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.

  • It is a bit vague. There are often principles assumed implicitly ("immutable") in addition to explicitly stated assumptions that are up for acceptance/rejection ("variable"). Self-contradictory means contradictory under the former principles already (for the set of all sets originally those of naive set theory), while contradictory is relativized to the variable assumptions. – Conifold Sep 15 '16 at 18:01
  • or in simpler terms, the contradiction issues from the thing itself (concept, proposition, whatever), not from some additional thing. if you have P and also !P you can derive an immediate contradiction, but it's not a self-contradiction – user20153 Sep 15 '16 at 20:15
  • If you're interested in this sort of thing, try reading Chaitin's terrific reprint book cs.auckland.ac.nz/~chaitin/ws.html It's pretty easy to follow, with only modest math background prerequisites. Sorry, I'm not seeing a free online pdf, but cannot prove one doesn't exist. – user19423 Sep 16 '16 at 2:06

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
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.

  • 1
    self-reference seems a separate topic. – user20153 Sep 15 '16 at 20:17
  • mmm, I'd call "petty god" an oxymoron rather than a self-contradictory concept. it's really two concepts. "set of all sets" obviously self-referential but I'm not sure the same can be said of all self-contradictions. unfortunately I cannot think of a counter-example off the top of my head. I could be wrong. – user20153 Sep 15 '16 at 21:08
  • Hi. I believe @mobileink is correct. When they teach you set theory, they provide non-trivial proofs that "the set of all sets" is self contradictory. Nobody relies on mere self reference in this regard. – Ram Tobolski Sep 18 '16 at 20:15
  • @John Your added examples are interesting, but not exactly related to the case of "the ser of all sets". The library catalog seems infinite, not paradoxical. Infinity makes it of course impractical, but infinity isn't a problem in set theory. As to self negation, there doesn't seem to be any of this in "the set of all sets". It is a strictly positive concept. – Ram Tobolski Sep 19 '16 at 13:28
  • @Ram Tobolski Please read the whole article about Russel's paradox in Wikipedia. en.wikipedia.org/wiki/Russell%27s_paradox Please check that it is included in the list of self-referential paradoxes en.wikipedia.org/wiki/Category:Self-referential_paradoxes – John Am Sep 19 '16 at 14:00

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

  • Hi Ram. your first sentence is very strong. these days there are many logics. if there is a genuine plurality of logics, (including logics that do not involve axioms) then I'm not sure all logical contradictions can be reduced to self-contradictions. thoughts? – user20153 Sep 18 '16 at 20:29
  • @mobileink As you may know, in so called "classical" logic, self contradictions (and only them) imply any other statements (p and not p => q, for any q). There are also "para consistent" logics, in which self contradictions do not imply any other statement. I suppose there could be logics in which some statements which are not self contradictions would imply any other statement. Such statements would be a kind of contradictions which were not based on self contradictions... – Ram Tobolski Sep 18 '16 at 22:11
  • aren't you equating "false" and "contradiction"? you do not need a contradiction to go from P to Q for any Q, all you need is P false. I'm not sure "ex falso quodlibet" and "ex contradictione quodlibet" are the same principle. the former only involves truth, the latter involves inference, at least insofar as arriving at a contradiction involves inference. Maybe I'm nitpicking, but I think the difference between truth-conditional logic and constructive (or inferential?) logic is Really Big. ;) – user20153 Sep 18 '16 at 23:43
  • @mobileink You are correct that falsity does materially imply any other statement. My answer, however, does not deal with material implication, but with logical consequence (entailment). Only contradictions, not falsities, entail any other statement. I have renamed "imply" to "entail" in my answer, to make this clearer. – Ram Tobolski Sep 19 '16 at 11:32
  • I don't see what "p" and what "not p" are . Does it mean existence? P is? P is not? "A human is big and small at the same time. Big in relation to an ant, small to an elephant." Here the subject is predicated with two opposite predicaments and there is no contradiction at all. Also: "I 'm alive today but in 40 years i will be dead". No self contradiction. Also i don't see how "there are no unicorns" and "I saw a unicorn at the park" are mutually contradictory because if the park is the "fantasy" park there is no contradiction. – John Am Sep 19 '16 at 13:12

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