one. It can be put to use in demonstrations.
In that respect it is infinitely better than what Cantor puts forth as a
definition in the style of Kant, who was a superb logical genius but who never
studied logic and therefore sometimes commits grievous faults of logic. Namely
Cantor tells us that if we think of a collection & then put out of view this and that
element of our idea, what remains is multitude. But that is no definition at
all. It is an experiment telling us granting for the sake of argument that it attains its aim
how to find and identify multitude. A definition
ought to be an analysis of a conception useful in demonstration. The
purely relative definition based on Balzano's definition of greater and less
does that. But it takes too narrow and formal a view of the matter. It fails
to show the immense logical significance of multitude. It also makes
it to be a purely relative between one collection & others. That is not true.