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Technology Stocks : Artificial Intelligence, Robotics, Chat bots - ChatGPT

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From: Frank Sully	5/4/2025 7:59:21 PM
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Off Topic Discussion (Continued) Paradoxes of Infinity Many people since Cantor's time (and even since the time of the Ancient Greeks) have been bothered by the strange consequences of infinite sets. At the turn of the 20th Century there was a revolution in the Set Theory underlying Mathematics. E.g., Russell's Paradox : Suppose a barber shaves only those who do not shave themselves. Who shaves the barber? These problems led to the Axiomatic Formulation of Zermelo-Fraenkel Set Theory. Using adjoining to these axiom is the more controversial Axiom of Choice: given any collection of sets, it is possible to define a new set with a choice of an element from each set in the collection. ZFC Set Theory is the standard framework for Modern Mathematics. But another field of Mathematics developed early in the 20th Century was Measure Theory and Lebesgue Integration. This Theory relies upon a desire to generalize the length of an interval to define the "length" or "size" (i.e., "measure") of an arbitrary subset of real numbers. It turns out that one can use the Axiom of Choice to prove the existence of the Vitali Set, a non-measurable set. Even more disturbing is the Banach-Tarski Paradox: using the Axiom of Choice it is possible to subdivide the solid ball of radius one into five subsets, which can then be rigidly rotated and translated and then reassembled into two identical solid balls of radium one. These subsets are non-measurable. For these reasons, some refuse to accept the Axiom of Choice. Other less severe consequences of infinite sets lead some to, e.g., refuse to accept the existence of irrational numbers. Here is a good PBS program on these issues. Watch the first 10 minutes - the last 3 minutes can be ignored. youtu.be

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