On Sun, Oct 20, 2013 at 06:22:15PM +0200, Bruno Marchal wrote: > > On 20 Oct 2013, at 12:01, Russell Standish wrote: > > >On Sun, Oct 20, 2013 at 08:52:41AM +0200, Bruno Marchal wrote: > >> > >>We have always that [o]p -> [o][o]p (like we have also always that > >>[]p -> [][]p) > >> > > > > > >There may be things we can prove, but about which we are in fact > >mistaken, ie > >[]p & -p > > That is consistent. (Shit happens, we became unsound). >

Consistency is []p & ~[]~p. I was saying []p & ~p, ie mistaken belief. > > > > >Obviously, one cannot prove []p & p, for very many statements, ie > > > >[]p & p does not entail []([o]p) > > []p -> [][]p OK? > Why? This is not obvious. It translates as being able to prove that you can prove stuff when you can prove it. If this were a theorem of G, then it suggests G does not capture the nature of proof. Oh, I see that you are just restating axiom "4". But how can you prove that you've proven something? How does Boolos justify that? > (and []p -> []p, and p -> p) + ([](p & p) <-> []p & []q) (derivable > in G) > Did you mean [](p&q) <-> []p & []q? That theorem at least sounds plausable as being about proof. > so []p & p -> [][]p & ([]p & p) > -> []([]p & p) & ([]p & p), > > thus ([]p & p) -> [][o]p (& [o]p : thus [o]p -> [o][o]p) > > > > >Therefore, it cannot be that [o]p -> [o]([o]p) ??? > > > >Something must be wrong... > > > > I hope I am not too short above, (and that there is not to much typo!) > > Bruno > And thus you've proven that for everything you know, you can know that you know it. This seems wrong, as the 4 colour theorem indicates. We can prove the 4 colour theorem by means of a computer program, and it may indeed be correct, so that we Theatetically know the 4 colour theorem is true, but we cannot prove the proof is correct (at least at this stage, proving program correctness is practically impossible). -- ---------------------------------------------------------------------------- Prof Russell Standish Phone 0425 253119 (mobile) Principal, High Performance Coders Visiting Professor of Mathematics hpco...@hpcoders.com.au University of New South Wales http://www.hpcoders.com.au ---------------------------------------------------------------------------- -- 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 http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out.