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In Reply to: And now.... posted by MattC on September 11, 2006 at 17:29:27:
A set can either be a member of itself or not. The set of all mathematical objects is a mathematical object so it's a member of itself. The set of all sets with only three members has an infinite number of members so it's not a member of itself.There must be a set of all sets that are members of themselves and therefore a set of all sets that are not members of themselves. Is that set a member of itself?
This is a problem which has no solution. I leads us to Godel's incompleteness theorem (that no mathematical system can be both consistent and complete) which is an even bigger problem.
Follow Ups:
"There must be a set of all sets that are members of themselves and therefore a set of all sets that are not members of themselves. Is that set a member of itself?"A set cannot contain two or more identical elements. So when a set contains all sets, does it become itself (all members of all sets, including itself) and therefore cannot be a set because it contains itself and thus contains an identical element?
Tom
"Nice guy. Really seems to care. About WHAT I have no idea."
-Rodney Dangerfield
The proposition as posed is known as Russell's paradox after Bertrand Russell and it is so serious that set theory had to be changed.One of the resulting set theories is the iterative conception of sets which contains axioms such as yours.
VTA adjustment is of little or no significance because VTA is continually changing and therefore will never be in adjustment. And with all due respect for Mr. Gandy and the Rega name, the paradox might rightfully be known as Gandy's VTA Paradox.
..or a few of the easier words at least.
.
Dave
Later Gator,
Crank up your talking machine, grab a jar of your favorite "kick-back", sit down, relax, and let the good times roll.The early bird may get the worm, but the second mouse gets the cheese.
Not being reminded of all the really stiff shots to the head (Dain Bramage) I've taken over the course of my existence, the sad realities of which I'm forced to revisit whenever I encounter anything along these lines, Say What ??????? ;-)
It follows from the truth of Mark's remarks thatIf there is no problem without a solution, then the moon is made of blue vinyl.
*************88
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