08-26-2016, 12:26 PM

secur wrote: "But L'Hospital's rule would make no sense if the product of the limits were always the limit of the product. Nit-picking? Perhaps. but obviously Christian never taught freshman calculus. Usually most of the class is going to be confused by this very misunderstanding."

I would explain it this way. There is a theorem in arithmetic that a point can simultaneously map to any set of points of any cardinality, provided that it is far enough away. The question arises of what happens when antipodal points approach the same cardinal set. We reach a limit in which as you say, the product of the limits were always the limit of the product.

One would find that this only works completely, in the topological domain of a space that is simply connected, like the 3-sphere.

In the incomplete space of quantum mechanics, one can prove anything at all. Even 1 = 2. Even the illusion that one can choose a proposition and its negation, at the same time. Quantum theorists would say that's just the way the world is. Relativists are more circumspect.

I would explain it this way. There is a theorem in arithmetic that a point can simultaneously map to any set of points of any cardinality, provided that it is far enough away. The question arises of what happens when antipodal points approach the same cardinal set. We reach a limit in which as you say, the product of the limits were always the limit of the product.

One would find that this only works completely, in the topological domain of a space that is simply connected, like the 3-sphere.

In the incomplete space of quantum mechanics, one can prove anything at all. Even 1 = 2. Even the illusion that one can choose a proposition and its negation, at the same time. Quantum theorists would say that's just the way the world is. Relativists are more circumspect.