On Wed, Nov 19, 2014 at 12:00 PM, Bruno Marchal <marc...@ulb.ac.be> wrote:
> > On 19 Nov 2014, at 16:44, Richard Ruquist wrote: > > > > On Wed, Nov 19, 2014 at 5:12 AM, Bruno Marchal <marc...@ulb.ac.be> wrote: > >> >> On 19 Nov 2014, at 05:18, meekerdb wrote: >> >> On 11/18/2014 4:57 PM, LizR wrote: >> >> On 19 November 2014 06:45, meekerdb <meeke...@verizon.net> wrote: >> >>> On 11/18/2014 5:00 AM, Bruno Marchal wrote: >>> >>> >>> On 17 Nov 2014, at 21:13, meekerdb wrote: >>> >>> On 11/17/2014 2:55 AM, Bruno Marchal wrote: >>> >>> The bible explains better (if we assume it is correct) >>> >>> >>> And if it isn't correct it doesn't explain anything. Which is why >>> science seeks to test correctness prior to explanatory power. >>> >>> >>> Ideally, or FAPP, perhaps. >>> >>> But fundamentally, science cannot test correctness, not even define it >>> properly. >>> >>> >>> Are you saying that a theory cannot be tested an found incorrect?? >>> >>> >> I would think the obvious way to parse what Bruno has said here is >> "science cannot show that something is correct". >> >> >> Is that right, Bruno? >> >> >> Yes. >> >> >> >> Of course empirical tests are better at showing a theory is wrong than >> showing it's right, which is Popper's observation. >> >> >> Indeed. >> >> >> >> I'm curious as to how you define correctness properly? >> >> >> I can't do it for myself, nor can any machine do it for herself. But a >> "sufficiently strong" machine can do it for a lesser strong machine. You >> can define arithmetical truth and PA's correctness in the set theory ZF for >> example. In that case "correctness" is defined in the manner of Tarski: p >> is correct if it is the case that p is satisfied by this or that >> mathematical structure, (for RA and PA, you can use the usual (N,+, *) >> structure, and with computationalism, that arithmetical truth (not >> definable in arithmetic) is enough). >> >> > This sounds like a description of which mathematical theories suggest the > existence of higher more-correct selves. > > > Not more correct, but knowing much more things. ZF knows that PA is > consistent, and ZF knows much more than PA about arithmetic, although of > course we still don't know if ZF knows the truth or the falsity of Riemann > hypothesis, but few doubt that ZF has any doubt about it. > > Note that ZFC (ZF + the axiom of choice) does not know any more than ZF. > The axiom of choice has no consequences for arithmetic. (That is not > entirely easy to prove, but is a good exercise if you know Gödel's > constructible sets). > > By Gödel's theorem, arithmetical truth is no exhaustible, so all machines > are superseded by other machines, and in fact this remains true for the > machine invoking Oracles (divine being which are supposed to know the > answer of Pi_1, Sigma_2, ... questions (by divine I just mean here non > computable, yet well definite in the standard model of arithmetic (true or > false). > > ZF + kappa knows much more thing than ZF. In fact ZF + kappa believes that > ZF is consistent. And ZF+kappa believes vastly much more than ZF about > arithmetic, but is still under the jug of incompleteness, and the > hypostases apply to PA, ZF, ZFC, ZF+kappa, etc. (Assuming ZF+kappa is > consistent). > > You can't be "more correct", as you are correct, or not. But the spectrum > of what you can believe in arithmetic can be very different. The whole of > the computable, Turing universality, is equivalent with Sigma_1 complete. > RA is already sigma_1 complete, and is quite humble in her arithmetical > knowledge. From PA and the extension, you have the Löbianity (PA is not > only sigma_1 complete, but PA knows that it/he/she is sigma_1 complete, and > it knows the plausible reason why it has to be humble with respect to the > arithmetical truth, on which it can only point, without explicit > definition). > > Sigma_1 completeness, the ability to prove all true sentences having the > shape ExP(x), with P recursive/decidable, is universal with respect to > computability, but is very humble with respect of provability, and there is > no "universal provability" notion: all provability predicate or machine can > be extended (even mechanically) to a more powerful machine, where > powerfulness is measure in term of classes of arithmetical propositions. > There is just no complete theory, definable by a machine, for the > arithmetical reality. Gödel's and Tarski's theorems makes the arithmetical > truth quite transcendental for the machines. > > Another post worth saving from the obscure Archives. But it sounds like you are defining differing Logic-Arithmetic-Dependent LAD higher worlds in relation to our Sigma_1 complete world. Does the RA world control gravity? PA-biology? ZF-Intelligence? ZF+kappa-Theology? etc. Richard > Bruno > > > > > Richard > > > >> Bruno >> >> >> >> >> >> >> Brent >> >> -- >> You received this message because you are subscribed to the Google Groups >> "Everything List" group. >> To unsubscribe from this group and stop receiving emails from it, send an >> email to everything-list+unsubscr...@googlegroups.com. >> To post to this group, send email to everything-list@googlegroups.com. >> Visit this group at http://groups.google.com/group/everything-list. >> For more options, visit https://groups.google.com/d/optout. >> >> >> http://iridia.ulb.ac.be/~marchal/ >> >> >> >> >> -- >> You received this message because you are subscribed to the Google Groups >> "Everything List" group. >> To unsubscribe from this group and stop receiving emails from it, send an >> email to everything-list+unsubscr...@googlegroups.com. >> To post to this group, send email to everything-list@googlegroups.com. >> Visit this group at http://groups.google.com/group/everything-list. >> For more options, visit https://groups.google.com/d/optout. >> > > > -- > You received this message because you are subscribed to the Google Groups > "Everything List" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to everything-list+unsubscr...@googlegroups.com. > To post to this group, send email to everything-list@googlegroups.com. > Visit this group at http://groups.google.com/group/everything-list. > For more options, visit https://groups.google.com/d/optout. > > > http://iridia.ulb.ac.be/~marchal/ > > > > -- > You received this message because you are subscribed to the Google Groups > "Everything List" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to everything-list+unsubscr...@googlegroups.com. > To post to this group, send email to everything-list@googlegroups.com. > Visit this group at http://groups.google.com/group/everything-list. > For more options, visit https://groups.google.com/d/optout. > -- You received this message because you are subscribed to the Google Groups "Everything List" group. 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