gjk@talcott.UUCP (Greg Kuperberg) (03/16/85)
> What Lambert says (I think) is that it is conceivable > (but unlikely) that somebody comes up with an obviously valid inference > rule that would enable us to decide CH from the ZFC axioms. > I do not agree, since the current inference rules already allow you to > construct models in which CH holds and models in which it doesnt hold. > Stronger inference rules deciding CH would thus lead to a contradiction. > Something that is conceivable to me is that one might come up with new > axioms for Set Theory that are *obviously* valid and would decide CH. I don't see the contradication. With the old inference rules, both models are valid, while with the new inference rules, only one of them is. What is wrong with that? Another question is, how much would these new rules of inference be worth? This is a much more drastic change than new axioms; the meanings of "proof" and "decidable" would change as well. Of course, one could throw away all the rules of inference and make completely new ones. Imagine for example the "Rich Rosen" rules of inference: Rule 1: In a proof, any WFF with an odd number of "~"'s (negation operators) can follow any other WFF, and in fact such WFF's can be the beginning of a proof. Rule 2: In a proof, any WFF with an even number of negation operators can never appear. Rule 3: If G is a WFF and G is the last line of some proof, then G is said to be "true". If G is a WFF and ~G is the last line of a proof, then G is said to be "false". Of course this is a very silly set of inference rules. Nevertheless, Goedel's theorem is false in this world, which is perhaps a desirable thing. And CH is decidable from the standard axioms of set theory (in fact it is decidable even without those axioms). But I must leave with a question: Could one conceivably *generalize upon*, rather than change, the current rules of inference, and make Goedel's theorem false? Remember, if a statement and its negation are both true, then Goedel's theorem is vacuously true. --- Greg Kuperberg harvard!talcott!gjk "No Marxist can deny that the interests of socialism are higher than the interests of the right of nations to self-determination." -Lenin, 1918
play@mcvax.UUCP (Andries Brouwer) (03/18/85)
In article <359@talcott.UUCP> gjk@talcott.UUCP (Greg Kuperberg) writes: >> ... the current inference rules already allow you to >> construct models in which CH holds and models in which it doesnt hold. >> Stronger inference rules deciding CH would thus lead to a contradiction. > >I don't see the contradication. With the old inference rules, both models >are valid, while with the new inference rules, only one of them is. What >is wrong with that? > What is a valid model? It is one satisfying the imposed axioms. Inference rules tell you how to get valid statements from other valid statements, but have no direct bearing on models, except in that one never could accept a rule of inference when there exist models satisfying the input formulas but not the output formulas of the rule. That is why CH cannot be decided by stronger inference rules.