Dear Bruno, While I am VERY impressed by your reasoning, I must insist that it is necessary to make this aspect of COMP falsifiable. How is it decided empirically that entity X can not prove that P? This reminds me of the statement "all crows are not non-black". It seems to me that we are putting ourselves in the impossible position of having to prove a negative.
It is one thing to be able to point to mathematical proofs germane to mathematics proper but when we are trying to create models that are to be quantifiably predictive, we simply can not postulate such entities as "Platonia" and Arithmetic Realism as a basis. The same argument that we apply against "intelligent design theories" can be used against such non falsifiable assumptions. Kindest regards, Stephen ----- Original Message ----- From: "Bruno Marchal" <[EMAIL PROTECTED]> To: <[EMAIL PROTECTED]> Cc: <[EMAIL PROTECTED]> Sent: Monday, March 01, 2004 6:45 AM Subject: Re: Definitions and Argument (was RE: Penrose, wave function collapse and MWI) At 14:36 28/02/04 +0000, Brett Hall wrote: >I think this clarifies things a little. My original way of writing what I >interpreted the incompleteness theorem to be was to say that there exist (in >sufficiently complex axiomatic) systems in which there are true propositions >without proof. I think that this is misleading on reflection. It is more >accurate to say that there exist in complex axiomatic systems (like, for >example, arithmetic) propositions (or well formed formulae) that cannot be >proved either true nor false, that is, which are undecidable. > >Such propositions do have a truth value - the law of the excluded middle >still holds - so, being reasonable we have to assume that there do indeed >exist statements that are true but unprovable. You should say: unprovable ... *by the system in question* Let us call S the system in question (S for some consistent System). Godel's proof entails that there are true proposition P that S cannot prove. This means also that if you add P as axiom to S, you will not get a contradiction (if not, S would prove the false statement "not P"), this means that S+P is (another) consistent system S'. And, obviously S' can prove P, because P is an axiom for S' so that S' proves it in one line. >The other half of this way of >speaking is to say that 'there also exist false statements that are >unprovable' (but this, technically - is quite redundant as proving a >statement is false is the same as proving as true the negation of that same >statement). > >Does this make sense? Sure. But when you talk about a non provable proposition, you should always mention who (or which system) cannot prove it. Godel really has shown that formal provability is a relative notion (relative to the system considered). This is in total contrast with formal computability which seems to be an absolute notion not depending on any system (and that's the basic conceptual motivation for Church Thesis). As I said in another post, there is a case for *absolutely undecidable* statement. But this is trickier and is related to some formalization of "informal provability". If you are interested look at the reference: REINHARDT W.N., 1985, Absolute Version of Incompleteness Theorems, Noûs, 19, pp. 317-346. REINHARDT W.N., 1986, Epistemic Theories and the Interpretation of Gödel's Incompleteness Theorems, Journal of Philosophical Logics, 15, pp. 427-474. (From Conscience and Mecanisme, where I make the case that some form of comp are *absolutely* undecidable). Bruno http://iridia.ulb.ac.be/~marchal/