Re: Gödel
"It may be a surprise to a lot of people to learn that not all things are either true or false. There are some things that are neither true nor false, but in a third state of "undecideable", and you can prove that."
It certainly surprises me. That any proposition is either true or false is a basic assumption. Without it we couldn't do much. We wouldn't even know if the square-root of 2 were irrational or not.
"Consider the set of all sets that do not contain themselves. Now note that this set contains itself if and only if it does not contain itself. This is a breakdown of so-called "naive set theory" which motivated changes to both set theory and to logic, and indicated the inherent inconsistency of all logical systems."
It did not indicate the inherent inconsistency of all logical systems, just of naive set theory. No one knows if the currently accepted axioms are inconsistent or not, nor will we ever know.
Morgan
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