Well, I know nothing about design theory (the theory of combinatorial designs?), but I can assure that you can encode it in set theory (I have never heard of a mathematical concept that couldn't be formalized in set theory, and if I did, I would take it kinda like the way a physicist would when hearing to someone say that had a perpetuum mobile).
And since all the foundations can be encoded in each other, you can also encode it in type theory or whatever.
The way I think about it, combinatorics and set theory are separate layers. So combinatorics is not about set theory in the same sense that emacs is not about Lisp, or in the same sense that Lisp is not about the Pentium III.
The foundation language of set theory is like a fully-implemented and trustworthy compiler (of course, the set theory that people do these days is as informal as other advanced math). The difference between programming and math is that in math, stubs don't have to be filled in later, because they are "obvious". While programmers get a lot out of a complete implementation, mathematicians don't get much out of a complete formalization.
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I just thought: probably a lot of model theory is about models of set theory, so maybe this is what the author of the Wikipedia article had in mind.
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krja
(no subject)
Date: 2006-11-19 08:01 am (UTC)And since all the foundations can be encoded in each other, you can also encode it in type theory or whatever.
The way I think about it, combinatorics and set theory are separate layers. So combinatorics is not about set theory in the same sense that emacs is not about Lisp, or in the same sense that Lisp is not about the Pentium III.
The foundation language of set theory is like a fully-implemented and trustworthy compiler (of course, the set theory that people do these days is as informal as other advanced math). The difference between programming and math is that in math, stubs don't have to be filled in later, because they are "obvious". While programmers get a lot out of a complete implementation, mathematicians don't get much out of a complete formalization.
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I just thought: probably a lot of model theory is about models of set theory, so maybe this is what the author of the Wikipedia article had in mind.