Here’s an exercise in separating fundamentals from conventions: if we held a technical summit with all of the galaxy’s advanced alien civilizations, what would we find we had in common, and what would we find inscrutable?
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I'm probably under-educated in this area, but I do not understand how does constructivism work if we go beyond finitary objects. How does one produce an example of an infinite, non-recursively-enumerable or non-arithmetic set of integers?
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Constructivists accept that a thing exists if one can produce a constructive proof of its existence. The constructive reals can be defined by a function that produces a successively tighter bounds on the exact value and is proven to converge, even though computing it would take
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The Axiom of Choice shouldn't be viewed as true or false, but as a tool to explore the space of non-constructively provable propositions. Anything that can be proven without AC ought to be, as a proof is more powerful as it represents a computation.
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