On Sunday, December 16, 2018 at 11:27:50 AM UTC-6, Bruno Marchal wrote: > > > > On 15 Dec 2018, at 00:00, Brent Meeker <[email protected] > <javascript:>> wrote: > > > > > > > > On 12/14/2018 2:59 AM, Bruno Marchal wrote: > >>> On 13 Dec 2018, at 21:24, Brent Meeker <[email protected] > <javascript:>> wrote: > >>> > >>> > >>> > >>> On 12/13/2018 3:25 AM, Bruno Marchal wrote: > >>>>> But that is the same as saying proof=>truth. > >>>> I don’t think so. It says that []p -> p is not provable, unless p is > proved. > >>> So []([]p -> p) -> p or in other words Proof([]p -> p) => (p is > true) So in this case proof entails truth?? > >> But “[]([]p -> p) -> p” is not a theorem of G, meaning that "[]([]p -> > p) -> p” is not true in general for any arithmetic p, with [] = Gödel’s > beweisbar. > >> > >> The Löb’s formula is []([]p -> p) -> []p, not []([]p -> p) -> p. > >> > >> > >> > >>> > >>>> For example []f -> f (consistency) is not provable. It will belong to > G* \ G. > >>>> > >>>> Another example is that []<>t -> <>t is false, despite <>t being > true. In fact <>t -> ~[]<>t. > >>>> Or <>t -> <>[]f. Consistency implies the consistency of > inconsistency. > >>> I'm not sure how to interpret these formulae. Are you asserting them > for every substitution of t by a true proposition (even though "true" is > undefinable)? > >> No, only by either the constant propositional “true”, or any obvious > truth you want, like “1 = 1”. > >> > >> > >> > >> > >>> Or are you asserting that there is at least one true proposition for > which []<>t -> <>t is false? > >> You can read it beweisbar (consistent(“1 = 1”)) -> (consistent (“1=1”), > and indeed that is true, but not provable by the machine too which this > provability and consistency referred to. > >> > >> > >> > >> > >>>> > >>>>> Nothing which is proven can be false, > >>>> Assuming consistency, which is not provable. > >>> So consistency is hard to determine. You just assume it for > arithmetic. But finding that an axiom is false is common in argument. > >> Explain this to your tax inspector! > > > > I have. Just because I spent $125,000 on my apartment building doesn't > mean it's appraised value must be $125,000 greater. > > ? > > > > > > >> > >> If elementary arithmetic is inconsistent, all scientific theories are > false. > > > > Not inconsistent, derived from false or inapplicable premises. > > In classical logic false entails inconsistent. Inapplicable does not mean > anything, as the theory’s application are *in* and *about* arithmetic. > > If we are universal machine emulable at some level of description, then it > is absolutely undecidable if there is something more than arithmetic, but > the observable must obey some laws, so we can test Mechanism. > > > > > > > > > >> > >> Gödel’s theorem illustrate indirectly the consistency of arithmetic, as > no one has ever been able to prove arithmetic’s consistency in arithmetic, > which confirms its consistency, given that if arithmetic is consistent, it > cannot prove its consistency. > > > > But it can be proven in bigger systems. > > Yes, and Ronsion arithmetic emulates all the bigger systems. No need to > assume more than Robinson arithmetic to get the emulation of more rich > system of belief, which actually can help the finite things to figure more > on themselves. Numbers which introspect themselves, and self-transforms, > get soon or later the tentation to believe in the induction axioms, and > even in the infinity axioms. > > Before Gödel 1931, the mathematicians thought they could secure the use of > the infinities by proving consistent the talk about their descriptions and > names, but after Gödel, we understood that we cannot even secure the finite > and the numbers with them. The real culprit is that one the system is rich > enough to implement a universal machine, like Robinson Arithmetic is, you > get an explosion in complexity and uncontrollability. We know now that we > understand about nothing on numbers and machine. > > > > > > >> Gödel’s result does not throw any doubt about arithmetic’s consistency, > quite the contrary. > >> > >> Of course, if arithmetic was inconsistent, it would be able to prove > (easily) its consistency. > > > > Only if you first found the inconsistency, i.e. proved a contradiction. > > > That is []f, that does not necessarily means arithmetic is inconsistent. > The proof could be given by a non standard natural numbers. > > So, at the meta-level, to say that PA is inconsistent means that there is > standard number describing a finite proof of f. And in that case, PA would > prove any proposition. In classical logic, proving A and proving ~A is > equivalent with proving (A & ~A), which []f, interpreted at the meta-level. > Now, for the machine, []f is consistent, as the machine cannot prove that > []f -> f, which would be her consistency. G* proves <>[]f. > > It is because the domain here is full of ambiguities, that the logic of G > and G*, which capture the consequence of incompleteness are so useful. > > Bruno > >
In writing [ https://codicalist.wordpress.com/contents/ ], the idea at the beginning and now is that math and matter (code and substrate, myth and hyle, ...) are married like yang and yin. After reading Galen Strawson, and a few more recent "panpsychists" coming along - that there are psychical (experiential) states of matter (in addition to physical states) is the best understanding of consciousness we have so far - tips the balance towards matter's role as "head of the household" (so to speak). - pt -- 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 [email protected]. To post to this group, send email to [email protected]. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
