Bertrand Russell, Set Theory and Russell's Paradox - Professor Tony Mann

[brandonpeck]

Best explanation I've heard so far!

[NebulousEnigma]

Here are some paradoxes I thought up:

Don't follow this order
The set of all things not in this set
Don't take this advice
This statement is inconsistent
This is not a fact
This sentence does not contain accurate information
This sentence claims the exact opposite to what it says
I am skeptical of this sentences claims
I disagree with this sentences claims
This sentence does not describe itself
This is an exaggeration
This statement is inaccurate
Being uncertain as to whether or not you have uncertainty about this sentence
over self confident in thinking you are over self confident
This is unbelievable
This is a Reference to the things that are impossible to reference
If you refer to the things that are impossible to refer to then you have refereed to them, thus they are not something impossible to refer to. Thus there is no such thing as something impossible to refer to. This however means that you have not actually refereed to something impossible to refer to, but rather something possible to refer to.

Definition denial syndrome (DDS) is the fictitious mental condition that I invented that describes the state wherein the sufferer does not believe they have DDS
Do you believe you have DDS?
if you do not believe you have DDS when you know the definition then you must be deluded.
If you believe you have DDS, then you don't have DDS, However, if you truly believe that you have DDS and you understand the definition then you must be deluded. This is paradoxical because no belief is justified, you are deluded whether or not you believe.
...A man believes he has at least one delusion, is he deluded? If he is deluded, then he is not deluded because his belief is accurate. If he is not deluded them he is deluded because he holds a false belief

[dAvrilthebear]

I should say that this was well presented, thank you!

[hexonatapeloop]

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[friendlybanjoatheist]

"Cantor's Theory of Infinite Sex"? Where can I learn more about that? :-)

Good lecture. Thanks.

[orhununyeri]

Somehow i feel like there must be a connection about this paradox with the Heisenberg Uncertainty Principle.

[spencer]

I think the heteroligous analogy is a good way to think about wave quantum duality. Its just a totally different set of events from those two things.

[joshuaronisjr]

The difference between the village, and the set or the adjectives examples, is that while the village example was a statement, the other two were questions.
You can say the statement is nonsense, and you're done.
But the question you can't answer.

[SeanMauer]

I call these "oscillating paradoxes" and I wonder if there's a connection to the way electromagnetic waves propagate.

[noname-icmx]

good explanation of russell's paradoxe

[donmrdiego]

What if a barber comes from a different village to shave the original barber? Then the original barber will be part of the ones who do not shave themselves, and the barber from the other village decides not to be part of such set, grows his beard long, banishes the original barber for being so conceited by trying to shave himself... and charges double to the inhabitants of the village. And sends flowers to Russell, for there is no such paradox, it seems to be a linguistic issue, not mathematical. Well, I am no philosopher, nor mathematician, how does this help the world?

[chrisg]

1) The two paradoxes, the barber and the Russell set, are alike in being symbolisable as a compound relational proposition: (x) Sbx <=> ¬Sxx. For all x, whether men in the village or sets, the barber b shaves x if and only if x doesn't shave himself, the set  b includes x if and only if x doesn't include itself. On substituting the letter b for x we derive the formal contradiction: Sbb <=> ¬Sbb, Formally or structurally there's no difference, and could just as well symbolise any other pair of subject-verb-object clauses linked by the functor <=>.  There's nothing special about sets or barbers, it's all about the relationship between two distinct sentences. 

2) We can also generate other paradoxes with this structure, for example:
(x) Sbx <=> ¬Sxb    the barber who shaves all and only those who don't shave him
(x) Sxb <=> ¬Sxx    the barber who is shaved by all and only those who don't shave themselves
(x) Sbx <=> ¬Sbx    the barber who shaves all and only those whom he doesn't shave

3) In each case the apparent contradiction, whether in the premise (as in the third of the above examples), or the derived expression Sbb <=> ¬Sbb, can be resolved by noting that the individual sentences or constituent propositions occur at different places in the compounds and therefore refer to different times. This logical form could be made explicit with grammar, such as tense markers, or time vocabulary such as "yesterday". The barber shaves all and only those today who didn't shave themselves yesterday. Does he shave himself today? Yes if he didn't yesterday, no if he did. Logical. Mutatis mutandis for the Russell set, which includes now all and only those sets which  previously didn't include themselves. We just need to start taking account of relative times in these mathematical expressions, as we already do to some extent with the use of brackets for the operations of arithmetic and algebra.

[xoppa]

To sum up the lecture...the Russel set paradox is a much more serious problem than the Barber paradox. The conclusion of the barber paradox is simple, no such village exists since it would imply a contradiction (and we don't accept contradictions existing in reality). But in set theory we want sets of the form { x | P(x) } to exist, where P(x) is a well defined property involving x. Then we have a problem when we define the set of all sets that do not contain themselves, i.e. R ={ x | x ∉ x }. By the axioms of naive set theory such a set exists, and then we immediately get the contradiction R ∈ R <--> R ∉ R. From a false statement you can derive anything, so such an axiom system for set theory is useless. ZFC axioms fix this problem using a modified axiom of predication.

[nullvoid]

Paradoxes goes to infinity and infinity is non deterministic. So, Universal set is an infinite set which may contain itself and make itself another infinite set, and well that's a paradox.
What am I even trying to stipulate here???
And that's another Paradox!!!

[eXtremeDR]

The common false assumption regarding paradoxes is that most people assume there are only two possible states: true or false. Correct would be: true, false, both and none. Which is itself a paradox. 

[MrClaverp]

The set of all sets contains all numbers and symbols but does not contain those numbers and or symbols that identify it as the set of all sets. Because such numbers and or symbols cannot be verified, therefore, the set of all sets has non-verifiable meaning.

[ravitheja]

The truth is false. Is it true or false?

[DavidBadilloMusic]

Sometimes I feel like all these paradoxes that come out of recursive concepts are given an additional "allowance" that could never ever be reproduced in any analogous form in real, palpable life. Why is it permitted to think of a Set that contains ALL sets, including itself, but in real life there could never ever be a container that also contains itself. Could there ever be a bucket so big that contains all the buckets of the universe including itself? I highly doubt it. In physical terms, it doesn't make sense. Why do we grant these liberties to conceptual analogies just because they are conceptual? Shouldn't all science, including mathematics, be derived from the physical universe? Just because I can say "this is my cubical sphere" doesn't mean that the phrase is meaningful or can be reproduced in real life. Like when you can represent some artifacts in graphical form in such a way that they generate an optical illusion. That doesn't mean that the idea the illusion generates in our mind can be replicated in the real world.

[morgengabe]

The last one is just a funny coincidence. Not a real paradox, imo.
At the very least you'd need a clearly conflicting definition of what it means for a word to describe itself. Perhaps more importantly, you'd need a better starting point than "the word short describes itself" Oh yeah? I'd wager that the dictionary consists of more words that are equal in length or shorter. Someone crack python out and verify for us though.