Definition talk:Relativisation

I see what you are achieving here, but it is IMO very risky to merge the $\in$ from the formal language and the standard $\in$ used for quantifying (unless this is specified on some definition page, of course). Are you going to write a page formally defining the language of set theory or will it stay implicit until I have asked about it enough to be able to write it myself? I used to have a digital copy of Takeuti but my pc crashed last week so I can't get to it atm. --Lord_Farin 21:03, 6 August 2012 (UTC)


 * I could write up the page. What should I call it?  Definition:Language of Set Theory or put it somewhere under Definition:Formal Language? --Andrew Salmon 21:27, 6 August 2012 (UTC)


 * Definition:Language of Set Theory would be fine; please try to use the terminology and symbology from Definition:Language of Predicate Logic and try to crank out a reference to the signature being only the binary relation symbol $\in$ (or $E$ or whatever). --Lord_Farin 21:34, 6 August 2012 (UTC)


 * Please be aware that I am writing Induction on Well-Formed Formulas at the moment, it might come in as a handy reference in proofs. --Lord_Farin 21:35, 6 August 2012 (UTC)

What does stuff mean?
My initial take on this is that the following need to be clarified:


 * 1) "Let $p$ be a well-formed formula." In what context? In order to be a well-formed formula, you need a formal language for it to be a well-formed formula of. It is presumed it's "some version" of predicate calculus, but the precise dialect needs to be made precise - there are many ways of formulating such.
 * 2) What is $A$?
 * 3) What are the symbols allowed in this particular version of predicate calculus? Is $\iff$ a meta-symbol or is it a symbol of the formal language?

Once this is established this page will make a lot more sense on its own. --prime mover 21:58, 6 August 2012 (UTC)


 * Which is precisely what I have asked Asalmon to write and I think he is doing it now. --Lord_Farin 22:00, 6 August 2012 (UTC)
 * $A$ (I think) is a first class citizen like $x$; I think the typographical distinction has to do with their conceptual rôles being different. But again, this is plain guessing/intuition, it's not my page. --Lord_Farin 22:03, 6 August 2012 (UTC)


 * $\iff$ is a meta-symbol. It's reducing any well-formed statement $p^A$ to other statements not using $p^A$.  It's supposed to denote definitional abbreviation. --Andrew Salmon 22:08, 6 August 2012 (UTC)


 * Is it OK to copy/paste information from Definition:Language of Predicate Logic to Definition:Language of Set Theory, namely the information about quantifiers and connectives?


 * I think simply referring to Definition:Language of Predicate Logic suffices, something like:

"The language of set theory is the language of predicate logic endowed with the binary relation symbol $\in$."


 * and then some elaboration on that we'd like to consider some connectives to be def. abbrev. --Lord_Farin 22:23, 6 August 2012 (UTC)


 * For completeness' sake, it may be better/good to simply copy the stuff as well. There is no harm in duplicating stuff when the page is so foundationally important. --Lord_Farin 22:25, 6 August 2012 (UTC)


 * Just occurred to me that you shouldn't forget the equality relation... --Lord_Farin 22:40, 6 August 2012 (UTC)