# Gödel Numbers and Lojban: Difference between revisions

We can construct Gödel numbers for any Lojban text. Therefore, if we can prove that the lojban grammar & morphology is expressible using lojban MEX 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 na nei, mind you, correctness on the syntactical view is meant).

• 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 & morphology seems to be parsable using DFAs (Deterministic Finite Automaton), 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 & morphology is not fully parseable using DFAs. And because we already know that the grammar is (it is 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.
• xod:
Very interesting; the limits of Lojban! I lack the mathematical power to join the discussion, but I am extremely interested in the conclusions!
• Jay Kominek:
I believe Richard Curnow has mentioned before that the DFA for handling Lojban morphology has somewhere between 900 and 1000 different states, and can't reliably be human-generated. Instead, he describes it with a NFA (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).
• Correct me if I misunderstand, but the conclusion would be that Lojban's grammar and morphology cannot be expressed by 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, Lojban 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...
• rab.spir:
• 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 Lojban to be different? 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.
• xod:
An aside into provability: how can you say "true statements which cannot be proven"! That's completely meaningless.