Definition talk:Formal Language

What is a structure and what does it mean for a structure to contain something? I would assume that in the meta-theory, this is a tuple, no? --Dfeuer (talk) 23:42, 6 September 2013 (UTC)


 * In defining a formal language, we can practically only talk in natural language (for no other language has been specified yet). To enforce a "tuple" as the carrying structure is not necessary at this point. We just need something, anything, that consists of these three parts and has them distinguishable.


 * I have changed "contains" to "comprises" to emphasize that there is nothing else in this structure. By all means, feel free to think of the structure as a tuple -- the intuition remains the same. &mdash; Lord_Farin (talk) 09:21, 7 September 2013 (UTC)


 * One of the initial challenges I had when I first started posting up the fundamentals of maths was: how not to be circular. An ordered tuple is defined as a set with a particular structure. A set is defined via ZFC or whichever axiomatic structure. ZFC itself contains an axiom which accepts a concept derived from logic. In order to define that logic rigorously, we need to specify some sort of formal language so to do. The rigorous definition of that formal language requires, in this page, the use of the definition of "ordered tuple".


 * I broke this chain by starting with the axioms of natural deduction as the "most fundamental" truths (notwithstanding Euclid), and my initial analysis used nothing more than reasoning based on that. A more rigorous treatment is required (as has been pointed out a few times), but such a treatment does require machinery of mathematics which is not, at that level, available. In particular, the step from propositional to predicate logic needs the concept of infinity to be able to justify.


 * In the work which I originally wrote (back in 2003 -- 2005) I used a double-pass (and indeed triple-pass) approach, where the original proplog theory was revisited with just enough of the concepts of basic number theory / abstract algebra having been deduced so as to be able to discuss set theoretical / proplogical concepts allowing the concepts of enumeration.


 * This to a certain extent mirrors the approach taken by, where the concept of a general cartesian product of sets (and consequently set union / intersection) are not even raised (for more than a binary operation) until chapter 18, that is, after the natural numbers have been introduced in chapter 16 (which are brought in via the mechanism of the naturally ordered semigroup). --prime mover (talk) 09:59, 7 September 2013 (UTC)


 * Oh, and in answer to the initial question, could we replace "structure" by "object"? I was the one who originally used the word "structure". Apologies. It had not occurred to me that it would need defining. Should we define "structure"? "A structure is an object with internal structure." (Hell, that ain't the blues ...) --prime mover (talk) 10:03, 7 September 2013 (UTC)


 * I trust in the ability of the reader's mind to intuitively grasp things at this level. We have to start somewhere. We could just as well have written "anything that consists of three identifiable parts" and I contend that "structure" sounds better. Everybody knows what we mean anyway (and here I write "know" in the sense of "deep down, intuitive knowledge"). &mdash; Lord_Farin (talk) 11:22, 7 September 2013 (UTC)