In this video, the mathematician Gregory Chaitin states that "the notion of the set of all sets is self-contradictory". What does "self-contradictory" mean? Is it different from "contradictory"? There are set theories in which there exists a set that contains every set as an element.
Self contradictory is something that contradicts its own self. A set of all sets is self contradictory because a set cannot contain its own self normally, so it cannot contain "all sets". Its a self reference paradox.
EDIT: To locate the self reference element check this example:
Suppose that every public library has to compile a catalogue of all its books. Since the catalogue is itself one of the library's books, a librarian should include it in the catalogue for completeness. But what would happen if there were a restriction that when the book is a catalogue it should include a list of its contents? This will lead to an infinite repetition of the catalogue name with its contents.
Book list -------- a, b, c, book list (a, b, c, book list( a,b,c, book list( -> inf ) ) )
So it is not pure self reference that makes the paradox (it only sets the basis for the paradox to appear) but the self-negating possibility of a self referential statement. I can say "I am alive" but i cannot say "I am not alive".
A snake does not have a problem eating snakes or tails but if it eats its own tail it is self consumed. This leads to the Russel's paradox , where a universal set can contain itself but it would lead to another set of sets that "do not include themselves" which is paradoxical for a universal set.
All logical contradictions are based on self contradictions.
A self contradiction occurs when there is a statement p such that both p and not p hold.
p and not p
A statement, or a set of statements, is called self contradictory iff (*) it entails a self contradiction.
s => p and not p
A predicate Q, or a description "the Q", are called self contradictory iff any attempt to use them will entail a self contradiction. This is the sense in which "the set of all sets" has been proved to be self contradictory.
Qa => p and not p
The Q exists => p and not p
Finally, for a use for "contradictory" without "self": we say that statements (or sets of statements) p1 and p2 are mutually contradictory iff neither is self contradictory, but their conjunction (p1 and p2) is self contradictory. For example, the statements "there are no unicorns" and "I saw a unicorn at the park" are mutually contradictory.
p1 =/> p and not p
p2 =/> p and not p
p1 and p2 => p and not p
(*) iff = if and only if