[Zhiliudian], Please Maintain Basic Humility Toward Classical Mathematical Theories!
Today a [Zhiliudian] wrote "I will not praise Russell's paradox, I will only question this paradox." From this I can see that he fundamentally doesn't understand the background or significance of Russell's paradox, committing the same error as Yunguzi questioning Marx, or as Dacai Shifu and various Chan Masters questioning Miss Chan -- that is, criticizing others without even understanding what they're saying. Is this phenomenon being so widespread perhaps a bit abnormal? First let me quote a passage from his post:
"Why would anyone praise Russell's paradox? I think the question itself is erroneous. 'Let T be the collection of all sets that are not elements of themselves, i.e., T := {x | x not in x}. The question is: does T belong to T?' To define a set one must define a definite property; all objects possessing a certain property belong to a certain set. In this proposition, 'element of itself' is not a clear property. It can be a pronoun. Suppose 'itself' equals natural numbers, then this statement becomes the set of all elements other than non-natural numbers, and then you give a concrete example and absolutely no paradox will appear. Therefore, the paradox appears because human language is imprecise. Once 'itself' is made concrete, the paradox vanishes."
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What this person says here has a much clearer and more detailed explanation in any proper book on set theory. Before Russell's paradox appeared, the general assumption was that any given property could yield a set. Russell's paradox precisely showed that the concept of set cannot be used arbitrarily, which led to the axiomatization of set theory. Axiomatized set theory long ago eliminated Russell's paradox -- this is the most basic common knowledge!
"It is generally acknowledged that Russell's paradox straightforwardly and directly uses the diagonal method to construct a self-contradictory set T containing a logical contradiction, dispensing with all mathematical technicalities, fully displaying the 'secret weapon' that plays the key role in Cantor's proof that the set of real numbers is uncountable and in Cantor's theorem -- the logical contradiction in the paradox -- making it a fine piece of mathematical art successfully applying the diagonal method to expose defects in existing theoretical systems. Therefore, even as people were sad and dejected, they kept praising Russell's paradox."
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Russell's paradox was merely an occasion that triggered the axiomatization of set theory. After axiomatization, there is simply no place for Russell's paradox in set theory. Professional mathematicians don't concern themselves with this issue anymore. If they were to be concerned, everyone would rather devote all their time to the axiom of choice and wouldn't spare a single second for the so-called Russell's paradox! As for popular books that use Russell's paradox to make money -- that has nothing to do with mathematical theory. Furthermore, "Cantor's 'secret weapon' that plays the key role in the proof of uncountability of the set of real numbers and Cantor's theorem -- the logical contradiction in the paradox" -- this again is something only someone very poor in mathematics could say. Cantor's proof uses nothing more than a simple proof by contradiction. Proof by contradiction is not a paradox. This is such simple common knowledge it needs no elaboration.
In summary, when everyone discusses classical theories, please first understand them properly. Classical theories are classical precisely because they have been questioned by billions of people. The probability of finding an error in a classical theory is one in a billion. The logic of classical theories does not go wrong -- just like Newton's logic was not wrong; it's merely that its scope of applicability was somewhat restricted. Study the classics well, first thoroughly master the classics -- otherwise you'll only end up as a laughingstock!