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This list is a small subset of all currently known mathematical axioms. Mathematicians have long pondered whether all of mathematics could, by deductive reasoning, be proved to be reducible to a finite, complete, consistent set of axioms. Giuseppe Peano attempted to achieve that goal for number
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theory. David
Hilbert pursued the same goal for all of mathematics. However, Godel's Incompleteness Theorems proved that neither Hilbert's goal, nor even Peano's less ambitious goal, could ever succeed. Godel's Incompleteness Theorems are widely, though not universally accepted.
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This is a list of some significant mathematical axioms as that term is understood in mathematics, by
Knowledge page. In epistemology, the word axiom is understood differently; see axiom and self-evidence. Individual axioms are almost always part of a larger axiomatic system.
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My confusion was that, as a non-mathematician beginning my study of epistemology, I thought that perhaps it had been proved that all of mathematics was founded in a "list of axioms," and that this page was that list. However, I now believe I was wrong, as detailed below.
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This is a list of axioms as that term is understood in mathematics, by
Knowledge page. In epistemology, the word axiom is understood differently; see axiom and self-evidence. Individual axioms are almost always part of a larger axiomatic system.
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It struck me that perhaps the first few sentences of this page could be more clear, depending on whether I understand Godel's
Incompleteness Theorems.
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I'd appreciate some feedback as to whether my proposed edit (below) is correct, and if correct, whether it would be useful on this page.
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I'm not a mathematician. I'm merely beginning a study of epistemology for personal interest.
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This is my first Talk entry. I apologize if I'm violating a Talk convention with this post.
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Shouldn't axiom of
Kolmogorov be part of this list ?
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Perhaps the first sentence of this page is misleading
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