From: KL

To: Editor

Date: 09 Feb 2004

In Mr. Date's answer to the questions regarding Russel & Gödel, being a "logician trainee"  (when I'm not a DBA on some platforms which pretend to be relational, I'm a philosophy student) I have some comments.

Mr. Date writes: "There's no need to invoke the relational model, or DBMSs, or second-order logic, in order for this depressing result to apply!"

I do agree that there is no need to invoke the relational model or a DBMS to get the Gödel result, even if I like the idea of trying to express the Gödel result in such a model or system. The second-order logic is of importance, however: as Gödel proved in his PhD dissertation in 1930, the first-order logic is both complete and consistent. It is, however, impossible to implement arithmetic in first order logic. Gottlob Frege, in his Begriffschrift (1879) and in his Grundgesetze der Arithmetik (1893), tried to build arithmetic from logic. For arithmetic, we need at least second-order logic. Frege did manage to do this, but the Frege system did contain a paradox, which was discovered by Russel in his letter to Frege. Russel and Whitehead tried to get around this paradox by introducing the theory of types. The idea, basically, is that type 1 expressions can speak about atomic facts (i.e. not other propositions), type 2 expressions can speak about atomic facts and type 1 expressions, type N expressions can speak about atomic facts and up to type N-1 expressions. So, an expression like "a set which does or does not  contain itself" is typeless and therefore nonsensical in the Russel/Whitehead framework. Here, Gödel's paper On formally undecidable propositions (1931) proves that a logical system which is strong enough to deal with artihmetics (say, in the form of Peano axioms or in the form of the PRINCIPIA MATHEMATICA) will always have undecidable propositions if it is consistent. The theorem therefore shows that it cannot be mathematically proven that mathematics is consistent, even not by using tricks like the Russel/Whitehead type theory. If the consistency of mathematics can be proven, it won't be by mathematical methods.

C. J. Date Responds: It’s nice to see that somebody takes this stuff seriously! Regarding first- and second-order logics, I’d just like to add that we can’t even define equality if we limit ourselves to first-order logic (never mind arithmetic). But we don’t let this depressing fact stop us from exploiting the notion of equality (never mind arithmetic). Regarding “expressing the Godel result in a relational framework”, you might like to take a look at my book RELATIONAL DATABASE WRITINGS 1991-94 (Addison Wesley, 1995), pages 86 and 118-119.

Posted 04/09/04