> On 15 Dec 2018, at 00:00, Brent Meeker <meeke...@verizon.net> wrote: > > > > On 12/14/2018 2:59 AM, Bruno Marchal wrote: >>> On 13 Dec 2018, at 21:24, Brent Meeker <meeke...@verizon.net> 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 > And even then there might be a question of the rules of inference. > > 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 https://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 https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
