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First Order Logic
Sheesh, good question. I started on this in Category:Predicate Calculus but got seriously bogged down. My exposition was taken mainly from Jerome Keisler's "Mathematical Logic and Computability" which jumps in at a rather deeper level than I was able to cope with - but so many other works seem incoherent on the subject. Take in from whatever direction you want. The important thing is to get something documented, the details can be tidied up as and when we have an idea of where it's going. --Prime.mover 05:27, 25 August 2009 (UTC)
In response to ...
- Hi, another very general logic query. We presently have formal logic divided into a bunch of categories (some of whose borders are blurry), like predicate logic, propositional logic, mathematical logic and so on. I would propose we either subsume the categories into a mathematical logic supercategory, or eliminate them altogether in favor of a mathematical logic category alone. I'm not sure, e.g., mathematicians really refer to predicate logic as such, but rather to "first-order logic without equality," which is on a continuum with a bunch of other logics (like first-order logic with equality, or second-order logic). My intention is to tackle the important basic results of first-order logic with equality (the logic that set theory takes place in) and then branch out a bit to other areas of mathematical logic like proofs of the incompleteness theorems. Mag487 07:25, 26 August 2009 (UTC)
Good question, to which I'm not sure I can provide an adequate answer ... Several points:
1. A page is not restricted to being in just one category, so if there is a definition which is in that blurry area between one category and another, there's no problem in putting it in both.
2. I think it's better to have more categories rather than less, strangely enough it actually makes it easier to find things - as long as the thing you're looking for is in as many categories as is relevant. I'm prepared to be overruled on this one, but it's not a big question as we can always rearrange it all once we have an idea of what the structure of this is really going to be.
3. I'm just about up to speed on the difference between first-order logic without and with equality (Keisler is fairly clear in his exposition here) but terminology varies depending on whom you read. I believe it's important to implement both definitions, perhaps with a redirect - but we need to decide which way the cat's going to jump.
4. One of my plans was to establish the set-theoretical proof of the existence of the natural numbers. I started out with the exposition of the Definition:Naturally Ordered Semigroup and demonstrated that such a structure is unique up to isomorphism. Then I put down the demonstration that this structure possessed all the necessary properties held by the natural numbers, and took it from there. The next stage is to show that this structure has the Peano properties, and then to show that the definition of Definition:Ordinal also fits in with this definition. Or something. I likewise wanted to get to the incompleteness theorems, but I got bogged down on predicate logic - it's an aspect of mathematics that I have had to teach myself.
5. I also dabbled with documenting the Bourbaki exposition of mathematics, but that didn't look like it was going to gel.
So, in short, feel free to take it away and do with it what you like. The only worry I really have is making sure that every page has a link back to somewhere else - every term used should refer back to some definition somewhere. I think the really basic "axiomatic" terms have all been defined, from my (probably abortive!) work-in-progress on foundations of mathematics, so there should be some page somewhere with something on that can be linked to. If the page has a different name from what you expect it to be, feel free to add a redirect.
Best of luck ... I'll be lurking, but I don't contribute much at the moment, I'm busy on another project. --Prime.mover 18:56, 26 August 2009 (UTC)
When you put comments for a step in the equation template, it's probably best not to put them in parenthesis, just for stylistic uniformity. Nice proofs btw. --Cynic (talk) 03:49, 30 August 2009 (UTC)