On Friday, December 14, 2018 at 5:00:33 PM UTC-6, Brent 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. > > > > > 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. > > > 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. > And even then there might be a question of the rules of inference. > > Brent >
I have read in various texts that at some point matter (all there is in the universe) may reach a point of inconsistency: All matter itself would just disintegrate. That's all, folks! - 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.
