{"id":23418,"date":"2013-06-27T10:00:27","date_gmt":"2013-06-27T14:00:27","guid":{"rendered":"http:\/\/blogs.ams.org\/mathgradblog\/?p=23418"},"modified":"2014-07-02T09:53:40","modified_gmt":"2014-07-02T14:53:40","slug":"russell-naive","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/mathgradblog\/2013\/06\/27\/russell-naive\/","title":{"rendered":"Russell Was Not Naive"},"content":{"rendered":"

\"images-2\"<\/a>As one peers at a radiating metropolis of buildings piercing the night sky, it is easy to imagine the thought and engineering that produced such a modern wonder.\u00a0\u00a0\u00a0 The intricate detail and unique designs appeal to the artistic senses.\u00a0 Before each building\u2019s architectural beauty is manifested, a strong foundation must be poured.\u00a0\u00a0\u00a0 Such a concept is inherent throughout physical law and abstract mathematics.<\/p>\n

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In the trenches of abstract mathematics\u00a0set theory\u00a0forms its foundations.\u00a0\u00a0\u00a0\u00a0Georg Cantor defined a set as, \u201ca gathering together into a whole of definite, distinct objects of our perception or of our thought- which are called elements of the set.\u201d\u00a0\u00a0 This definition is known as a na\u00efve approach to set theory and has (as will be shortly described) cracks.\u00a0\u00a0 One such man to find these cracks was Bertrand Russell.<\/p>\n

In na\u00efve set theory, the idea that any set can be formed without contradiction is an illusion.\u00a0\u00a0 Russell showed this with the following Paradox:<\/p>\n

Russell\u2019s Paradox (1901): <\/b>Suppose there exists a set A such that the only elements of A are the set of all sets that do not contain themselves.\u00a0 Does A contain <\/b>itself?<\/p>\n

One can see the apparent contradiction that arises as a consequence of the na\u00efve set approach.\u00a0 If A contains itself, then it is in contradiction with the description that A only contains sets that do not contain themselves.\u00a0\u00a0\u00a0 On the contrary, if A does not contain itself, then this is also a contradiction since A contains all of the sets that do not contain themselves.<\/p>\n

Another version derived from Russell\u2019s Paradox is the Barber Paradox.\u00a0 Basically this paradox proposes that, if there exists only one Barber in a small town that exclusively shaves men that don\u2019t shave themselves, then who shaves the Barber? \u00a0 The same self-contradiction emerges as with the original version of Russell\u2019s Paradox.<\/p>\n

These self-contradictions required the development of a more formal set theory.\u00a0 This was accomplished by two mathematicians, Ernst Zermelo (1908) and Abraham Fraenkel (1921), who developed an axiomatic approach to set theory.\u00a0 Zermelo-Fraenkel set theory avoids the contradiction of Russell by ruling it out using defining axioms.<\/p>\n

Set theorists in logic and mathematics continue probing the foundations \u00a0looking for new cracks to patch.\u00a0 Some of the cracks are not easily visible and remain hidden waiting for the right approach to expose them.\u00a0\u00a0 Lurking behind the shadows, you may find one waiting to be discoverd.\u00a0 What do you think about the foundations of mathematics and set theory?<\/p>\n

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As one peers at a radiating metropolis of buildings piercing the night sky, it is easy to imagine the thought and engineering that produced such a modern wonder.\u00a0\u00a0\u00a0 The intricate detail and unique designs appeal to the artistic senses.\u00a0 Before … Continue reading →<\/span><\/a><\/p>\n

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