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From: Ross A. Finlayson on 10 May 2005 12:52 Hi Graham, I'll post this to sci.logic. I guess besides the notion of a theory that can be complete, the axiom-free set theory with dually minimal and maximal ur-element can be used as a way to solve other paradoxes related to mathematical logic, primarily Cantor/Burali-Forti. I discuss that further in the last year or so on sci.math and sci.logic. I should read a copy of that book we discuss, that sounds agreeable. Thanks a lot for your replies. Ross F. On Tue, 10 May 2005, Graham Priest wrote: > Sure, Ross. Please feel free. But for the reasons I stated, I do not > intend to enter into correspondence about it. > > Best wishes, > > Graham > > > >Hi Dr. Priest, Graham, > > > >Hey, thanks a lot for your reply. > > > >One of the things I like about the null axiom theory is that I think it's > >possible to avoid the consequences of Goedelian (in)completeness, of the > >first (variously) kind. That is where Goedel says any _axiomatization_ is > >incomplete, a lack thereof is not. > > > >I wonder if you would consider it acceptable if I posted your reply to > >sci.logic, the other participants would find it of interest. I would like > >to post your reply to sci.logic. > > > >This weekend I went on a jetboat trip in the Hell's Canyon of Snake River, > >Idaho USA. > > > >Thank you very much, > > > >Ross F. > >On Wed, 4 May 2005, Graham Priest wrote: > > > >> Dear Ross, > >> > >> Many thanks for your letter, and the invitation to say something on > >> sci.logic. I'm afraid that I never take part in internet discussion groups > >> as a matter of principle - the principle being that I just don't have the > >> time. > >> > >> Good luck with your non-standard set-theory. We could certainly do with a > >> few new ideas! > >> > >> Best wishes, > >> > >> Graham > >> > >> > >> > >> At 03:48 AM 5/4/05 -0700, you wrote:
From: george on 11 May 2005 11:17
Bhupinder Singh Anand wrote: > As I argue elsewhere, Goedel's reasoning > can be taken to establish Can, only if you're stupid enough to keep beating your head against that particular stone wall. This has been refuted 69 times now. > that we can constructively, and in an > intuitionistically unobjectionable > manner, establish that an arithmetical > relation R(n) holds for any > given natural number n, No, you can't, and more to the point, THIS IS WHY PEOPLE SHOULD NOT attempt this AT HOME! By At Home, I mean, IN NATURAL LANGUAGE! The point being that YOU HAVE GOT YOUR QUANTIFIERS BACKWARDS. GIVEN ANY NATURAL NUMBER n, we can establish intuitionistically and constructively that R(n) holds, i.e., that n does not encode a proof of a contradiction from PA. This IS NOT the same as saying that we can establish intuitionistically that "R(n) holds for any natural number n"! We CAN establish all the INDIVIDUAL pieces constructively but there is NOTHING intuitionistic or constructive about calling INFINITELY many int/con tasks ONE int/con task! Intuitionistic and constructive logic are every bit as bereft of INfinitary inference rules as classical! > but that there is no algorithmic way of > verifying this assertion In that case, there was no algorithmic way OF DOING IT IN THE FIRST PLACE. There are infinitely many DIFFERENT algorithms for confirming R(n) but there is no ONE algorithm that does them all. > (in other words, any Turing machine that > computes R(x), treated as a Boolean function, > will go into a > verifiable, non-terminating, loop for some > natural number n). That loop is NOT verifiable for ALL n by any particular TM! For every n, there are ALL 3 of: a) a TM that decides R(n) (This is trivial when you think about it; either the TM that always/immediately halts false or the one that always/immediately halts true MUST be right; this is why applying TMs to individual problems as OPPOSED to families of problems is silly to begin with); b) a TM that verifies that some-TM-other-than-(a) Loops INSTEAD OF deciding R(n), and c) a TM that loops while trying(and failing) to determine (b). Everybody KNOWS already that every INDIVIDUAL finite n can be treated constructively. THE ONLY thing that matters is whether you can have ONE FINITE thing handling INFINITELY MANY DIFFERENT infinite cases, and there is NO interpretation of ANYthing that Godel says that suggests that this is possible when the axiom-set is PA and the problems are proofs of non- contradiction. > > On this view, WHAT view? You have not coherently articulated any view. > the particular argument between Priest and Cartwright, > Tennant, et al, becomes vacuous; we simply define any > predicate [P] of a formal language L as well-defined if, This is IDIOTIC. You canNOT HOPE to generalize over "formal languages" in general! ANYthing can be a formal language! IF we are talking about classical first-order languages (which is SURELY the case you BETTER get right FIRST before moving on to anything more complicated), ALL predicates are well-defined! What makes something a predicate is PURELY syntactic! If the predicate is NAMED then it is defined! > and only if, given any element > s in a domain D over which the > variables of the language are > well-defined under an interpretation M, This is, I repeat, IDIOTIC: whether the PREDICATE is well-defined has ABSOLUTELY NOTHING to do with any possible DOMAIN OR INTERPRETATION! You can vary the domain and the interpretation ARBITRARILY! IF the predicate was well-defined then it will NOT STOP being so JUST Because you moved to a different domain! If it was NOT well-defined then it will not START being so just because you decided to think about it under a different interpretation! > there is always an effective > method for determining whether P(s) holds > or not in M for the > interpreted predicate P. This surely has a HELL Of a lot more to do with M than it does with P. > > (Note that, given an interpretation M of > a formal theory L, we can > always - according to a reasonable > reading of Priest - define the > domain D as consisting of all > elements that satisfy the axioms of L > under any interpretation. That is idiotic. I don't know what Priest was smoking but that is simply incoherent, unless he is defining "interpretation" a hell of a lot more narrowly than everybody else. ABSOLUTELY EVERYTHING can be an element of the domain under SOME interpretation. Absolutely NOTHING can satisfy the axioms of L under ANY interpretation: OBVIOUSLY, NO MATTER WHAT an element is, I can define an interpretation over a domain of those elements that DOESN'T satisfy L, unless L is tautologus (i.e. empty). > The fact that some D-elements are dissimilar > may need qualification when defining satisfaction > in M but, prima facie, this should not lead to any > irresolvable issues.) Prima facie, the issue is that you still haven't gotten it through your head that interpretations are unconstrained. All this talk about domains is irrelevant in this context. > > The existence of functions in real and complex analysis that are > continuous, but not uniformly continuous, and of sequences that are > convergent, but not uniformly convergent, indicates that such a > definition of a well-defined predicate can be mathematically > constructive, and intuitionistically unobjectionable. > > We can then define [P] as well-defining a mathematical object (i.e., a > syntactic totality), say {x: P(x)}, in M if, and only if, there is an > algorithm (effective uniform method) for identifying elements such that > P(x) holds in M. > > In the absence of such an algorithm, we may still have that, for any s > in D, it is effectively decidable whether or not P(s) holds in M, but > we can no longer treat all such elements as a syntactic totality that > can be referred to within, or by, the language without inviting > inconsistency. > > On this view, the 'Domain' referred to by Priest need not be a > syntactic totality in the sense of being a mathematical object as > defined above. However, it can still be referred to, albeit loosely, as > a semantic totality for expressive, and philosophical, purposes, so > long as we do not associate any algorithmic properties with it without > a formal proof. > > Regards, > > Bhup |