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For a good concrete example, I would suggest that you look into synthetic differential geometry. This is an axiomatic approach to differential geometry which takes place in a smooth topos. The theory is very beautiful and intuitive, and allows you to rigorously reason using infinitesimals. Since this is a purely axiomatic theory, you can come up with a ...


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Have you seen Ivan Dotu and Pascal Van Hentenryck, Scheduling social golfers locally? They claim 45 golfers in groups of 5 can play 7 days; 54 golfers in groups of 6 can play 6 days; 50 golfers in groups of 5 can play 8 days; and some other results that lie outside the bounds of your question. But they don't actually present any instances of the ...


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Edit: Yep, as pointed out in the comments, my proof is wrong. I totally expected something like this. The fatal problem is that I missed some counterexamples in the general case. I'm leaving this answer up, because as a rule I don't delete posts just for stuff like this, but that's what you get for trying to dive in head-first, I guess. /edit I know this ...


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The interplay between CT and MT is pretty well established. The term to search for is locally accessible categories. Another subject to look at may be topos theory, again with plenty of material online. A complete, or even a very partial list of applications of CT and MT will require a lot of bytes. MT has applications in algebra and in analysis, and that ...


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I don't know a proof for the implication you stated when we are talking about the simple Bunyakovsky conjecture. ( It seems in the beginning that the implication is true by construction of two polynomials $P$ and $Q$ such that $P-Q=2$ and apply the conjecture but it's not obvious like i explained below) The simple Bunyakovsky conjecture states that: any ...


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You ask: Can the proof of one conjecture be considered a proof of the other conjecture? Results of the form "well-known conjecture $\varphi$ implies well-known conjecture $\psi$" are indeed interesting and important. For example, Wiles didn't directly prove Fermat's Last Theorem; what he actually proved was the modularity theorem for semistable ...



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