-=Dark_Angel=-, P.I.26 Mar 2005

No, you haven't grasped it.

I'm not talking about a "valve that allows water to pass when it is closed, even as much water as when it is open".

For all I care, that valve doesn't exist. It's irrelevant.

(Look at the words I am writing, and use your brain to understand them.)

I am talking about two SETS of valves. One is the SET OF ALL POSSIBLE OPEN VALVES, and the other is the SET OF ALL POSSIBLE CLOSED VALVES.

Here is what I am saying. This is what my IMPORTANT THEOREM is saying:

Take any OPEN VALVE in the SET OF ALL POSSIBLE OPEN VALVES. Call this valve Ted. Then there is some CLOSED VALVE in the SET OF ALL POSSIBLE CLOSED VALVES, called Roger, who is leaky -- and therefore lets through exactly as much water as Ted.

Ted and Roger are not the same valve. Clearly they aren't, because Ted is OPEN and Roger is CLOSED.

I am also saying the following. Take any CLOSED VALVE in the SET OF ALL POSSIBLE CLOSED VALVES. Call him Gretchen. Now there is an OPEN VALVE in the SET OF ALL POSSIBLE OPEN VALVES, called Nancy, who is blocked, and therefore only lets through as much water as Gretchen.

Clearly, Nancy and Gretchen are not the same valve, because Gretchen is closed, and Nancy is open.

Do you understand? Do you understand how this is utterly different to talking about a SINGLE VALVE which lets through as much water open as it does closed?

Here is a COROLLARY: to the theorem: There is a one-to-one correspondence between the SET OF ALL POSSIBLE OPEN VALVES, and the SET OF ALL POSSIBLE CLOSED VALVES. Take any valve, open or closed. Then there is a valve in the other set that lets through just as much water.

THEREFORE: The set of all possible open valves, collectively, lets through exactly the same amount of water as the set of all possible closed valves.

THEREFORE: Closed-valvedness in itself is not a determinant of how much water a valve lets through.

THEREFORE: Closed-valvedness does not inhibit the flow of water.

Do you understand now? You haven't understood before, although you've repeatedly claimed to. Do you get it now? I don't care if you agree or disagree. I just want to know if you understand what I am saying.