User talk:Qedetc

Welcome
Welcome to ProofWiki! Since you're new, you may want to check out the general help page. It's the best first stop to see how things are done (next to reading proofs, of course!). Please feel free to contribute to whichever area of mathematics interests you, either by adding new proofs, or fixing up existing ones. If you have any questions please feel free to contact me, or post your question on the questions page.

Here are some useful pages to help get you started:
 * Community Portal - To see what needs to be done, and keep up to date with the community.
 * Recent Changes - To keep up with what's new, and what's being added.
 * Check out our house style if you are keen on contributing.
 * Main Page talk - This is where most of the main discussions regarding the direction of the site take place. If you have any ideas, please share them!

Cheers, prime.mover (talk)

Links and Categories
Please try and add a) links to definitions and any other proofs that a page depends on, and b) the category that you consider your proof to belong. That's the whole point of this site - being able to link each concept in a proof back to its source. As it is, I'm going through and adding a MissingLinks template and a category that a proof seems to belong to, but I'd rather be getting on with doing other stuff.

If the concepts and proofs required for an exposition do not exist yet (or you just can't find them) add a link anyway (it will appear red) rather than just ignore it. Please pretty please with a cherry on top. --prime mover 15:38, 2 June 2011 (CDT)


 * Thank you for your response. Please be aware that your work is valued, (particularly by me because it's areas of mathematics with which I am unfamiliar - all my postgraduate education has been strictly self-driven) but I for one am not confident that any cross-linking I do is going to be at all accurate. I understand your reluctance to get bogged down in the fiddly detail of all this when all you want to do is lay the results down - but be aware that if we do it, we may get some of the tying-together wrong!


 * I notice that there is significant crossover of terminology between this area of formal language theory / model theory which you're in the process of working on, and the point-set topology that I'm also working on (educating myself as I go). So significant is this, that I'm now wondering about the best way of establishing that linkage. No immediate urgency for that, it's something to put on the back-burner unless you have something solid in this area to document. --prime mover 16:18, 2 June 2011 (CDT)

To jump in -- I have no great aspirations to write new stuff at the moment so I'm quite happy to follow up and add in links where I can (though, as you know, they ought to be checked). So long as you're not too unhappy with my alterations, I'll go on linking stuff in with the results I'm aware of --Linus44 18:09, 2 June 2011 (CDT)


 * ... and one more thing: note that links should not contain any LaTeX. If you want its presentation style to do so, fair enough - but think of a pithy way to interpret that link into English. Again, if it's too much trouble then do what you wanna do, but be aware that any link with LaTeX in it will be impossible to search on so we'd have to change it. --prime mover 00:15, 3 June 2011 (CDT)


 * Picking a page name is an art. My personal taste is for short names, the pithier the better, a bit like a newspaper headline. The trouble with that is that their intent can be confused. Not completely necessary to completely define the result in the page name, just enough to identify it. No matter, if the page name is crap it can always be moved to a new one. Variable names in page names - if you can't think of a better way, no worries. --prime mover 15:09, 3 June 2011 (CDT)

Crossover between formal language theory and topology
Well for one, there's ultraproduct which uses the concept of ultrafilter. Then there's the Compactness Theorem which may (or may not) have a topological isomorphism going on.

It sounds plausible, because point-set topology is concerned with the nature of sets which are or are not an element of a subset of the power set of a given set, and model theory does the same thing, by assigning a "true / false" or "satisfiable / unsatisfiable" label against a set which could be considered as being "open / closed" in the topological context. BUT ... do these topological concepts continue to hold in model theory?

Specifically, if it can be shown that the union of any collection of "satisfiable" models is also satisfiable, and the intersection of two satisfiable models is also satisfiable, then there's your isomorphism.

I haven't investigated any of this, I'm just rattling my fingers under the effect of late-Saturday-night chemicalisation, if you get my drift. --prime mover 17:06, 4 June 2011 (CDT)

I'll use this as an excuse to ramble then, and throw a bunch of text at you that I'm not sure of the value of. :D

First, I'll just throw this link at you http://en.wikipedia.org/wiki/Ultralimit and mention that there's some use of these ultrafilter based constructions in large-scale geometry stuff (like geometric group theory). I think the construction of asymptotic cones was developed or popularized or something like that by a model theorist. I don't know how deep those connections are though.

Perhaps closer to what you're thinking of is http://en.wikipedia.org/wiki/Type_%28model_theory%29#Stone_spaces. It's possible to define a topological space where, loosely speaking, points correspond to descriptions of elements under some fixed theory, and isolated points correspond to descriptions which must be satisfied by some element in each model of the theory. I'm badly paraphrasing one way of stating the omitting types theorem, which seems to be close to what you're speculating about. Separately I've seen these sorts of spaces used for quantifier elimination proofs. So there's definitely some useful interplay between topological thinking and what's going on in model theory. I guess this is further suggested by the fact that in continuous versions of logic / model theory (where you're allowing truth values in e.g. a bounded interval; this includes usual true/false valued logic as a special case), these kinds of type spaces can be given metrics, so they really do present some notion of "closeness" between descriptions of elements.

Though, the above isn't really on the level of theories like you seem to be suggesting, since it involves a fixed theory. Unfortunately, although an arbitrary intersection of satisfiable theories is always satisfiable (any model of one theory is a model of the intersection), a general union of satisfiable theories is not necessarily satisfiable, for example a theory which contains a sentence stating that there are exactly $n$ elements cannot be consistent with a theory which asserts that there are exactly $n+1$ elements. More generally, if $T$ contains a sentence $\phi$, then it is always inconsistent with the theory $\{\neg\phi\}$. So there isn't a topological space whose open sets are the consistent theories in some language.

I'll stop rambling now since I'm not sure how much of that is even readable. Qedetc 19:03, 4 June 2011 (CDT)


 * On the button. Exactly the sort of thing I was thinking about. Now I know there's something out there, I can check it out at leisure. Thx. --prime mover 02:46, 5 June 2011 (CDT)

Reinvention of Propositional / Predicate Calculus?
I've read a few books on logic on various levels, and one of the things that I've noticed is that they all say very much the same sort of thing but with different words, symbols and pictures. Ultimately it all boils down to the same thing.

I'm worried that we're about to have the same thing happen on this site. In particular the pages supporting the definition of "type" seem to repeat in a different form the same concepts that already exist in the Mathematical Logic section. Not quite sure what to do about it though, because most of this stuff is what I've taught myself so I don't know what the currently "fashionable" approach is, beyond the fact that modern scoffers sneer that Bourbaki's out-of-date and their approach is complete rubbish (because, ahem, we don't really understand it).

So what I'm trying to say is, careful with that axe, Eugene - see whether what's already there can be amended to hold your approach to this without reinventing everything and having two completely separate threads that say the same sort of thing in different language. --prime mover 14:30, 5 June 2011 (CDT)

This is definitely a valid concern, and I'm worried about it myself. I'm also concerned if I eventually want to add any continuous model theory stuff, because then the requirements for signatures and structures change a bit, and most of the ways I've speculated about so far for how that would be handled seem like they would be confusing for everyone else.

I think I hooked up most of the terms on the page Definition:Type (Model Theory) that needed defining with definitions that were already there. I've been putting off linking "satisfiable" to things because Definition:Satisfiable redirects to a page that defines it in terms of Definition:Boolean Interpretation. The definition I'm used to uses an assignment of variables to elements and then evaluation. It is really all the same, but the different levels (the boolean one feels more "proposition calculusy" to me than "predicate calculusy") and flavours of content (computability theory, proof theory, model theory) make it sound different.

I'd prefer not to pave 20 different roads to the same place, and I probably have in a few spots. I guess part of it so far has been that I'm still getting used to jumping on other people's work. I'm a bit weary of all the interconnections, and I'm not sure what the best way to add to or alter a definition is if it seems a bit off from the context I'm writing in.

Qedetc 15:50, 5 June 2011 (CDT)


 * Are you familiar with object-oriented programming? I wonder whether to use an approach similar to that, whereby there is a "basic" definition for the "proplog" stuff which has a "subclass" for "predlog" which has extra attributes. If you start with a definition which has the absolute raw basics in it, you can add a page (or if it works better, a subsection of the same page) which adds the appropriate extra bells and whistles for the extra requirements. See how Definition:Partition (Set Theory) and Definition:Partition (Topology) have been configured (although that's a lame example, can't think of a better one offhand). --prime mover 16:08, 5 June 2011 (CDT)