definition of semantics ; Wikipedia on Model_(logic)

[gusl]

I'm kinda proud of my wiki page on semantics (which is really a pretty stubby index to other wiki pages on all kinds of semantics), especially my mathematical definition.

I find it strange that Wikipedia has no article called Model_(logic). It makes no sense for them have an article on model theory if they don't define "model". Also, correct me if I'm wrong, but saying that model theory is about the "representation of mathematical concepts in terms of set theory" is total BS.

As important as it is to study the representation of mathematical concepts, I don't think it's a well-defined area of study/research, and if it were, it should be called "formalization studies" or (imagining a good future) "mathematical knowledge representation". Also, you can construct mathematical objects with whatever foundation you want. Why do so many people have a fetish for set theory?

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UPDATE: Who the hell wrote that the semantics of I-F logic is in terms of "zero-sum games"? These are win-lose games! Sigh... Wikipedia...

Answer: a troll, of course.

Comments

[mdinitz.livejournal.com]

I think people like set theory because ZFC has worked so well. Not many people really object to ZFC anymore, other than constructivists who are loud at CMU but not a very large portion of the general mathematical community. Set theory is something that intuitive to most mathematicians, and it seems to work pretty well, so why not use it?

[gustavolacerda.livejournal.com]

But in this context, the point is to define models (combinatorial structures). So why think about foundations at all?

[mdinitz.livejournal.com]

I literally don't know anything about model theory, so I'm going to have to bow out of this conversation. And I'm all in favor of not going down to the foundations when you don't have to, which is why I never do. But certainly in general set theory plays an incredibly important role in combinatorics. The most ubiquitous combinatorial structure, graphs, are inherently defined in terms of sets (a base set V and a subset E of VxV), and the cooler generalizations of graphs (e.g. matroids) are even more directly just set structures.

[gustavolacerda.livejournal.com]

I would say that set theory is the language in which we choose to talk about combinatorics. There may in fact be good reasons why it's the standard choice. But saying that combinatorics is about set theory is like saying that sociology is about English.

OTOH, I sometimes defend the view that *everything* is about combinatorics.

[mdinitz.livejournal.com]

SoI guess even though I'm a pseudo-combinatorialist, I have thhe other view. I think that graph theory is really nothing but the study of certain sets, and I'm even more sure that matroid theory is just the study of certain set systems. I haven't thought about it, but probably even design theory is just sets too. Sets seem to me to be much more fundamental than combinatorics.

[gustavolacerda.livejournal.com]

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.