{"id":131,"date":"2023-07-03T03:02:39","date_gmt":"2023-07-03T02:02:39","guid":{"rendered":"https:\/\/blogs.imperial.ac.uk\/cocteaupedia\/?p=131"},"modified":"2023-07-03T12:34:07","modified_gmt":"2023-07-03T11:34:07","slug":"russels-paradox","status":"publish","type":"post","link":"https:\/\/blogs.imperial.ac.uk\/cocteaupedia\/2023\/07\/03\/russels-paradox\/","title":{"rendered":"Russell’s Paradox"},"content":{"rendered":"

1. The earliest from: Burali-Forti\u2019s paradox<\/em><\/h3>\n

Though this form is only a subset of a general Russel’s paradox (from self reference), I have collected it for it involves an interesting concept in sets: The Ordinals.<\/p>\n

We define a set to be an ordinal if it is hereditarily well-founded and hereditarily transitive. Hereditariness is defined as the specific characteristic of the set is inherited by every element of the set, and transitive is defined as the element of the element of a set is still the element of this set.<\/p>\n

The direct observation of an ordinal O<\/em><\/strong> is that any element, sub element…etc. are all an element of O<\/em><\/strong>. Further more, any smallest indivisible element inside O is an element of O. <\/strong><\/em>Importantly, we immediately recognise that the set of all ordinals is, itself an ordinal (Let’s call it U <\/strong><\/em>).<\/p>\n

This property of ordinals make it possible to give well-defined orders in them. for example, consider an order type that rank each indivisible element, combined with the order type that rank the cardinality of sets (in terms of indivisible elements). In this way, U<\/strong><\/em> should be in a highest rank, higher than any<\/strong><\/em> element in U<\/strong><\/em>, includes itself.<\/p>\n

2. Naive set theory and Russell’s paradox<\/em><\/strong><\/h3>\n

The way that Russell spotted the paradox was through the unrestrictive comprehension axiom. This says that two formula f<\/em> and g<\/em> are identical iff f(x)<\/em>= g(x)<\/em> for any x<\/em>. In this way we can construct a set {x: f(x)<\/em>} to collect all x satisfying formula f(x)<\/em>.<\/p>\n

Russell took f(x)<\/em> as x<\/em> is not an element of\u00a0x<\/em>. Then R<\/em>= {x: f(x)<\/em>} seems to be both an element and not an element of this set. This leads to a contradiction<\/p>\n

3. Solutions to Russell’s paradox<\/em><\/strong><\/h3>\n

Let us rewrite what Naive set theory gives us in a tidier mathematical form:<\/p>\n

\"\"<\/p>\n

Russell’s response is call typed theory. This came from vicious circle principle stating that ” no proposition function can be defined in prior to specifying the function’s scope of application. Thus (1) should be modified:<\/p>\n

\"\"<\/p>\n

Here S<\/em> should be restrict x<\/em> s.t. \u03b5f <\/em>is not impredicative<\/strong>. R<\/em> is defined in such a way that \u03b5f <\/em>is impredictive so according to (2) \u00a0no such a set A<\/em> exist.<\/p>\n

Typed method, effectively, simple denies the set described by “all possible…., universal…” to be a set. However, there are other responses to the paradox. For instance, the axiom schema of Separation states that:<\/p>\n

\"\"<\/p>\n

This is in similar manner to (3) but it does not directly state what set A<\/em> should be. But now take f<\/em> to be the one in Russell’s paradox gives the result of A<\/em> does not exist.<\/p>\n

Other arguments wanted to do it in different manner. Intuitionism and Para-consistency seems to be admit the existence of true contradiction and they altered some basic sentential logic. What are their relation to dialectics? I will come to that later.<\/p>\n

[Reference: Stanford Encyclopedia of Philosophy: Russell’s Paradox]<\/p>\n

 <\/p>\n","protected":false},"excerpt":{"rendered":"

1. The earliest from: Burali-Forti\u2019s paradox Though this form is only a subset of a general Russel’s paradox (from self reference), I have collected it for it involves an interesting concept in sets: The Ordinals. We define a set to be an ordinal if it is hereditarily well-founded and hereditarily transitive. Hereditariness is defined as […]<\/p>\n","protected":false},"author":1741,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[17,16],"tags":[],"class_list":["post-131","post","type-post","status-publish","format-standard","hentry","category-logic","category-set-theory"],"yoast_head":"\nRussell's Paradox - Cocteaupedia<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/blogs.imperial.ac.uk\/cocteaupedia\/2023\/07\/03\/russels-paradox\/\" \/>\n<meta property=\"og:locale\" content=\"en_GB\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Russell's Paradox - Cocteaupedia\" \/>\n<meta property=\"og:description\" content=\"1. 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