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Somewhere (?) in the writings of Gian-Carlo Rota, I recall a statement that old-fashioned Aristotelean syllogisms are not used in modern mathematics. I know of one gaudy counterexample, and wondered whether there are others.

The major premise is Matiyasevich's theorem, proved in 1970:

All recursively enumerable sets are Diophantine.

The minor premise is a discovery in the 1930s, I think by several people including maybe, Kleene, Turing, and Church:

Some recursively enumerable sets are non-recursive.

(Matiyasevich built on work of Julia Robinson, Hillary Putnam, and Martin Davis, done over a couple of decades.)

The conclusion crosses the 10th item off of Hilbert's famous list of problems:

Some Diophantine sets are non-recursive.

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    They are *used* every day. They are just not *mentioned*.2012-05-18
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    @AndréNicolas does that logic he used follow.2012-05-18
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    bad example to use. To be fair this is used every second that it's actually hard to think like that.2012-05-18
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    This is just the fact that $A\subseteq B$ and $B\subseteq C$ imply $A\subseteq C$...2012-05-18
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    No, @MarianoSuárez-Alvarez It is $A\subset B$ and $A\cap C\neq \emptyset$ implies $B\cap C\neq \emptyset$2012-05-18
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    I would argue that almost every "informal proof" (as opposed to formal proofs in a formal language, like the kind that can be machine-verified) is in fact, a syllogism, or strings of syllogisms parsed together. The verbiage added to make these "readable" often obfuscates this.2012-05-18
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    Every normal space is regular; some normal spaces are nonmetrizable; hence some regular spaces are nonmetrizable. There are lots of these.2012-05-21
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    @CarlMummert : But the example I gave was a proof of a really substantial result that possibly couldn't be expressed more simply. Are there others like that?2012-05-21

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