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text: '#Controversy: Mathematicians Divided! Counter-Intuitive Results, and The History of the Axiom of Choice#\nMathematicians have been using an intuitive concept of a set for probably as long as mathematics has been practiced. \nAt first, mathematicians assumed that the axiom of choice was simply true (as indeed it is for finite collections of sets). \n\n[-https://en.wikipedia.org/wiki/Georg_Cantor Georg Cantor] introduced the concept of [-transfinite_number transfinite numbers] \nand different [-4w5 cardinalities of infinity] in a 1874 \n[https://en.wikipedia.org/wiki/Georg_Cantor%27s_first_set_theory_article paper] (which contains his infamous\n[-https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument Diagonalization Argument]) \n and along with this sparked the introduction of [-set_theory set theory].\n In 1883, Cantor introduced a principle called the 'Well-Ordering Princple'\n(discussed further in a section below) which he called a 'law of thought' (i.e., intuitively true). \nHe attempted to prove this principle from his other principles, but found that he was unable to do so.\n\n[-https://en.wikipedia.org/wiki/Ernst_Zermelo Ernst Zermelo] attempted to \ndevelop an [-axiom_system axiomatic] treatment of set theory. He \n managed to prove the Well-Ordering Principle in 1904 by introducing a new principle: The Principle of Choice.\nThis sparked much discussion amongst mathematicians. In 1908 published a paper containing responses to this debate,\nas well as a new formulation of the Axiom of Choice. In this year, he also published his first version of \nthe set theoretic axioms, known as the [-https://en.wikipedia.org/wiki/Zermelo_set_theory Zermelo Axioms of Set Theory].\nMathematicians, [-https://en.wikipedia.org/wiki/Abraham_Fraenkel Abraham Fraenkel] and \n[-https://en.wikipedia.org/wiki/Thoralf_Skolem Thoralf Skolem] improved this system (independently of each other)\ninto its modern version, the [-https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory\n Zermelo Fraenkel Axioms of Set Theory].\n\nIn 1914, [https://en.wikipedia.org/wiki/Felix_Hausdorff Felix Hausdorff] proved \n[https://en.wikipedia.org/wiki/Hausdorff_paradox Hausdorff's paradox]. The ideas\nbehind this proof were used in 1924 by [-https://en.wikipedia.org/wiki/Stefan_Banach\nStefan Banach] and [-https://en.wikipedia.org/wiki/Alfred_Tarski Alfred Tarski]\nto prove the more famous Banach-Tarski paradox (discussed in more detail below).\nThis latter theorem is often quoted as evidence of the falsehood of the axiom \nof choice.\n\nBetween 1935 and 1938, [-https://en.wikipedia.org/wiki/Kurt_G%C3%B6del Kurt Gödel] proved that\nthe Axiom of Choice is consistent with the rest of the ZF axioms.\n\nFinally, in 1963, [-https://en.wikipedia.org/wiki/Paul_Cohen Paul Cohen] developed a revolutionary\nmathematical technique called [-forcing_mathematics forcing], with which he proved that the \naxiom of choice could not be proven from the ZF axioms (in particular, that the negation of AC\nis consistent with ZF). For this, and his proof of the consistency of the negation of the \n[-continuum_hypothesis Generalized Continuum Hypothesis] from ZF, he was awarded a fields medal\nin 1966.\n\nThis axiom came to be accepted in the general mathematical community, but was rejected by the\n[-constructive_mathematics constructive] mathematicians as being fundamentally non-constructive. \nHowever, it should be noted that in many forms of constructive mathematics, \nthere are *provable* versions of the axiom of choice.\nThe difference is that in general in constructive mathematics, exhibiting a set of non-empty sets\n(technically, in constructive set-theory, these should be 'inhabited' sets) also amounts to \nexhibiting a proof that they are all non-empty, which amounts to exhibiting an element for all\nof them, which amounts to exhibiting a function choosing an element in each. So in constructive \nmathematics, to even state that you have a set of inhabited sets requires stating that you have a choice\nfunction to these sets proving they are all inhabited.\n\nSome explanation of the history of the axiom of choice (as well as some of its issues)\ncan be found in the \npaper "100 years of Zermelo's axiom of choice: what was the problem with it?"\nby the constructive mathematician \n[-https://en.wikipedia.org/wiki/Per_Martin-L%C3%B6f Per Martin-Löf]\nat [-http://comjnl.oxfordjournals.org/content/49/3/345.full this webpage]. \n\n(Martin-Löf studied under [-https://en.wikipedia.org/wiki/Andrey_Kolmogorov Andrey Kolmogorov] of\n [-5v Kolmogorov complexity] and has made contributions to [-3qq information theory], \n[-statistics mathematical_statistics], and [-mathematical_logic mathematical_logic], including developing a form of \nintuitionistic [-3sz]).\n\nA nice timeline is also summarised on [-http://plato.stanford.edu/entries/axiom-choice/index.html#note-6\nThe Stanford Encyclopaedia of Philosophy].',
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