Russell's paradoxic headscratcher

[5 Dimensional Nick]

Dear Nexuser's,

Was just giving an impromptu maths lessons in my party (on Russell's paradox: Russell's paradox - Wikipedia) [i am fun to be with at parties, swear down ] and a very bright 16 year old perked up with a question that hadn't occurred to me before. Is there an example in reality for this set. I had assumed that because it was logically contradictory that such a set could not exist in reality but now thinking about it but its very definition it exists but is there a possible physical example for this set. The universe is a strange place and I could not find an answer or even this question on Google.

Ideas?

X

5D

 

[Nathanial.Dread]

I believe the classic real-world example of Russell's paradox is the barber who shaves all men in a town that do not shave themselves.

Blessings
~ND

 

[5 Dimensional Nick]

thanks,

lol. had heard that but never clicked it was analogous to Russell's paradox.

 

[pitubo]

Reality is assumed to be self-consistent. When symbols are used as an abstraction of underlying reality, we want to assume that the self-consistency holds. Some philosophical interpretations of mathematical logic disregard or even openly discard the reference to underlying reality. This opens the door to inconsistencies in the symbolic edifice.

In fact the dominant stream in mathematics is called formalism and it states (per wikipedia) that "the truths expressed in logic and mathematics are not about numbers, sets, or triangles or any other contensive subject matter — in fact, they aren't "about" anything at all. They are syntactic forms whose shapes and locations have no meaning unless they are given an interpretation (or semantics)."

There are some interesting pages discussing paradoxes, set theory and foundational logic. You can find a few if you search for "Russels's Paradox Brouwer". L.E.J. Brouwer founded a school of thought named intuitionism: "The fundamental distinguishing characteristic of intuitionism is its interpretation of what it means for a mathematical statement to be true."

I like Wittgenstein's observation quoted on the wiki page you referenced (albeit misspelled, here is the corrected link):

The reason why a function cannot be its own argument is that the sign for a function already contains the prototype of its argument, and it cannot contain itself. For let us suppose that the function F(fx) could be its own argument: in that case there would be a proposition 'F(F(fx))', in which the outer function F and the inner function F must have different meanings, since the inner one has the form O(f(x)) and the outer one has the form Y(O(fx)). Only the letter 'F' is common to the two functions, but the letter by itself signifies nothing. This immediately becomes clear if instead of 'F(Fu)' we write '(do) : F(Ou) . Ou = Fu'. That disposes of Russell's paradox. (Tractatus Logico-Philosophicus, 3.333)

If your friend is interested in foundational logic, you could point him to Wittgenstein's Tractatus. Attentively reading it cover to cover IMHO classifies as a "heroic dose"

BTW, the barber example is also mentioned and discussed on the wiki page for Russell's paradox.

 

[anon_003]

I watched a snippet of a video lecture on youtube. The professor in it had another example. Link

(Professor is Tony Mann)

In it he brings up the Grelling Nelson Paradox, which is analogous in many ways to the Russell Paradox.

He defines 'heterologous' as an adjective that doesnt describe itself. The first examples of words that would be considered heterologous were long (because it is only 4 letters despite its meaning) and monosyllabic (because it is more than one syllable).

He went on to pose a question. Does heterologous describe itself? Is 'heterologous' itself heterologous? If it is, then it doesn't describe itself, so therefore it does describe itself. If it isnt, then it does describe itself, which consequently means it doesnt describe itself. Ad infinitum.

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This could have many applications in the real world. Perhaps the nature of existence. Assuming something came from nothing, then is nothing something? If nothing is something, then that something is nothing. etc

 

[hixidom]

I watched a snippet of a video lecture on youtube. The professor in it had another example. Link

(Professor is Tony Mann)

The lecturer says that the Barbershop Paradox simply proves that the hypothetical place with the barber does not exist... What does that imply about the more general paradoxes which apply to this reality?

Otherwise, here's what I would think is the most general paradox: X and not-X.
I said it, but that doesn't mean it's true. I simply used the english language in an invalid way. I guess all paradoxes result from such mis-usage. In the case of Russell's Paradox, it is resolved in more advanced formalizations of set theory which came later. I think that such advances in language either improve the restrictions on what can be said or they provide a way of synthesizing concepts that were previously contradictory. For example (by Hegel), "being" and "nonbeing" are mutually exclusive without a higher level concept ("becoming" ) which synthesizes them.