@@ Line 1: / Line 1: @@
+#REDIRECT [[File:lenumakfaprami.pdf]]
-'''<- [[jbocre: Lojbanic Qabbalah|Lojbanic Qabbalah]]'''
-Now we can construct Gödel numbers for any [[jbocre: Lojban Description|Lojban Description]] text.
-Therefore, if we can prove that the lojban grammar &amp; morphology is
-expressible
-using lojban [[jbocre: mekso EX|mekso EX]] pertaining to the Gödel numbers, then,
-according to Gödel's
-proof (1931), it would be proven that there exist lojban sentences that
-cannot be proven grammatically/morphologically correct using a proof
-expressed in lojban.
-(not like ''[[jbocre: na nei|na nei]]'', mind you, I really mean correctness on the
-syntactical view.) ''OK, but ''na nei'' was the kind of sentence whose
-equivalent Gödel use to blow up sign systems '''semantically'''...''
-And given that Lojban's grammar &amp; morphology seems to be parsable using
-DFAs, I am pretty sure that it is expressible with arithmetical
-statements and therefore MEX.
-Of course, this is an annoying conclusion, because we feel that lojban
-is very able to fully express its own syntax. But on the other hand, if
-this conclusion proves to be false, then it would mean that Lojban's
-grammar &amp; morphology is *not* fully parsable using DFAs. And because we
-already know that the grammar is (it is [[jbocre: YACC|YACC]]-defined), it would mean
-that the morphology is problematic. ''(which I was already pretty sure
-of, dunno why ;-)''
-''Although I may have not been precise enough for a mathematical proof,
-I am sure that lojban's claim about being a "logical language" should
-entail discussions involving Gödel's statements about logic.''
-'''Very''' interesting; the limits of Lojban! I lack the mathematical
-power to join the discussion, but I am extremely interested in the
-conclusions! --xod
-I believe [[jbocre: Richard Curnow|Richard Curnow]] has mentioned before that the DFA
-(Deterministic Finite Automaton) for handling [[jbocre: Lojban morphology|Lojban morphology]] has
-somewhere between 900 and 1000 different states, and can't reliably be
-human-generated. Instead, he describes it with a NFA (did I just forget
-the acronym? argh) (Non-Deterministic Finite Automaton) and then
-converts the NFA to a DFA. Lemme tell 'ya, the NFA is hideous, too. (the
-whole point of this is to point out that while the grammar can be
-proven, nobody is seems sure about the morphology yet) --[jbocre: Jay
-Kominek|jay]
-''Correct me if I misunderstand, but the conclusion would be that
-Lojban's grammar and morphology cannot be expressed by [[jbocre: mekso|mekso]] or any
-other formal system, though the grammar and morphology could still be
-fully expressed using informal statements in Lojban.''
-Of course, as well as with informal statements in any other language.
-But the point is, [[jbocre: Lojban Description|Lojban Description]] is supposed to be parseable using computers,
-which means that its syntax ought to be fully expressible in computer
-terms, and thus mekso.
-It would be a major lose (correct me if I'm wrong) in regards to
-lojban's goals...
-''So this seems to be saying that either Lojban's morphology is vague in
-some area, or there are grammatically correct sentences which cannot be
-proven to be grammatically correct. Why do you assume it is the first?
-Number theory, a very powerful language indeed, contains true statements
-which cannot be proven. Why should we expect [[jbocre: Lojban Description|Lojban Description]] to be
-different?<br />Consider that these sentences which cannot be proven to be
-grammatically correct are going to be huge, monstruously long sentences.
-I don't think this defeats any of Lojban's goals. --[[jbocre: rab.spir|rab.spir]]''
-----
-''An aside into provability:''
-Hey, how can you say "true statements which cannot be proven"! That's
-completely meaningless. --xod
-Gödel settled this in 1931.
-[http://kilby.stanford.edu/~rvg/154/handouts/incompleteness.html] --
-nitcion.
-Yes, but this is rather tough.  A useful analogy can be found at
-[[jbocre: Goedel's Record Player|Goedel's Record Player]].
-More information and clear explanation about this stuff can be found in
-''[[jbocre: Gödel, Escher, Bach]]'' by Hofstadter (see previous paragraph)
-----
-The class of languages to which Lojban's grammar belongs is well
-understood. Assuming you can properly decompose a string of sounds or
-symbols into their separate words, there is no string of words for which
-membership in the language is undecidable. Whether or not that holds for
-the morphology seems to be unproven at this point. Maybe the LLG should
-investigate this and maybe stick their seal of approval on something as
-the official machine description of the morphology? Maybe I should snarf
-the lexer out of the official parser and see what it looks like...
---[[jbocre: Jay Kominek|Jay]]
-The lexer in the official parser is weak: about all it can do is divide
-compound cmavo.  It is ''not'' a full implementation of the morphology
-algorithm.
-We can decide whether the string is grammatically correct using
-[[jbocre: YACC|YACC]]. However, in the situation which started this discussion, we
-don't have the option of using [[jbocre: YACC|YACC]]. You have to use the numerical
-rules, expressed in [[jbocre: mekso|mekso]], which determine whether a sentence is
-grammatically correct. The implication of this is that there exist
-monstrous sentences, probably involving [[jbocre: mekso|mekso]] which include the
-Godel numbers of other [[jbocre: Lojban Description|Lojban Description]] sentences, which cannot be proven to be
-grammatically correct using those mekso rules. If you don't believe me,
-read [[jbocre: Gödel, Escher, Bach]] and all will become clear. --[[jbocre: rab.spir|rab.spir]]
-...And is it possible to change the title of this discussion? '''Done'''
----------
-Now that I finally have my own copy of [[jbocre: Gödel, Escher, Bach]], I've
-looked into how Gödel-incompleteness works - and I don't believe
-[[jbocre: Lojban Description|Lojban Description]] is affected by it. Mathematics (and particularly systems such
-as Typographical Number Theory) attempt to determine which statements
-are true and which are not in a formal way; these systems are incomplete
-because it is possible to create statements which imply their own
-falsehood.
-[[jbocre: Lojban Description|Lojban Description]], on the other hand, does not attempt to distinguish the true from
-the false mathematically. It only distinguishes the grammatically
-correct from the grammatically incorrect. For Gödel-incompleteness to
-apply to [[jbocre: Lojban Description|Lojban Description]], there would need to be a sentence for which
-demonstrating it to be grammatically correct somehow required that it is
-grammatically incorrect - ''without any regard to the meaning of the
-sentence.'' I don't think that grammar alone can imply anything of the
-sort.
-Additionally, the link {img
-src=http://kilby.stanford.edu/~rvg/154/handouts/incompleteness.html}
-points to a theorem saying that the correctness of a theorem in a
-sufficiently powerful system cannot be recognized by a Turing Machine.
-We know that a [[jbocre: YACC|YACC]] grammar ''can'' be recognized by a Turing
-machine. I believe this proves that Lojban's grammar alone is not
-"sufficiently powerful" to make statements about itself - which is a
-good thing, because that is the job of semantics - and so is not
-Gödel-incomplete.
-[[jbocre: rab.spir|rab.spir]]
-Erm. Ok. Again, forget YACC, think LALR(1). Turing machines can
-determine if any sentence does or does not belong to a LALR(1) language,
-in polynomial time, given memory proportional to sentence length. The
-question, then, is whether or not [[jbocre: Lojban Description|Lojban Description]] can express a Turing machine
-for recognizing itself. C is capable of constructing a Turing machine
-for recognizing itself, and thousands of other languages, and it has
-significantly fewer primitives than Lojban. (Further, C can express
-Turing machine rules for manipulating the semantics of itself) I'm
-confident that if you were bored enough, you could describe, using
-[[jbocre: Lojban Description|Lojban Description]] (and maybe even just MEX), a Turing machine which recognized
-Lojban. Maybe, though, I'm misinterpreting what the conundrum is.
---[[jbocre: Jay Kominek|Jay]]
-Yes, and that is why neither C nor [[jbocre: Lojban Description|Lojban Description]] is Gödel-incomplete at this
-level. Number theory is incomplete because it can mathematically
-determine whether statements are true or false, which necessitates that
-there are statements whose truth cannot be determined by number theory.
-[[jbocre: Lojban Description|Lojban Description]] cannot mathematically determine whether statements are true or
-false. Its mathematics only go as far as recognizing whether a statement
-is grammatically correct or not. Hence, the mathematical structure of a
-statement cannot make a statement about itself, and so no Gödel
-sentence exists in Lojban. Similarly, it can be determined in C whether
-a given C program (as long as it doesn't do weird stuff with the
-preprocessor) will compile, because the act of being compiled does not
-allow the program to make statements about itself.
-You could go to a higher level and find incompleteness. For example, you
-could set up a sub-grammar in MEX - not just the [[jbocre: number subgrammar|number subgrammar]]
-we already have, but one which would sort out true mathematical
-statements from false ones. This grammar would ''not'' be LALR(1). Then
-if you tried to describe this system using itself, you would have a
-Gödel-incomplete system. The system would '''be''' number theory, except
-with [[jbocre: Lojban Description|Lojban Description]] cmavo instead of mathematical symbols.
-On the same level, C (or any computer language) ''is''
-Gödel-incomplete when it comes to determining whether a program will
-halt, because there the mechanism used to determine whether the program
-halts is the same mechanism that the program uses.
-It comes down to the grammatical/semantic distinction which is mentioned
-at the very top of this discussion. If you take the semantics out of
-[[jbocre: Lojban Description|Lojban Description]] (and assume that semantics includes the meaning of statements
-made with mekso), you cannot express anything. So, in order for [[jbocre: Lojban Description|Lojban Description]]
-to make statements about itself, it has to do so at the semantic level.
-If you create a mathematical system which can process [[jbocre: Lojban Description|Lojban Description]] at the
-semantic level, you have an AI, and no longer something which is defined
-by the language itself; and to confuse this AI you might only need to
-say ''na nei''.
-(If I were Hofstadter, I would be able to phrase this much more
-clearly.)
-[[jbocre: rab.spir|rab.spir]]
-Well, I've acquired GEB, will read that, and from there investigate the
-original papers as needed to grok this topic, and then return. :)
---[[jbocre: Jay Kominek|Jay]]

Difference between revisions of "Gödel Numbers and Lojban"

Revision as of 17:40, 4 November 2013

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