> On 13 Dec 2018, at 21:24, Brent Meeker <[email protected]> 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! If elementary arithmetic is inconsistent, all scientific theories are false. 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. 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. > >> >> >> >>> which in tern implies that no axiom can ever be false. >> >> Which is of course easily refuted. >> >> >> >>> Which makes my point that the mathematical idea of "true" is very different >>> from the common one. >> >> “BBBBBBB” is true just in case it is the case that BBBBBBB. > > But you can't know whether it is the case that 10^10000 + 1 is the successor > of 10^1000 independent of the axioms, i.e. you assume it. That is the best we can do in science. Bruno > > Brent > >> >> I am not sure, but the point is that no machine can prove []p -> p in >> general. And the machine can know that, making her “modest” (Löbian). >> >> Bruno >> > > -- > 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. -- 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.
