On Nov 18, 6:43 am, Graham Cooper <grahamcoop...@gmail.com> wrote: > On Nov 18, 3:11 am, George Greene <gree...@email.unc.edu> wrote: > > > > > > > > > > > On Nov 17, 3:50 am, Graham Cooper <grahamcoop...@gmail.com> wrote: > > > > FROM AXIOMS you DERIVE THEOREMS! > > > True. > > > > Nobody CARES if SomeModel |= 'this is not derivable from your > > > axioms" > > > > looks TRUE! > > > Sure they do. > > > > IT's not DERIVABLE FROM THE AXIOMS! END OF STORY! > > > That's NOT the end of the story! > > > It is true that the name of the room is "sci.logic" and that WE > > therefore (even if no one else is) > > might be entitled to care about&only about what follows from the > > axioms. But the PROBLEM is > > that the primary USE for formal proof is IN *MATH*. > > THEREfore, EVEN though the sign on the door says sci.logic, it is > > MATHEMATICIANS > > who matter more. > > > In the relevant case of models and of |= , the MOST relevant model > > around here is > > N, is THE (true&actual) NATURAL NUMBERS. > > IT DOES matter a WHOLE HECK of a lot if something that is true about N > > is NOT > > formally derivable from some decent (i.e. recursive) set of axioms. > > That is VERY important! > > PEOPLE CARE about proving things ABOUT N, *NOT* just about proving > > whatever > > follows from some axioms. For starters, how would you decide WHICH > > axioms were > > IMPORTANT? It is NEVER JUST about the axioms themselves! YOU ALWAYS > > need SOMEthing > > OUTside of logic motivating your investigation! You are always USING > > logic to help you reason > > ABOUT something ELSE! > > > > Nothing Mathematically INCOMPLETE ABOUT IT! > > > It is true about all these recursively enumerable formal theories that > > THEY ARE incomplete *ABOUT* N. > > You are sort of right, however, in that first-order-logic ALSO has a > > COMPLETEness theorem. > > A small and reasonable set of rules of inference REALLY IS sufficient > > to derive&prove, formally, > > EVERY theorem that is true in ALL models of the axioms. > > But there is no decent set of axioms or rules that is sufficient to > > derive&prove every (first-order) sentence that is true *in*N*, > > that is true of the natural numbers. THAT is how FOL gets to have > > BOTH a "completeness" AND an "incompleteness" > > [meta]theorem. > > > Prolog > > just doesn't have anything to do with this. Prolog can't even do > > complete FOL. > > Prolog APPROXIMATES first-order negation-AS-failure. > > See this is the problem. > > You just BAFFLE WITH BULLSHIT every assertion made that doesn't match > your Library Of Logic Facts! > > SomeModel |= this-is-not-derivable-by-axiom-set(A1) > > --> A1 is incomplete. > > *IF* you had any credible mathematical capacity, > you would ACKNOWLEDGE THE ARGUMENT FIRST. > > Your UNPROVABLE THEORY - which is 100 YEARS OLD AND GROWING > > is merely SELF-CONSISTENT that is why you ARGUE with ATTACK at the 1st > opportunity because > > if the OPPOSITE ASSUMPTION is allowed your entire LIBRARY OF LOGIC > goes up in a puff of smoke. > > Herc >
There's a name for this in encryption.
If your system is strong, you can make the algorithm public.
Your logic library is so weak, you do not allow it to go under inspection by assuming anything otherwise.
You have to HIDE YOUR ENCRYPTION ALGORITHM - DEFEND GODELS PROOF with high level nonsense and ad homs
George will swear black and blue just for writing "ASSUME:X" if he knows the proof by contradiction already!