> On 16 Dec 2018, at 19:24, Philip Thrift <[email protected]> wrote: > > > > 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).
I will take this seriously the day someone provide just one evidence for Matter (not matter). Until this is provided, I feel that it is an empty speculation which has been used as a simplifying hypothesis in the Natural Science, useful as such in physics, but should not be taken as granted when we do metaphysics, unless experimental evidence. It is important to understand that there are no evidences for primary matter (Matter), only for matter. Once you grasp that all computations are realised in arithmetic, that is enough for not taking Matter from granted in mechanist philosophy/science. Unless a proof is given that Nature does not conform toi the reducible notion of Nature entailed by Mechanism, Matter (with the big M) will remain an unclear philosophical speculation, and it should not be used as a dogma, as this is no more science. Bruno > > - 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] > <mailto:[email protected]>. > To post to this group, send email to [email protected] > <mailto:[email protected]>. > Visit this group at https://groups.google.com/group/everything-list > <https://groups.google.com/group/everything-list>. > For more options, visit https://groups.google.com/d/optout > <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 [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.
