Spoken language would be inscrutable. Different physiologies would lead to an inability to speak or even hear each others’ phonemes. Written language would be inscruible, though we might find analogs of nouns, adjectives, and verbs.
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But mathematics would be shared. After translating syntax and symbols, we’d find we had exactly the same constructive axioms, and agree that other axioms are controversial. We’d have the same theorems and the same proofs. They’d rever a Pythagoras and a Leibniz.
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Replying to @TimSweeneyEpic
We already have several axiomatic systems, most notably ZF vs ZFC (as the axiom of choice is/has been controversial). It would be interesting to discover what axioms other intelligent species came up with.
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Replying to @Lucas_Trz @TimSweeneyEpic
I could never understand why some people would reject AC. It's equivalent to "Cartesian product of a collection of non-empty sets is non-empty". They must have a very weird intuitive concept of was a set is, if it does not satisfy this axiom.
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It's like looking looking at that cartesian product of an infinite number of infinite sets with all possible combinations of their elements and saying "There is nothing here." "Wait, don't you see all these?" "Oh, no, they are not sets. No sets here." "Huh?"
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I can understand finitists who say "We restrain our reasoning to finite collections only. Infinite collections are a pure imagination and we could not be sure of anything if we admitted them". But when someone, after accepting the axiom of infinity and the power set axiom, ...
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... rejects the axiom of choice, I just cannot understand what do they mean by "set". Like they are partially blind and just cannot see some sets for some reason, or do not recognize them as sets.
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Replying to @vreshetnikov @Lucas_Trz
The core distinction is between constructive logics (in which proofs of existence are guaranteed to produce an example) and non-constructive logics which can prove something exists without any clue as to what it is.
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Replying to @TimSweeneyEpic @Lucas_Trz
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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Replying to @vreshetnikov @Lucas_Trz
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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forever if the exact value were desired. The constructive reals are the subset of the reals that can be computed in this way. Constructive logic is less powerful than full logic. See the Curry Howard correspondence for the neat programming connections.
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